.J
Higher Genus Moduli Spaces in Closed String
Field Theory
by
Sabbir Ahmed Rahman
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 1997
©
Massachusetts Institute of Technology 1997. All rights reserved.
.....................
Department of Physics
August 28th, 1997
Author .................................
Certified by.....
.............................
Barton Zwiebach
Associate Professor
Thesis Supervisor
Accepted by.......
....
....
................................
George F. Koster
Chairman of Graduate Committee, Department of Physics
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Higher Genus Moduli Spaces in Closed String Field Theory
by
Sabbir Ahmed Rahman
Submitted to the Department of Physics
on August 28th, 1997, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
We provide an overview of covariant closed string field theory, covering briefly the
geometry of moduli spaces of Riemann surfaces, conformal field theory in the operator
formalism and the Batalin-Vilkovisky formalism. Several important applications are
also described including connections on the space of conformal theories, quantum
background independence, the ghost-dilaton theorem, and string field theory around
non-conformal backgrounds.
The proof of the ghost-dilaton theorem in string theory is completed by showing
that the coupling constant dependence of the vacuum vertices appearing in the closed
string action is given correctly by one-point functions of the ghost-dilaton. To prove
this at genus one the formalism required to evaluate off-shell amplitudes on tori is
developed.
Higher order background independence conditions arising from multiple commutators of background deformations in quantum closed string field theory are analysed.
The conditions are shown to amount to a vanishing theorem for As cohomology
classes. This holds by virtue of the existence of moduli spaces of higher genus surfaces with two kinds of punctures. Our result is a generalisation of a previous' genus
zero analysis relevant to the classical theory.
The string theory operators a, CKand I are shown to be expressible as inner
derivations of the B-V algebra of string vertices. As a consequence, the recursion
relations for the string vertices are found to take the form of a 'geometrical' quantum
master equation, {B, B} + AB = 0, where 'B' is the sum of string vertices. That the
B-V delta operator cannot be an inner derivation on the algebra is also shown.
The set of string vertices of non-negative dimension is completed in a consistent
manner. As a consequence the string action takes the simple form S = f(B). That
the action satisfies the B-V master equation follows immediately from the recursion
relations for the string vertices.
The set of string vertices is then extended to include moduli spaces having all
integral values of g, n,
T > 0. It is argued that the string background, and also the
B-V delta operator should be identified with the space B3, 1 . This leads naturally
to the proposal that the manifest background independent formulation of quantum
closed string field theory is given by the sum of the completed set of string vertices
B=
,n,o satisfying the classical master equation {B, B} = 0.
Thesis Supervisor: Barton Zwiebach
Title: Associate Professor
Acknowledgments
It is not without some sense of relief that I can now look back upon what has certainly
been an eventful five years. Many people contributed towards making this period a
happier one for me, and it is gratifying to be able to take this opportunity to mention
some of them here.
First and foremost I must thank my supervisor, Barton Zwiebach, to whom I
remain deeply indebted. His continual guidance, help and encouragement has been
extremely valuable to me. He has suggested interesting problems to work on, has
egged me on when work seemed to be coming to a standstill, and has ploughed
through one incomplete draft after another until each paper was transformed into a
publishable form. Furthermore, he was always there for moral support at times when
it was greatly needed.
My heartfelt gratitude remains for Amr Mohamed, Mohamed Mekias, Ali Jouzou,
Hisham Hasanein, Ziad Azzam, Mustapha Khemira and Ibrahim Sanhouri, all of
whom took care of me like a younger brother during my early years at MIT, and who
continued to do so after graduating and moving away.
I thank the many wonderful friends and acquaintances I made through the MIT
Muslim Students' Association, too many to mention but among them being Suheil
Laher, Mahbubul Majumdar, Moinuddin Monem, Abdul-Wahab al-Othman, Fouzi
al-Essa, Rafat Ahmad, Ameur Mezaache, Ahmad Nassr, Wissam Ali-Ahmad, Samir
Nayfeh, Sadiki Mwanyoha, Seif-Eddeen Fateen, Bilal Mughal, Issam Lakkis, Jalal
Khan, Yassir Elley, Wasiq Bokhari, Farhan Rana, Asad Naqvi, Khurram Afridi, Adnan Lawai, Wasiuddin Wahid, Asim Khwaja, Zeeshan Khan, Hussam al-Gabri, Mohamad Akra, Yehia Massoud, Abdul-Aziz al-Jalal, Bashir Dabbousi, Babak Ayazifar,
Kashif Khan, Mohammad Ali, Ayman Shabra, Mohamed Saeed, Gassan al-Kibsi,
Asif Khalak, Muayyad Qubbaj, Ismail, Sabri, Asif Zobian, Isam Habboush, Sohail
Husain, Bhavesh Patel, Ahmad Mitwalli, Omar Baba, Mustapha Kamal, Husni Idris,
Timocin Pervane, Karim Hussein, Arifur Rahman and Shamim Javed.
I would like to acknowledge my colleagues and mentors in the Physics Departments
of MIT and Harvard University, among them Jeffrey Goldstone, Kenneth Johnson,
Roger Brooks, Samir Mathur, Edward Farhi, Roman Jackiw, Nihat Berker, Mehran
Kardar, Patrick Lee, John Joannapoulos, George Benedek, Felix Villars, Cumrun
Vafa, Michael Bershadsky, Ayman El-Desouky, Shamit Kachru, Rachel Cohen, Peggy
Berkovitz, Pat Sokolov, Jon Rodin, Daniel Kabat, Sen-Ben Liao, Ranganathan Krishnan, Witold Skiba, Csaba Csaki, Joshua Erlich, Alexander Belopolsky, Arthur Lue,
Dmitri Dolgov, Amer Iqbal, Dongsu Bak and Leo Pantelidas.
Very special thanks are due to Chris Isham at Imperial College, to Chris Lawrence,
Andrew Kao and my other badminton colleagues, and especially to Robert Dickinson
who, despite being a little confused himself at times, remains my closest companion
and unfailing source of metaphysical and psychological guidance.
Finally I cannot end without acknowledging the support of my wife and family
for helping me to pull through despite some difficult times.
to my parents
Contents
1 Introduction
1.1 Brief history of string theory ...................
....
1.2 Overview of string field theory ...................
...
1.3 Summary of results in this thesis ....................
.
1.3.1 Vacuum vertices and the ghost-dilaton ..............
1.3.2 Consistency of quantum background independence ........
1.3.3 String vertices and inner derivations ...............
1.3.4 Geometrising the string action ...................
1.3.5 Path towards manifest background independence ........
13
13
16
19
19
20
21
22
23
2 Review of covariant closed string field theory
2.1 Moduli spaces and string vertices ....................
.
2.1.1 Definition of the string vertices ..................
2.1.2 Schiffer variations .........................
..
2.1.3 Conformal field theory in the operator formalism .......
.
2.1.4 Moduli space forms and string multilinear functions ......
2.1.5 Symplectic structure and Batalin-Vilkovisky formalism . ...
2.2 Deformations and space of string backgrounds ..............
2.2.1 Connections on the space of conformal field theories ......
2.2.2 Special punctures and the operator ICK..............
2.2.3 Quantum background independence ................
2.2.4 Ghost-dilaton theorem ......................
2.2.5 String field theory around non-conformal backgrounds ......
25
25
26
28
30
34
35
41
41
43
47
50
53
3 Vacuum vertices and the ghost-dilaton
3.1 Vacuum vertices of genus g > 2 .................
3.2 Off-shell amplitudes in tori . . . . . . . . . . . . . . . . . . . .
3.2.1 Once-punctured tori from three punctured spheres .
3.2.2 Forms on the moduli space of punctured tori . . . . . .
3.3 The case of genus one . . . . . . . . . . . . . . . . . . . . . . .
3.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . ...
.
57
57
59
59
60
63
65
4 Consistency of quantum background independence
4.1 Review and notation .......................
4.2 Background independence revisited . . . . . . . . . . . . . . .
69
69
70
4.3
4.4
4.5
5
6
71
4.2.1 Field-independence of the density p ................
4.2.2 Origin of general symplectic connections .............
74
4.2.3 Background independence and As-cohomology ..........
75
The commutator of background deformations ..............
76
4.3.1 The commutator conditions ....................
76
4.3.2 Analysis of gauge freedom of B,, .................
78
4.3.3 Consistency conditions and recursion relations for moduli spaces 79
81
As-Cohomology classes and theory space geometry ...........
Appendix . .......
.. .. ...
...
.. ...
...
.. .. ...
84
String vertices and inner derivations
5.1 The BRST Hamiltonian Q and the moduli space 100,2 .........
5.2 0, K: and I - Explicit operator identifications ...............
5.3 Recursion relations and the string action .................
5.4 The B-V delta operator - an inner derivation? . . . . . . . . .
Geometrising the string action
6.1 The moduli spaces Bl and Bo ...........
.
.
6.2 The moduli spaces B, o ...................
6.3 The geometrised string action ................
.
. . . . .
........
7 Path towards manifest background independence
7.1 Completing the set of string vertices ...................
7.2 A postulate about sewing ........................
7.3 The sewing of B°0, and the B-V delta operator ..............
7.4 Manifest background independent formulation ..............
7.5 Conclusion ..................
..............
.......
.... . .
85
85
86
89
90
93
93
96
96
97
97
99
100
101
101
List of Figures
1-1
Riemann surfaces from Feynman diagrams . . . . . . . . . . . . . . .
2-1
2-2
2-3
2-4
2-5
2-6
2-7
The projections Pg,. --+Pg,* -+ Mg,n -- Mg,0. . . . . . . . . . . . .
The minimal area metric prescription . . . . . . . . . . . . . . . . . .
The cutting property of minimal area metrics . . . . . . . . . . . . .
Dissecting the tadpole graph...........
.. ... ... ... ..
The Schiffer variation ..............
.. .. .... .... .
The sewing procedure...............
.. .. .... ... ..
Family of local coordinates............
.. .. .... .... .
26
27
28
28
29
31
45
3-1
One-punctured tori from three-punctured spheres
66
............
Chapter 1
Introduction
1.1
Brief history of string theory
String theory has had an unorthodox and fascinating history. One may trace its origins back to the early days of dual models [1] in the late 1960s, which were proposed
in an attempt to describe the strong nuclear interaction. The 'duality' hypothesis was
the hypothesis - observed to hold approximately experimentally - that the s- and tchannel contributions to two-particle strong elastic scattering amplitudes are exactly
equal. This requirement was found to be satisfied by the now-celebrated amplitude
postulated in an ad hoc manner by Veneziano in 1968 which also correctly predicted
the general behaviour of the masses of the lightest hadronic resonances for given particle spins. The dual model appoach to strong interactions remained active for about
five years after its inception, eventually being abandoned due to various experimental
and theoretical problems. While the Veneziano model was in reasonable agreement
with high energy behaviour of the strong interactions, it disagreed in the region of
high-energy scattering at fixed angles. The latter was better explained by the parton
model, accounted for by the alternative theory of quantum chromodynamics (QCD)
which emerged around 1973, which uprooted dual models as a theory of the strong
interactions. But that is not all. Dual models generically predicted the existence of
a massless particle of spin two which was not observed experimentally. Furthermore,
dual models only seemed to make sense in 26 dimensions for the theory of bosons and
10 dimensions for the more realistic theory of bosons and fermions. The fermionic
theory was found to require the existence of two-dimensional supersymmetry which
led to the development of space-time supersymmetry, recognised to be a generic feature of all consistent string theories. The appearance of QCD which provided the
SU(3) component of the standard model, meant that the original motivation for dual
models was lost.
As it happens, it was not for two years after their discovery that dual models were
recognised as theories of extended one-dimensional objects, i.e. 'strings' [2, 3]. This
placed them outside the realm of ordinary quantum field theory which is based on
point-like elementary particles. String theory allows the incorporation of particles of
high spin without the ultraviolet pathologies of point-like theories. These divergences
were a problem in theories of quantum gravity which all contained a massless particle
of spin two, the graviton. The massless spin two particle of string theory seemed to
have couplings similar to that of the graviton, so the suggestion naturally arose that
string theory may actually be a consistent theory of quantum gravity. The typical
size of a string would have to be at around the Planck scale, which is so small that
the approximation to point particles would be a good one at the scales which are
currently accessible.
The extra dimensions of string theory brought to mind the Kaluza-Klein theory
which attempted to unify gravity and electromagnetism by curling up an additional
fifth dimension into a tiny circle. There was therefore the hope that the extra dimensions could possibly be a blessing in disguise and that string theory may actually
realise Einstein's dream of providing a unified theory of all the known interactions.
The extra six dimensions would be curled up into some tiny 'compactified' space,
leaving only the usual four-dimensional Minkowski spacetime observable. These ideas
were too radical for most, and only a few collaborators investigated them seriously in
the period from 1974 to 1984.
A series of remarkable discoveries during 1984-85 brought string theory back into
the limelight, and it has remained an active area of research for theoretical physicists
ever since. It became clear that there were only five consistent superstring theories,
each living in 9 + 1 dimensions. These are denoted by Type I (N = 1), Type IIA
and Type IIB (N = 2), and Heterotic Es xEs and SO(32) (N = 1), where N denotes
the number of supersymmetries. The Type I theory is based on unoriented open
and closed strings, while the others describe oriented closed strings. Also, the Type
IIA theory is non-chiral (parity-conserving), while the others are chiral. In order to
be realistic, the six extra spatial dimensions must be compactified into a space of
size comparable to the string scale. In particular, it was found that the Heterotic
string compactified on a 'Calabi-Yau' manifold (a three-dimensional complex Kihler
manifold with SU(3) holonomy) had some low energy features, such as low-mass
fermions which occur in families, which resembled the observable universe. Since
then, many string vacua (too many, perhaps) have been found which seem to be in
agreement with the standard model predictions.
Recent years have seen the appearance of another kind of 'duality' [4], quite distinct from the original one, which has resulted in another spate of important and surprising discoveries of an inherently non-perturbative nature. Indeed, it has been found
that the five consistent superstring theories mentioned above, previously thought distinct, are actually five versions of a single underlying theory perturbatively expanded
around five different 'backgrounds', each corresponding to a different point in 'theory space'. Three types of dualities have been identified, perhaps rather whimsically
called S-, T- and U-duality after the Mandelstam variables or channels for two-particle
scattering channels whose amplitudes were equivalent in the original dual models. Tduality is an equivalence between two theories related under exchange or mapping of
momentum- and winding-mode excitations associated with one or more compactified
dimensions. This duality is essentially perturbative in nature and was the first to be
discovered. S-duality identifies two theories related through inversion of the string
coupling constant. This equivalence between strongly- and weakly-coupled theories
generalises the electric-magnetic duality of Maxwell's theory of electromagnetism. Uduality is a combination of the ideas of S- and T-duality in that one theory is said to be
U-dual to another theory if the first theory compactified on a space of large (small)
volume is the same as the second theory theory at strong (weak) coupling respectively. The new insights into the interrelations between these theories have led to the
postulate of two additional backgrounds, an 11-dimensional theory called 'M-theory'
which would represent a strong coupling limit of the Type IIA and Heterotic E8 xEs
theories, and a somewhat more hypothetical 12-dimensional theory called 'F-theory'
which would give rise to the Type IIB theory on compactifying on a two-dimensional
torus T2 .
Other important advances have been made in parallel with the progress in string
dualities. Recent work has emphasised the importance of certain higher-dimensional
fundamental objects, termed 'p-branes' (strings being the particular case of 'onebranes'). These appear to be as fundamental as strings, except that a perturbation
expansion cannot be based around them. The special case of 'Dirichlet p-branes' or
simply 'D-branes' [5] which satisfy Dirichlet boundary conditions, have made possible the analysis of various non-perturbative phenomena using perturbative methods. Their most notable application has been in the study of black holes. By using
D-branes to count quantum microstates of classical black hole configurations, the
Bekenstein-Hawking prediction for their entropy has been successfully reproduced in
several cases. Other recent example of advances in string theory have come from areas
such as matrix models [6], Seiberg-Witten theory [7], non-commutative geometry [8],
and geometric engineering [9].
Despite new insights brought about by investigations into string dualities, the
overall picture remains incomplete. Dualities describe relationships between only a
few fixed points in the space of string backgrounds. Besides these isolated points, they
provide little or no further information about the global structure of theory space.
One of the most important goals in string theory then is to find a single global theory
which is formulated without explicit reference to a particularly background. Such
a 'manifestly background independent' formulation would not only provide detailed
information about the global structure of theory space, but would be expected to
unravel the mysteries of string dualities and may perhaps eventually provide insights
into how nature is moulded into a particular physical background. String field theory
seems to be a possible approach towards finding such a global theory, and much of
this thesis will be devoted towards finding such a manifest background independent
formulation for the bosonic closed string, which we expect will generalise in a natural
and straightforward way to the superstring.
In the remaining sections of this introductory chapter we will provide a brief
overview of the recent results in covariant closed string field theory. At the same time
we will summarise the work to be presented in this thesis.
1.2
Overview of string field theory
Based on the original work of Siegel [10], the full quantum field theory of covariant
closed strings was explicitly constructed by Zwiebach in [11]. The details of this
construction will be the subject of Chapter §2, but we offer here an overview of the
basic ideas before immersing ourselves into the technicalities.
Closed string field theory is somewhat unique in that it appears to be the first
field theory for which the sophisticated Batalin-Vilkovisky (B-V) field-antifield quantisation formalism (see §2.1.5) is both necessary and useful in its complete form. One
begins with a choice of conformal field theory background CFTmtt,,e with central
charge c = 26 which determines the physics. This is precisely counterbalanced by a
contribution of -26 to the central charge coming from the anticommuting (b, c) conformal ghost system, CFTghost, which arise on BRST quantisation. The 'ghost' field
c has primary dimension -1 while the 'antighost' field b is primary of dimension 2.
The complete theory obtained by coupling the matter and ghost sectors CFTmatter®
CFTghost has no central charge, and (modulo the value of the string coupling constant'), determines a string background.
Having chosen a background one can find the physical states and compute scattering amplitudes. In the usual BRST prescription the physical states are defined as
cohomology classes of the nilpotent BRST operator Q. This is called the 'absolute'
cohomology. However in closed string field theory, it is necessary for well-defined
scattering amplitudes that the physical states be identified with classes of the 'semirelative' cohomology which has the additional constraints that all states be annihilated by the operators bo and L o . (Practically speaking, the Lo condition need not
be imposed explicitly as states annihilated by bo with non-vanishing L o eigenvalues
are BRST-trivial).
Strings trace out worldsheets corresponding to Riemann surfaces [Fig. 1-1], so
scattering amplitudes are defined as a perturbation series over spaces of decorated
Riemann surfaces. One needs to define an action such that its perturbative expansion
correctly reproduces the string amplitudes for that background. To do this, one
introduces the string field IT), which is represented as a linear sum over the basis
states of the combined CFT, the coefficients being given by the fields and antifields
0' of the B-V formalism, Eqn.(2.1.58). The master action S[T] is a non-polynomial
functional of the string fields, Eqn.(2.1.67), obtained by performing the integration
of canonical forms over the string vertices V,,, each of which is a certain subspace
of the moduli space Pg,, of n-punctured genus g decorated Riemann surfaces (see
§2.1). The string vertices (§2.1.1) are constrained by certain geometrical recursion
relations Eqn.(2.1.1) necessary for the action to satisfy the B-V master equation,
Eqn.(2.1.47). A solution to these recursion relations is determined by the minimal
area metric prescription (see §2.1.1) which provides a one-to-one mapping between
Feynman graphs and Riemann surfaces.
The canonical forms (§2.1.4) are obtained by considering the tangent vectors responsible for deforming the moduli of the decorated surfaces. These deformations
'Refer to the discussion on the dilaton theorem, §2.2.4.
',T
1-
z
Figure 1-1: Riemann surfaces from Feynman diagrams. In this example two closed
strings interact via a one loop scattering process, tracing out a two-dimensional worldsheet conformally equivalent to a four-punctured genus one Riemann surface.
may be implemented using the Schiffer variations (§2.1.2). The operator formalism
of conformal field theory (§2.1.3), which to each surface assigns a state in the CFT
Hilbert space, can then be used to extract the forms required in defining the master action. The string vertices are chosen to satisfy certain other conditions which
guarantee a Hermitian master action with amplitudes which display factorisation and
unitarity.
In addition to the usual B-V structure on the functions of fields and antifields,
there turns out also to be a B-V algebra on the moduli spaces of surfaces, with the
antibracket and delta operator essentially being identified with the usual sewing of
surfaces as described in the operator formalism (§2.1.5). These algebras are homomorphic and one may construct explicitly the functional mapping from surfaces to
functions which implements the homomorphism, Eqn.(2.1.80). This interplay between
surfaces and functions has been fundamental to many of the recent developments in
string field theory.
Having successfully constructed the closed string field theory, attention shifted
towards the nature of its background dependence (or lack thereof). It had been
conjectured that string field theory was actually background independent, meaning
that if one were to rewrite the target space fields for one background in terms of
those for another, whilst still preserving the underlying symplectic structure, then
the action weighted measures for the former are mapped into that of the latter. Some
preliminary results had already been obtained by Sen [12-14], where analysis of the
effect on quadratic and cubic terms in the string action as well as on-shell S-matrix
elements gave evidence that the new theory formulated around a nearby conformal
background, obtained by deforming the original CFT by a marginal operator, was
the same as the original one.
In order to verify this conjecture, it is necessary to have some way of comparing
the actions at nearby backgrounds, which requires development of our understanding
of the local structure of 'theory space'. A theory space of CFTs is defined as a vector
bundle (V, rx,M) where M is a connected base manifold parametrising the CFTs and
rxthe projection from the fibres H. (representing the CFT state space) to the base for
each point x E M (corresponding to a particular CFT). There exist sections over the
bundle consisting of the assignment of surface states (E(x)l E V*®n for each surface
E. There are also (marginal) operators IO,) E 7-x for each vector in the tangent
space TM responsible for deformations in the theory space. This structure allows
the definition of a connection IN and a covariant derivative D,(F) on the theory space
(§2.2.1). The CFT deformations (see §2.2) are performed by integrating a 'special'
puncture (§2.2.2) over each surface minus its unit disks, at which a marginal operator
1o,) is inserted. The particular form of the variational formula Eqn.(2.2.13) used
to define the covariant derivative captures the notion that the information about all
theories in a theory space is contained in the data at any one point, making the space
more than simply a family of theories related by a smooth variation of correlation
functions.
Equipped with the theory space formalism, the conjecture of quantum background
independence (§2.2.3) was proven in [15]. Given the recursion relations for the string
vertices and the existence of connections on the CFT theory space, an antibracketpreserving map between the state spaces of two nearby conformal theories is explicitly constructed, Eqn.(2.2.40), which takes the corresponding B-V action-weighted
measures into each other. The Hamiltonian B,, Eqn.(2.2.49), implementing the deformation relating the theories is defined in terms of a set of interpolating 'B-spaces'
denoted B
which are moduli spaces of Riemann surfaces of genus g with n ordinary (symmetrised) punctures and one special (antisymmetrised) puncture. The
proof of quantum background independence provided a firm basis for the widely held
belief that string theories built from nearby conformal backgrounds merely represent
different states of the same underlying theory.
Related to the problem of quantum background independence is the ghost-dilaton
theorem (§2.2.4). The ghost-dilaton ID ) is a physical state which exists in every
bosonic string background and which only just fails to be either primary (it is not
annihilated by L 1 or L 1) or trivial in the semi-relative cohomology (ID) = -QIx)
but bo lX) # 0). The dilaton theorem [16-19] is an old result relating on-shell string
amplitudes with a single zero-momentum dilaton and other physical states to the same
on-shell amplitude without the dilaton. Simply stated, a background deformation in
the direction of the dilaton field is expected to cause a shift in the string coupling
constant r. The dilaton theorem was considered in a more geometrical context in
[20,21], where an attempt was made to show that integrating a ghost-dilaton insertion
over a bordered Riemann surface amounts to multiplying it by its Euler constant.
The ghost-dilaton theorem was considered in the framework of closed string field
theory and generalised to off-shell amplitudes in [22]. Here it was found that while a
deformation in the direction of the ghost-dilaton field changes the string background
(by shifting the string coupling constant), the corresponding CFT deformation is a
trivial one as a result of the ghost number anomaly. This surprising result dispelled
the popular belief that a string background is the same as a CFT background.
This discovery also revived the idea stated in [11] that there might exist string
backgrounds which are not conformal at all (see §2.2.5). Preliminary work in this
direction was done in [23] where higher order background deformations for the clasical
theory were considered and found to be implemented by moduli spaces of surfaces
containing more than special puncture. These new B-spaces are the basic objects
required to write the action for string field theory around non-conformal backgrounds,
which was duly constructed in the sequel [24].
This concludes our brief overview of the main aspects of covariant closed string
field theory. More details regarding the ideas which have been described above will
be given in Chapter 2. These ideas form the basis for the new results contained in
this thesis, which we will now summarise.
1.3
Summary of results in this thesis
In this section we shall describe in outline the new results which are contained in this
thesis and which will be expanded upon in detail in the following chapters.
1.3.1
Vacuum vertices and the ghost-dilaton
Whenever a string background contains a nontrivial ghost-dilaton state, a shift of
the string field along this state is expected to alter the value of the dimensionless
string coupling constant. This expectation, the so-called 'ghost-dilaton theorem' (see
§2.2.4), was recently established in [22] as a property of the covariant closed string
field theory action. More precisely, this work established the result only for the fielddependent terms in the string action. In Chapter 3 we complete the proof of the
ghost-dilaton theorem by showing that a shift by the ghost-dilaton also changes the
value of the string coupling in the field-independent terms of the string action. These
field-independent terms arise as vacuum vertices of nonvanishing genus g.
The effect of the ghost-dilaton shift is found by inserting a ghost-dilaton on the
vertices defining the string action. For each vertex, one integrates over the corresponding moduli space of surfaces the result of integrating a ghost-dilaton insertion
over each surface. For string vertices with external punctures it is found convenient
to place the antighost insertions for moduli changes on an external puncture. This
choice makes it clear that a suitable ghost-dilaton insertion computes the Euler number of the bordered surfaces comprising the string vertex. Complications arise for
the vacuum vertices, though the nature of these difficulties are different for the cases
g > 2 and the case g = 1. In general, there are no external punctures available to
support the moduli-changing insertions. Moreover, for genus one, the existence of
conformal isometries dictates that there is no integral over the position of the dilaton
insertion.
In §3.1 we demonstrate how the methods of [22] actually apply for g > 2 even
though the moduli-changing Schiffer vectors are supported on the puncture where
the dilaton is inserted. We find that integrating the dilaton over a vacuum vertex
correctly reproduces the Euler characteristic of the unpunctured surfaces.
In §3.2 we discuss the evaluation of general off-shell amplitudes for punctured
tori. This is necessary for the ghost-dilaton, since despite being a physical state
it has no primary BRST representative. We construct any once-punctured torus
with an arbitrary local coordinate around the puncture by sewing two punctures of
a particularly simple three-punctured sphere with an appropriately chosen sewing
parameter. We give explicit expressions for the Schiffer vectors which generate the
tangents to the moduli space of punctured tori with local coordinates. The results of
this section represent an extension to genus one of the techniques developed in [25].
In section §3.3 the above ideas are used to show that both the canonical string twoform in the direction of the ghost-dilaton ID) and the canonical string one-form in the
direction of IX) = -co 0) vanish identically on the moduli space of once-punctured
tori (here ID) = -QIx)). These results imply that the genus one field-independent
terms in the string action are unchanged by a shift of the ghost-dilaton.
1.3.2
Consistency of quantum background independence
It has been some time now since the background independence (see §2.2.3) of string
field theory was proven [15, 26]. The infinitesimal background deformations were
shown to be implemented as canonical transformations whose Hamiltonian functions
were defined by moduli spaces of punctured Riemann surfaces with a single special
puncture. It was realised in [23] that these B1 spaces represented only the first
order perturbations of the string background, and that higher order deformations
implied the existence of moduli spaces B2 , B3 ,... of surfaces having more than one
special puncture, antisymmetric under the exchange of special punctures. These were
duly constructed in [23, 24] for the classical case. Furthermore, these B-spaces have
precisely the properties required to formulate string theory around non-conformal
backgrounds, and such an action was constructed explicitly in [24]. The B-spaces
do not depend upon any particular choice of string background and as such would
be expected to play an important role in any manifestly background independent
formulation of the theory. Arriving at such a formulation remains one of the major
goals in string theory.
The recent work of Zwiebach [23, 24] dealt only with the classical closed string
theory, whereas we would of course eventually wish to deal with the full quantum
theory. Experience has shown that in string field theory, classical results usually
generalise to the quantum case without too many complications, and this is reaffirmed
in Chapter 4 where the quantum generalisations of Zwiebach's results are found.
The chapter is organised as follows. In §4.1 we review some properties of connections on the space of conformal field theories, and recall some basic facts about the
string vertices, introducing some notation for B-spaces which will be used throughout.
In §4.2, we follow [15] to derive the conditions for background independence when
the Batalin-Vilkovisky density function p is allowed some field-dependence. We then
review the origin of general symplectic connections, and give the form of the Hamiltonian B,,(F) implementing background deformations for general connections. We find
that background independence to first order amounts to a As-cohomology theorem
for the action, being the natural generalisation of the antibracket-cohomology of the
classical case. In §4.3 we consider the commutator of background deformations and
show that background independence to second order implies a higher cohomology
theorem. The consistency conditions are shown to be satisfied through the existence
of spaces B 2 with two special punctures of all genera. In §4.4, we use the language of
differential forms on the theory space manifold to efficiently derive As-cohomology
theorems to all higher orders for the string action, and extend the complex of B-spaces
to include positive-dimensional moduli spaces of Riemann surfaces for all genera and
all numbers of ordinary and special punctures compatible with the dimensionality
requirement.
1.3.3
String vertices and inner derivations
There has long been a desire reformulate string field theory in a simple and concise
form. It has already been seen in [15,23,26] and [24] (extended in Chapter 4), how one
might introduce the moduli spaces of decorated Riemann surfaces B",n of non-negative
dimension (g indicating the genus, n the number of ordinary punctures and ii the
number of special punctures), which implement the first and higher order background
deformations of the closed bosonic string. There is the hope however, not only of
completing this set but also of somehow including the remaining B-spaces which can
take all non-negative integral values of (g, n, i), and in some way incorporating these
into the action. The main obstruction to this had been the problem of interpreting
those objects of 'negative dimension' which cannot be thought of as moduli spaces of
surfaces in the usual sense.
A step in this direction was achieved in the work [24] of Zwiebach when the moduli
space B01,, naively of dimension 2 -1, was introduced, being a sphere with one ordinary
and one special puncture. Moreover, the antibracket sewing of this state was identified
with the action of the operator I responsible for changing an ordinary puncture into
a special puncture, and the associated function f(BL,1) = -B(2)
-(W121F)1l=>)2,
which acquires a ghost insertion, was absorbed into the string action. This remarkable
result demonstrated that the operator I, usually thought of as an outer derivation of
the B-V algebra of string vertices, could itself be represented by the action of one of
the string vertices, thus transforming it into an inner derivation of the algebra. As a
result he was able to simplify the recursion relations for the B-spaces, which took the
following form,
B-B- ±+ 2f{B,B} - V,
= 0.
(1.3.1)
In the above, B is a sum of moduli spaces of Riemann surfaces B = --,, B,. with
n ordinary punctures and A special punctures. In this summation n and A can take
all non-negative integral values with the exception of the spaces WB,o (which would
contribute constant terms to the action), B0,1 and B°,2 . Given the success had with
the operator I, one is led to ask whether the operators o and K: may similarly be
expressible as inner derivations of the B-V algebra.
In Chapter 5 we take the first step towards such a simplification. In particular,
we succeed in deriving identifications for the operators a, K and I, the auxiliary
vertices VO, 3 and o1, and the Hamiltonians Q and BF) in terms of the negativedimensional spaces Bo,1 and B°,2 (which we introduce). This implies the non-intuitive
result that the operators 0 and KI,previously thought to be outer derivations having
no obvious description in terms of string vertices, could possibly be expressed as inner
2
Recall that the dimension of Bf,
is 6g - 6 + 2n + 3ft.
derivations. The recursion relations then take the form of a quantum master action
for the B-spaces. The problem of simplifying the action turns out to be a little harder,
and will be addressed in Chapter 6. Chapter 5 is organised as follows.
In §5.1 we introduce the moduli space B ,2 which has dimension -2 and is identified
with the kinetic term Q of the action, which is also the Hamiltonian associated to the
BRST operator. In §5.2 we list the operator identities satisfied by 0, K: and 1. From
these, we succeed in deriving the unique set of requirements on the spaces B1.,
2 and
but
also
the
only
for
0,
K:
and
I,
identifications
not
B°,1 implying consistent operator
vertices V•,3 and 71. This demonstrates that these operators may be expressed as
inner derivations as we had hoped.
In §5.3 we state the resulting form of the quantum action around arbitrary string
backgrounds and the corresponding recursion relations. These take a completely
geometrical form in that they are described by a quantum master action for the string
vertices. By expanding them we verify that they agree precisely with the earlier form.
In §5.4 we attempt to express the B-V delta operator as an inner derivation as
was done for the other operators. This attempt fails as it has the implication of a
vanishing antibracket.
1.3.4
Geometrising the string action
One hopes eventually to be able to reformulate closed string field theory in a form
which is both simplified and which brings out the deeply geometrical underlying basis.
There were two main hurdles which needed to be overcome before such a goal could be
realised. The first of these was the need to find a geometrical description of the usual
string field theory operators 0, K: and I. The second was the need to complete the set
of string vertices WB,n by both introducing those vertices of non-negative dimension
which had for various reasons previously been excluded, and extending the set to
include the vertices of 'negative' dimension.
Some advance is made in this direction in Chapter 5 where it is shown there that
it is possible to express all three operators, 0, K: and I as inner derivations on the
B-V algebra of string vertices. Also, the recursion relations for the string vertices are
found to be given by a 'geometrical' quantum B-V master equation,
-{B,
B} + AAB = 0
B
2
(1.3.2)
S = S 1,o + f(B) .
(1.3.3)
while the action takes the form,
While this signals firm progress, it is not yet totally satisfactory seeing as B =
g,n,n Bn,B still excludes the non-negative dimensional vertices B1 o,, B0,0 and B o
(where n > 2). The main goal of Chapter 6 is to include these spaces in a consistent
manner ensuring that previous results (particularly quantum background independence [15], the ghost-dilaton theorem (see [22] and Chapter 3), and the recursion
relations [11]), are still satisfied. The structure of Chapter 6 is as follows.
1
In §6.1 we review the original reasons for excluding the spaces Bo,, B
3,
and aB
°
and explain why the existence of the space B ,2 (which is introduced in Chapter 5)
makes it consistent to reintroduce them. As a corollary we also see how a choice of
B °,2determines the choice of connection. In §6.2 we briefly explain why it is consistent
to introduce the moduli spaces Bo and thereby complete the set of string vertices of
non-negative dimension.
We then state in §6.3 the expression for the quantum action around arbitrary string
backgrounds. It takes a completely geometrical form expressed solely as a function
of the sum of string vertices. Furthermore, the recursion relations are contained in a
quantum master action for the B-spaces. As an application of these we end by listing
expressions for the boundaries of the newly introduced moduli spaces.
1.3.5
Path towards manifest background independence
Since the proof by Sen and Zwiebach [15, 26] of local conformal background independence of both classical and quantum closed string field theory, much effort has
been exerted towards constructing a formulation manifestly independent of the conformal background [27]. However, recent work on the dilaton theorem (see [22, 28]
and Chapter 3) by Belopolsky, Bergman and Zwiebach has shown that string backgrounds actually encode more data than conformal backgrounds, namely the vacuum
expectation value of the dilaton field which governs the strength of the string coupling. This discovery called into question the previously held assumption that string
field theories were built solely upon conformal field theories corresponding to classical
solutions. This led to a search for more general string field theories, towards which
strong progress has been made, notably in the recent work by Zwiebach [24], where
it was sketched how classical closed string field theory might be built around nonconformal backgrounds without reference to any conformal field theory. Background
independence was not quite yet manifest, as the string action was written as a function on the state space of an underlying two-dimensional quantum field theory which
represented the tangent space to the space of backgrounds at that particular theory.
However it did suggest that a manifestly background independent formulation was
within reach.
In Chapter 5 the operators a, K: and I are expressed as inner derivations of
the B-V algebra, while in Chapter 6 the recursion relations for the string vertices
are shown to take the form of a quantum B-V master equation as in (1.3.2), where
B = ",,,,, BW,n is now the sum of the string vertices for all non-negative integers
g, n,n except g = 0, n + Ai < 1. The action takes the simple form,
S = f(B).
(1.3.4)
In this geometrised form, the only background dependence is contained in the string
field I9)and the fermionic state IF)required to define the function f mapping the
moduli spaces of decorated Riemann surfaces to functions on the space of fields and
antifields.
In Chapter 7 which is more speculative than earlier chapters, we explore the possibility of introducing the remaining negative-dimensional moduli spaces, B00° , B0,1
and B',0 in order to complete the set of string vertices, and incorporate them into the
geometrised formulation of the theory. We argue in that the choice of string background is encoded into the space Bo0, and further that the B-V delta operator should
be identified with the antibracket sewing of B0°,I. The set of string vertices becomes a
background independent algebraic structure containing complete information about
the theory. This leads us directly to postulate the manifestly background formulation of quantum closed string field theory, where the 'geometrised' action is given by
the sum of string vertices B =
ng,n,a>o
,1, satisfying a classical master equation,
{B,B} = 0.
Chapter 2
Review of covariant closed string
field theory
2.1
Moduli spaces and string vertices
From an elementary point of view, in closed string field theory closed loops of string
trace out two-dimensional worldsheets embedded in an underlying twenty-six dimensional target space background [Fig. 1-1]. The Feynman diagrams of string theory
are therefore represented by Riemann surfaces of genus g with n punctures, where g
represents the number of internal loops for a particular scattering process and n the
number of external particles. Particular subspaces of the moduli spaces of these Riemann surfaces, called the 'string vertices', constitute the fundamental building blocks
of covariant closed string field theory. In this section we shall review the properties
of decorated Riemann surfaces and the string vertices in some detail. Let us first
introduce some notation.
The moduli space of unpunctured Riemann surfaces of genus g is denoted by
M 9 ,0 . The compactified moduli space Mg,0 is obtained by including degenerate surfaces. The compactified moduli space of Riemann surfaces of genus g with n punctures
is then denoted by Mag,. The moduli space Pg,. is the moduli space of punctured
Riemann surfaces with local coordinates defined up to a phase around the punctures.
An efficient way of defining these local coordinates is to place a simple closed curve
around each puncture. The coordinates may then be obtained up to a phase by conformally mapping the disk defining the coordinate curves to the complex unit disk
in such a way that the puncture is mapped to the origin. Finally, the moduli space
Pg,n is obtained from Pg,n by specifying the phase for each local coordinate, which
may be done by marking a point on each coordinate curve which will be mapped to
unity. There are natural projections Pg,n --+ Pg,, + Mg,, - Mg,o defined by forgetting about the marked points, the coordinate curves, and the punctures respectively
[Fig. 2-1].
g,n
I1
I
mg,o
i
Figure 2-1: The projections Pg,n -+ Pg,n -+ Mg,n
2.1.1
Mg, 0 .
Definition of the string vertices
The string vertices Vg,n are sets of decorated Riemann surfaces which form a subset of
g,,, and provide the main geometrical input to off-shell closed string field theory. The
surfaces in Vg,, must satisfy several consistency conditions. For manifest factorisation
and unitarity, the vertices must not contain degenerate surfaces. For covariance, the
vertices must be symmetric under the exchange of punctures. For consistent gluing
properties, the coordinate disks on each surface in Vg,n must not overlap. Also, for
hermiticity of the string field interactions, if a surface E is in V•,, then the mirror
image E* must also be contained in Vyg,. Finally, there is a very stringent consistency
condition on the string vertices which guarantees BRST invariance represented in the
following geometrical equation [29],
a
1
2
91 + 92 = 9
+nl
tn2
=nt•+
n 2>1(2.1.1)
Eqn.(2.1.1) applies for n > 3 when g = 0. This is equivalent to setting Vo,n - 0,
for n = 0, 1, 2. It also applies for all V9,, when g > 1 with the exception of the oneloop cosmological constant, V1,o. It is a consistency condition for sets of Riemann
surfaces with local coordinates, independent of a specific choice of conformal field
theory or string theory. The existence of vertices Vg,n satisfying this condition is the
main geometrical input to off-shell closed string field theory.
A simple solution to requirements mentioned above is provided by the minimal
area metric prescription of Zwiebach [11]. This shows us how to obtain a unique
Figure 2-2: The minimal area metric prescription applied to a four-punctured sphere.
All nontrivial closed curves must be at least of length 27r.
Feynman graph corresponding to any punctured Riemann surface. Furthermore, as
all conformally equivalent surfaces are mapped to the same unique Feynman graph, it
provides a single cover of moduli space which implies that the consistency condition
above is satisfied. The minimal area metric prescription is defined as follows.
For a given Riemann surface E, the minimal area metric is the (conformal) metric
of least possible area under the condition that all nontrivial closed curves on E be
longer than or equal to 27r [Fig. 2-2].
The metric satisfying these conditions is unique. It gives rise to closed geodesics
of length 27r that foliate the surface completely. Under the assumptions that for
each surface E the minimal area metric (i) exists, (ii) is complete (meaning that
the neighborhood of each puncture is isometric to a flat semi-infinite cylinder of
circumference 27r), and (iii) is smooth in some neighborhood of each puncture, one
can prove that the minimal area metrics satisfy the requisite conditions to define the
string diagrams of covariant closed string field theory.
A couple of results and definitions will be followed by a concrete definition of the
string vertices. Consider an annulus of saturating geodesics which define a foliation.
The height h of the foliation is defined to be the shortest distance between the two
boundaries of the annulus. Any foliation with smooth metric and height greater than
2xR is isometric to a flat cylinder of circumference 27r and height h. Cutting along the
middle geodesic of such a foliation always results in surfaces satisfying the minimal
area metric property if the metric on the new surface(s) is taken to be the restriction
of the metric on the original surface [Fig. 2-3].
The curve Co is defined to be the boundary of the maximal region foliated only
by geodesics homotopic to a puncture, and C, to be the saturating geodesic in the
cylinder a distance I away from Co.
The string vertices may be defined as follows. Consider a Riemann surface E of
genus g and n punctures (n > 3 for g = 0, and n - 0 if g = 1) equipped with
the metric of minimal area. Then E E Vg,, if and only if the heights of all internal
foliations are less than or equal to 2wr. If E is in Vg,, the coordinate curves are defined
to be the C, curves around each puncture.
The C, curves give rise to 'stubs' of length 7r at each puncture. The case of the
one-loop vacuum graph is special and does not seem to fit into the above framework
easily. The solution to this difficulty may lie in the use of cutoff propagators in
I
h/2
h/2
Figure 2-3: The cutting property of minimal area metrics. Cutting along the middle
geodesic of a foliation of length greater than 27r results in two new surfaces equipped
with metrics satisfying the minimal area condition.
external leg +
Vo.3
+
propagator
L ýL-p
17'11ý
oizc~
$`:
V~
Figure 2-4: Dissecting the tadpole graph.
combination with stubs (see Chapter 6).
The Feynman graph associated to a surface E can be reconstructed from knowledge
of its minimal area metric. Semi-infinite foliations correspond to the external legs of
the diagram. If there are no internal foliations of height bigger than 21r, then the
surface is an element of the interaction vertex V,,,, where n > 0 is the number of
infinite height foliations, and g is the genus of the surface. The internal foliations of
height greater than 2r give rise to the propagators. By cutting the string diagram
along closed geodesics a distance r from (i) the boundaries of each internal foliation
of length greater than 2,r and (ii) the boundary of each semi-infinite foliation, the
surface breaks up into a number of semi-infinite cylinders, the external legs, a number
of finite cylinders, the propagators, and a number of surfaces with boundaries, that
correspond to the elementary interactions. Each elementary interaction corresponds
to an element of the set Vy, where g is the genus of the surface with boundary and
n is the number of boundaries [Fig. 2-4].
There is therefore a one-to-one relationship between a Feynman graph and the
associated minimal area metric. This concludes our preliminary discussion of the
string vertices.
2.1.2
Schiffer variations
We will now give the form of the tangent vectors Vi in TpP9,, describing the possible
deformations of the decorated Riemann surface represented by the point P E Pg,n.
p'
z.·
I
Figure 2-5: The Schiffer variation.
These are constructed using Schiffer variations. These deform the surfaces by either
altering the complex moduli of the underlying Riemann surfaces, moving the positions
of the punctures, or changing the coordinate curves. Having explicitly given the form
of the tangent vectors Vi E Tp(Pg,n), the tangents to Pg.n, M,,, and Mg,o can be
deduced via the usual projections from Pg,n to those spaces.
Let P correspond to the decorated Riemann surface E with punctures pi and
coordinate curves yi where i = 1...n. The Schiffer variations are generated as follows.
Take an n-tuple of vector fields,
v = (v (1)IV(2), ..- •v(n)),
(2.1.2)
where v( i) is a holomorphic vector field defined in an annular neighborhood of the
coordinate curve -yi defining the disk Di surrounding the puncture pi. Then for each
of the punctures define the map,
zi , z' = zi + v(i)(z,) ,
(2.1.3)
which for sufficiently small e maps the coordinate curve yi into simple closed Jordan
curves '/. The deformed surface E' is constructed by gluing the interior of -Y to the
exterior of 7i for each puncture in the following manner.
Let p be a point on the boundary of E - E Di, and p' a point on the boundary
of the new disk defined by '-. The two points p and p' are identified thus [Fig. 2-5],
p E yi
+-+
P' E 7-
when
zi(p) = zi(p').
(2.1.4)
This is termed 'Schiffer's interior variation'. The new local coordinates are simply
the zi coordinates, with the new puncture p' located at zi = 0.
The curve used to perform the Schiffer variation can be deformed to another
curve homotopic to the original curve as long as no singularities in the vector field
are encountered. Also, for a given contour the most general situation which needs to
be considered is that of a Schiffer variation with a vector field that inside the countour
may become singular only at the puncture.
Let us list the conditions on the vector fields for obtaining the various types of
surface deformations.
The n-tuple of vectors v(0) comprising v does not change any data if it arises from
a single globally defined holomorphic vector field on the whole of surface minus the
punctures (singularities are permitted at the punctures). Any vector which is not of
this type must change the data. The following possibilities exist for vectors which do
not arise from a globally defined holomorphic vector field.
A vector field defined around a puncture which extends holomorphically up to the
puncture, where it vanishes, but does not extend holomorphically to the rest of the
surface, does not change the underlying moduli, nor the position of the puncture, but
does change the local coordinate around the puncture.
Similarly, a vector field that extends holomorphically into the puncture without
vanishing there, and which does not extend to the rest of the surface must change the
position of the puncture, unless the surface has continuous conformal self-mappings
(which only exist for n = 1, 2, 3 at g = 0 and for n = 1 at g = 1), in which case at
least the local coordinates are changed.
Finally, a vector field which extends holomorphically neither to the rest of the
surface nor to the puncture changes the complex moduli of the underlying surface
(except for genus zero when no such moduli exist). The Weierstrass gap theorem
shows that if we specify the leading singularity z- n , with n > 0, of a vector field
around a point p (with z a local coordinate) there are 3g - 3 values for n such that
the vector cannot be extended holomorphically to the rest of the surface. Every
vector v = z - n , with n one of the special values, must generate a nontrivial tangent
in T(Mg,o) and the tangents associated with all the special values of n give a basis
for T(Mg,o).
The n-tuple of vector fields v on a surface corresponding to a given tangent vector
in Pg,n can therefore be generated by the sum of (3g - 3) variations of the type,
(vk ) , 0, 0,-,
0),
k =1,2,-, 3g - 3
(2.1.5)
all based on the same puncture (and each changing a complex modulus), and n
variations of the type,
,
(00,... ,v(k),..., 0)
k=1,2,-. -,n
(2.1.6)
each around a different puncture (changing the position of the puncture and the local
coordinates). Any other n-tuple v' representing the same tangent in Pg,n can only
differ from v by an n-tuple t arising from a globally defined holomorphic vector field
on the surface minus the punctures.
2.1.3
Conformal field theory in the operator formalism
The usual formulation of covariant closed string field theory requires a choice of
conformal background representing a classical solution around which the full theory
is developed. An extremely powerful formulation of conformal field theory which will
100i,i E
I
ooi,j
L
L ooi-J
oi,j
Figure 2-6: The sewing procedure.
be the basis of our work is given by the operator formalism first introduced in [30,31],
whose basic properties we now recall.
Consider two decorated Riemann surfaces E1 E •g ,i,and E E pg2,n2. One can
create a new surface by the process of 'sewing' the i-th puncture of El to the j-th
puncture of E2. This procedure is carried out by removing the relevant coordinate disk
from each surface and then identifying the boundaries via the analytic prescription
ziwj = 1, where zi and wj are the local coordinates associated respectively with the
i-th puncture of El and the j-th puncture of E2 [Fig. 2-6]. The resulting surface
is denoted by EC00i,jE2 E Pg+g
9 2,n+n2-2 . Analogously, one can sew together two
punctures of a single surface EE P•g, to create a new surface ooij E Pg+l,n-2In the operator formalism a conformal field theory is simply a way of associating
to any surface EE P,n (where n > 1), a unique element (EI E 7V*® " where W-is some
complex vector space and 7-*its dual. This is done in a way which is compatible
with the sewing procedure just describe so that there exist an operation 00,j which
acts on states in such a way that (E•Ioi,j(EY
and (0looi,j = (Eooij
2 I = (10ooi,j21
are satisfied.
More specifically, we define the (symmetric) reflector state (R12 1 associated to a
twice-punctured sphere with local coordinates zl = z and z2 = 1/z about z = 0 and
z = oo respectively. Given a basis |li,) of states in W7,we can define a bilinear form
(a metric) on W by,
gij = (R12li)141)j) 2 ,
(2.1.7)
which we assume has an inverse gij. These raise and lower indices according to
Jqi)
-=
gij[ij) and 1IQi) _ giji). A dual basis for 7* may be defined as follows,
1(0'-- (R 12 1(D)
2
,
(2.1.8)
such that (Vl1j) = 6ý. Then we may write,
(R12
=
gij 2(VJ 1(0I
(1 .
(2.1.9)
The ket reflector IR 12 ) is similarly defined by,
IR12) = 1i)9
1
(2.1.10)
j)2
Their contraction produces the relabelling operator,
(R12 1R23 ) = 311,
(2.1.11)
The sewing conditions are represented in terms of the ket reflector as follows,
and
(E100i,jE21 = (1I((E2lRninj)
(Ecoo,jl = (EIRij).
(2.1.12)
where ni and nj number the spaces corresponding to the i-th puncture on El and
j-th puncture on E2Other useful states are the out vacuum 1(01 which is a sphere with a single puncture
at the origin with local coordinate z, the in vacuum 10)1 - 2(01R12), and also the
standard three-punctured sphere (V123 (z)I which has punctures at 0, 00 and w with
local coordinates z, 1/z and z - w respectively.
Consider a state IA) E 71, If it is an eigenstate of Lo and L0 with eigenvalues
a and d respectively, then IA) has conformal dimension (a, -). If in addition IA) is
annihilated by L, and Ln for all n > 0 then it is said to be primary. The field A(z)
associated to IA) is given by,
A(z) = (V123 (z)IA) 3 1R24 ).
(2.1.13)
The state IA) can be recovered from the field A(z) through IA) = limz-+o A(z)lO).
There are also ghost fields c(z) and ?(z) and antighost fields b(z) and b(z) which
are primary of dimension (-1, 0), (0, -1), (2, 0) and (0, 2) respectively. They have
the mode expansions,
b(z)
,
c(z) =
n
b,
(2.1.14)
n
and their complex conjugates. One defines the ghost number operator G by,
G = 3 + [ (cobo - boco) + E(cnb, - bncn) + c.c..
(2.1.15)
n=1
In Zwiebach's conventions the in vacuum has vanishing ghost number, GIO) = 0, and
the out vacuum has ghost number six, (01G = 6(01, so that any correlation function
must have six units of ghost number between each corresponding vacuum pair to be
non-vanishing. Note that any ghost/antighost oscillator adds/subtracts one unit of
ghost number. The ghost number of a surface state (El of genus g with n punctures
is given by,
G (')
(E
i=l1
= (6g - 6+6n)(E,
G(')I)
i=1
= (6g - 6 + 6n)E).
(2.1.16)
We also introduce the BRST operator,
Q=
dz
2i
1ddz
Cd(z)(Tm(z) ++ 2 Tgh(z) +
2fi
2i
1
( )(Tm(Z)+ Tgh(Z))
2
(2.1.17)
(where Tm (z) and Tgh(z) are respectively the matter and ghost components of the
stress tensor and f dz/2wiz = f d-/21i-Z = 1), which satisfies for all (CE,
Q
(El
=( 0 .
(2.1.18)
i=1
The ghost/antighost modes and the BRST operator have the following properties
with respect to the reflector states,
(c + c(2))IR12 ) = 0,
(R12 (c ) ++ C(2) = 0,
(b()- b 2))IR 12) = 0,
(R12 l(b 1) - b ) = 0,
(Q(1) + Q(2))IR 12) = 0,
(R12j(Q( 1) + Q(2)) = 0.
(2.1.19)
Infinitesimal deformations of surfaces states (e.g via Schiffer variations) are generated by the Virasoro operators. These may be defined in the operator formalism in
terms of a standard twice-punctured Riemann sphere which has coordinates 1/z at
oo but coordinates f (z) = z(1 - Elno Enz n ) at 0. For the positive modes we have,
(f-(z)(1),
1(2)
00
R 23 ) = 1 +
E (En)
nLI +
Jn)
O(+2 ),
(2.1.20)
+ O( 2 ),
(2.1.21)
n=O
(with an obvious notation), and for the negative modes,
1
(1)
2) = 1 +
(f(z)(R23)
(n_
L
+ nL_.)
n=O
These satisfy the Virasoro algebra (without central extension),
[Li,Lm] = (n - m)Ln+m,
[Ln, m] = (n - m)L!+m,
[L, im] = 0.
(2.1.22)
If we apply a Schiffer variation defined by the vector v = (v(1 ),7... v(n)) to a surface
E, the deformation of the corresponding state (EI is given by,
6,(• = -(ZIT(v),
where T(v) is defined in terms of the stress tensor T(z) = (V123 (z)IL (3)10)aR
3
the vector fields (and their complex conjugates) by,
(
T(v) = i=-1
T(i)(z)v(i)(z)
i +f
T()(i(i)(i) •ij ..
(2.1.23)
2 4)
and
(2.1.24)
(2.1.24)
The integrals are performed along contours surrounding the punctures lying in the
domain of definition of the local coordinates and the n-tuple of vectors. Using the
mode expansions for the stress tensor and the vector fields,
T(z)
,
=
v(z) =E
n
,.
(2.1.25)
n
we can write T(v) explicitly in terms of the modes,
(L(') (')+L(''))
T(v) =
(2.1.26)
i=-1
2.1.4
Moduli space forms and string multilinear functions
We will now construct a set of differential forms on the space Pg,n which can be
naturally integrated over subspaces of the minimal area section. The degree of these
forms will go from zero to infinity, since the space Pg,n is infinite dimensional.
We will need to make use of the antighost analogue of (2.1.24), so we introduce
the notation,
n
dzi
)b(Zi)(z)v`(zi) +
b(v)0 =
(i) (
f)i)
zi
-) .(2.1.27)
i-=1
A differential form .[J]gn, labelled by n states IB 1) .. IBn) and of real degree k, on
the tangent space Tp(P 9,n) is then defined by,
B
V]g
,
Vk(,...
) -- (-2ri)-3 ( 3g-
+n ) (Ep| b(vi)
... b(vk) B 1) ... IBn) ,
(2.1.28)
where Vi E Tpp 9,,. It is convenient to introduce the notation,
(Q[k]gnj
n
(-27ri) - (3 g- +")(
EpI b(v i ) ... b(Vk),
(2.1.29)
so that ]g'" = (q[k]'gnBl) ... jBn). Additionally, we shall often write Q(r)g,n instead
of l[6g-6+2n+r]g,n. Note that the ghost number of (El is (6g - 6 + 6n), while the
antighost insertions contribute -(6g - 6 + 2n + r) and the string fields contribute
=1 G(Bi). A total ghost number of 6n between the vacua (six for each Hilbert
space) is required for the form to be non-vanishing. This means that OQr)gn vanishes
identically unless Ej
(G(Bi) - 2) = r.
There is an important relationship between the BRST operator Q and the exterior
derivative 'd' on Pg,, when acting on the above forms,
!,A(Q[k]g,n
9g,n
1
1
Q(i) = (_)kj
Jd
9
n
d ([k-1]gn
1
=
k(_)kLAn (k-1l]gnlj,
"9,n
where Ag,n C Pg,n and we have made use of Stokes' theorem.
(2.1.30)
The forms defined above all require n > 1 as the operator formalism does not
explain how to associate states with unpunctured surfaces. This is overcome by using
the idea used in [30] to calculate partition functions.
Given a vertex Vg,o c MA,o (g 2 1), one finds a corresponding subset V,, 1 C M,, 1
which projects down to Vg,o on ignoring the extra puncture. The forms Q(r) g,o on
Tp(Mg,o) are then defined by,
O(r),0(,,
,
V6g-6+r) = (-2ri)(
1(r0-2),1IO0)(V,
... , V6g-6+r)
= (EPlb(vi) ... b(v6 g-6+r)J0),
where the surface Ep E (Vg,1) projects into the surface Ep E M 9g, and the vectors
Vi project to Vi as we forget about the puncture.
We can now introduce the string multilinear functions. The multilinear function
associated to the string vertex Vg,, is defined to be,
{B 1 ,
, B,} = ]Vn
(2.1.32)
.)g,
It is simply an integral of a form of degree (6g - 6 + 2n) over the subset of surfaces
defining the string vertex, using the minimal area section. Given an input of n string
fields, the multilinear product outputs a number which is in general a complex-valued
function of the set of target space fields. The functions defined above are multilinear
and graded-commutative, vanish unless Ei(G(Bi) - 2) = 0, and are intrinsically
Grassman even.
For g = 0 and n = 0, 1, 2 the multilinear functions are taken to be,
{ - }o= 0,
{B}o_= 0,
and
{B, B 2 }o = (Bi coQIB 2 ).
(2.1.33)
The string multilinear functions will be used later to define the master action.
2.1.5
Symplectic structure and Batalin-Vilkovisky formalism
We review here briefly the formalism of [32]. Consider a symplectic manifold M with
coordinates {z i } = (z ' ,... , z2), (not necessarily a Darboux frame). The symplectic
two-form is,
w = -dz i wij dz .
(2.1.34)
A Hamiltonian vector field correponding to the function F(z i ) is given by,
VF(Z)
where wikwkj
= J
=
W,'
(2.1.35)
F,
and we use the notation,
FF,
az
i
and
F=
Ozi
F,
(2.1.36)
where the two are related by,
F•
= (-)i(F+1)- F ,
(2.1.37)
where the indices i and F denote Grassman gradings. We introduce a volume element,
2n
Jdz',
dp(z) = p(z)
(2.1.38)
i=1
where p(z) is a density function introduced by Schwarz [33]. This allows us to define
the divergence of a vector field v = VivV ,
(2.1.39)
div v = 1(-)i-(pv') .
p
The B-V delta operation on a function F is then related to the divergence of the
corresponding Hamiltonian vector field,
1
1
AF = 2-divvF =
2
2p
+
(pwij
F).
(2.1.40)
The antibracket is then defined by,
(-)F{F,G} = A(F - G) - (-)G(AF) . G - (-)FF . (AG),
(2.1.41)
The p-dependence of the antibracket cancels out so that,
{F, G} = W(VF, VG) = VG(F)
= G~i wi• _F"
= G w23 3 F .
(2.1.42)
The operators A and {.,-} satisfy a Leibnitz rule,
A{F, G} = {AF, G} + (_)F+1{F,AG}.
(2.1.43)
In B-V quantisation, by imposing A2 = 0 we obtain the condition,
dw = 0.
(2.1.44)
This in turn implies the Jacobi identity for the antibracket,
(_)(F+1)(H+1){{F, G}, H} + cyclic = 0.
(2.1.45)
Suppose that we have an action S(z) which satisfies the B-V master equation,
Aes/h = 0,
(2.1.46)
or alternatively,
-{S, S} + hA
2
= 0.
(2.1.47)
In order to quantise the theory corresponding to S(z), we first choose a (k, n - k)dimensional Lagrangian submanifold L of M, such that w(v, i) = 0 for any pair of
vectors (v, 9) E TPL tangent to L at p. This corresponds to fixing a gauge. We then
introduce a volume element on L,
dA(el,...,e,) = dpI(el,...,en, fl,V..., fn)1/2 ,
where (el,...,e,) is a basis of TPL and w(ei, fj ) = 65.
(2.1.48)
The quantum theory corre-
sponding to the action S is then defined by the path-integral,
(2.1.49)
LdA eS/h.
We now translate the above into the language of covariant closed string field
theory. We are interested in the subspace - C W consisting of those elements of W
annihilated by L o and bo . These are rotationally invariant states which do not care
about the particular phase at the coordinate disks. One can define a symplectic form
on 1 as follows,
w(A, B) = (R'IC(2)IA) B)2
12A)1B) 2 ,
(2.1.50)
where (R'~2 is the projection of (R12 into X
'. In component notation we may write,
(012
1
.
iIWi42 (4PI(2.1.51)
Correspondingly, one can define the sewing ket,
IS 1 2) =
b-(1)IR' 2)
Ii)1(-)J+1iJj)
2,
(2.1.52)
where IR12) is the restriction of IR 12) to 'U. We have the contraction,
(R' 2 R23 )
(2.1.53)
31
where i•3 relabels the state space from one to three and also projects into the Lo = L0
subspace. This implies that,
(2.1.54)
(wU
12 1S23) = 3P 1 .
The matrices wi and w'3 are given explicitly by,
= (-)+1(icl
W
-(i Jbo Ii) .
),
(2.1.55)
These satisfy all the expected properties of the symplectic form on a supermanifold
(parametrised by coordinates z i, say), in that the form w = -dz'wi 3dzj is odd,
nondegenerate and closed. Also, the following properties hold,
E(Wij) = E(w23 ) = i + j + 1,
i = -_(-_)i7i
=ij=
(_(_+1)(J+1)W2i.
(2.1.56)
(2.1.57)
Let us recall some basic properties of the dynamical string field IX). Firstly, it is
a linear superposition of basis states of the underlying CFT,
j)
3
=
(2.1.58)
,
Di)
where the target space fields Oi are complex Grassman numbers with ghost number
gt9(0i) = 2 - Gi where Gi is the ghost number of JIDi), so that IT) is Grassman even
and has ghost number two. The string field must satisfy the subsidiary conditions,
bo IT) = LoI) = 0,
(2.1.59)
(for well defined string field vertices), and must be real,
(2.1.60)
(1 >I))= -IF)>.
The formalism naturally contains a B-V algebra structure. If we construct a set of
basis vectors IIi)
blj i) for 7-1 then,
(2.1.61)
.
j)) = -
w•(i'),
The string field may then be expanded as,
)=
Z II),)i =
i
E (1¢s)¢+ + I's)V),
(2.1.62)
g(da)_<2
where /"s are fields, and b are antifields. The second sum runs over a complete
i are the analogues of ei and fi of (2.1.48).
symplectic basis of W, where the I(I) and IV)
The coordinates i' are the analogues of the coordinates z' of the supermanifold. The
antibracket of two functions F and G of the fields and antifields is defined as,
,{Ff
a F
G}I
rE
G
F Vwijwj G.
(2.1.63)
which is actually the antibracket of (2.1.42) (with coordinates ~i rather than zi). We
may also write,
F, G} = ()G+
1
-
F) ~ G
S 12 ),
(
where
-
.
(2.1.64)
Similarly, the B-V delta operation acting on a function F is defined by,
F
d1
a 08
OF = -
1+
1r •
(pW )z F
(2.1.65)
2p
which is the delta operation of (2.1.40). One usually chooses a basis for which p is
field independent in which case one may write,
AF =
1
2
)F1
)F+1
eF
81T)i 8 X)2
IS12).
(2.1.66)
The closed string action is given by the following linear sum of the string field vertices,
z
2
- g K2g- +n
n!
S-=
,(2.1.67)
g,n>0
where the string field vertices are defined by {•9n}, = {J, - - ,
h is the Planck
h}g,
constant and K is the string coupling constant. Except where explicitly stated, we
shall use units in which r -= 1 throughout. It follows from the properties of the
dynamical string field IT) that the string field vertices, and hence the full string field
theory, are hermitian. Note that the kinetic term may be expressed in terms of the
symplectic form,
1
1
(2.1.68)
So,2 = 2 (TIcOQI)= 2 (wi121Q( 2) I)11T) 2 .
The string action defined above satisfies the B-V master equation as is required for
consistent quantisation.
There also exists a B-V structure on the moduli spaces of surfaces which we now
describe. It was explained in §2.1.3 how two punctures may be sewn by identifying
their boundaries according to the prescription zlz 2 = 1 where zl and z2 are the coordinates around the sewn punctures. It is also possible to rotate the coordinates around
one puncture through an angle 0 E [0, 27] before sewing and this is implemented via
the identification z1 z 2 = eOi. The process of 'twist-sewing' two surfaces is performed
by sewing the surfaces with an integration over all twist angles 0 E [0, 27r].
Given two moduli spaces A 1 C P ,l,n
and A 2 C Pg2,n2 symmetric under the ex9
change of labels at the punctures, one defines an antibracket {1Az, A 2 } C Pgl+g22,nl+•-2
by twist-sewing fixed-labelled punctures on every surface in A 1 to every surface in
A 2, and symmetrizing over the (nl + n 2 - 2) remaining punctures. Let [A] denote the
orientation of TrA for any space A. Then the orientation of {A41,A 2 } is defined by
[{A 1}), -, {A 2 1}] where {A1} and {A 2 } denote the set of basis vectors in TE{.A 1, A 2}
arising from [A1] and [A2 ] respectively, and A is the tangent vector associated to
changes in the twist angle. This moduli space antibracket, like the usual B-V antibracket, is odd, graded-commutative, and satisfies a Jacobi identity,
{A
1,
A 2 } - _(_)(AI+1)(A2+1){A 2 , A 1 },
(_)(A1+1)(A3+1)
A, A2}, A3} + cyclic = 0,
(2.1.69)
(2.1.70)
where A, and A 2 in the exponents denote the dimensions of the spaces.
Similarly, given a symmetric moduli space A C Pg,n, one defines a delta operation
AA C Pg+1,n-2 by twist-sewing two fixed-labelled punctures on every surface in A,
and multiplying by a factor of one half. The orientation of AA is defined to be
[', {A}].This moduli space delta operation, like the usual B-V delta operation, is
nilpotent and an odd derivation of the antibracket,
AA2 = 0,
(2.1.71)
A{A 1, A2}= {AA 1, A2} + ()A1+l{A 1, AA2 .
(2.1.72)
For any moduli space A C Pg,n, dA denotes the boundary of A. Given a
point p E 9A, a set of basis vectors [v1 ,.... , vkJ defines the orientation of &A if
[n, v1 ,...., Vk], with n a basis vector of TpA pointing outwards, is the orientation of
A at p. The boundary operator is nilpotent, is an odd derivation of the antibracket,
and anticommutes with the delta operation,
a2A = 0,
(2.1.73)
a {A1, A2}= {0A2,
AA1+
(2_)Al+1{A
2A2
,
1,)
(2.1.74)
AOA = -aAA.
(2.1.75)
If one defines a complex P = g,nPg,, as the formal sum of the spaces Pg,n where
(n > 3 for g = 0, n > 1 for g = 1, and n > 0 for g 2 2), whose elements are of the
form Ej aiAg,,,,n where the ai are real and the spaces Ag
9 ,n, C Pgi,ni do not include
surfaces arbitrarily close to degeneration, then the antibracket, the delta operator,
and the boundary operator become well-defined over this complex by defining them
to act as linear or bilinear operators over the summation.
If we define a special vector in P by,
S=
E
(2.1.76)
hgg,,,
g,n
where again (n > 3 for g = 0, n > 1 for g = 1, and n > 0 for g _ 2), then the
recursion relations (2.1.1) for the string vertices can be written,
&V+ 1 {V,
(2.1.77)
} + hAV = 0.
In the above we have defined Vo0, = 0 for n = 0, 1 and 2.
Let us discuss briefly the sewing of forms. The sewing ket IS12) implements at the
level of states the action of twist-sewing at the level of surfaces. If it is used to sew
two punctures on separate surfaces we have,
/A2
(L
1 (Q(kl)gl,nJ
2
(-(k 2 )9 2 n 2 i 1) 12)
=
k2
•
((kl+k2-1)gi+g2,ni+n2-2
,
Ai
1
(2.1.78)
Similarly, if the sewing ket it is used to sew two punctures on the same surface we
have,
f
1-
(k)gnIS12
= )k
3...n ((k
-
1)g+1ln
-2
.
(2.1.79)
These results can be used to prove the existence of an important homomorphism
between the B-V algebra of functions and the B-V algebra of moduli spaces. The key
to this homomorphism are the string functionals which provide a canonical mapping
from moduli spaces of surfaces to functions of the fields and antifields. The functions
then form a representation of the B-V algebra of surfaces.
Let Ag,n denote a subspace of P9 ,n of real dimensionality (6g - 6 + 2n + k). The
f(kA)
is defined as,
associated string functional f(Ag) is defined as,
n
f(A ,(k) _--n!1I(k),n
(2.1.80)
,n
This is a very natural operation as we are simply integrating the canonical forms on
moduli space over the moduli space of surfaces. From ghost number considerations
we have,
(2.1.81)
for k 0.
f(A(k)= 0,
The definition of functionals over the complete complex P of surfaces follows from,
f (Eaa
i
For A, B E
~,
f (A(ki)
=
(2.1.82)
i
the following representation identities hold,
f(AA) = -Af(A),
(2.1.83)
f({A, B}) = -{ f(A), f(B)},
(2.1.84)
f(oA) = -{S0 ,2 , f(A)}l.
(2.1.85)
These identities define the homomorphism between the B-V algebra on moduli spaces
of decorated Riemann surfaces and the corresponding algebra of string functionals.
2.2
Deformations and space of string backgrounds
In the operator formalism, a CFT is defined as a way of assigning a state (EI to each
decorated Riemann surface E in a way compatible with sewing (2.1.12). The CFT
may be deformed by deforming these states in some way, (El -+ (El + 6(EI. This
results in a new CFT if the algebra of sewing is preserved,
6 (5,ooi,jZ21 =
(6(EI)(E21Rij) + ( 1I(6(E2)IRij)
2
+ (E1I(E21(6IR•)) .
(2.2.1)
One can then introduce the concept of a space of CFTs by taking a base manifold M
and assuming that for each point X E M is associated a CFT with a state space W.,
having basis Iki(x)) and coordinates Vi, so that there is an assignment of a surface
state (E(x)l to each surface E. It is assumed that this theory space has the structure
of a vector bundle and that the (E(x)l are smooth sections.
2.2.1
Connections on the space of conformal field theories
A way of deforming the theory is to integrate the insertion of a dimension (1, 1)
primary state (0) over some region of each surface,
21ri
(; z10) dz A dz,
rE-Uioi
(2.2.2)
where (E; zI is the state corresponding to the surface with an additional puncture
at position z. The integration is well-defined as the integrand is a two-form on the
surface when O(z, -) is of primary dimension (1, 1). The integration is performed over
the region outside the coordinate disks Di. Under this condition we have 6(R 12 1
6|R 12 ) = 0 since the coordinate disks cover the entire surface.
Another type of deformation is given by the action of a linear operator on the
state space '7 applied to each surface state as follows,
(El = -(E
Zw(i),
and
IR 1 2) = 1(W( 1) +w(2))R
(2.2.3)
) .
12
i=1
Although this constitutes a valid deformation, it does not change the corresponding
underlying CFT as it amounts merely to a change of basis of the state space X-.
General deformations will therefore be taken to be of the form,
==(E
27ri
L J-
-UDi
(E; zlO) dz A d- - E(E|I
A sewing relation f(E; zlO)dz A dz =
i==1
w().
f(Elo(i)(z, ~) dz A dT means that
(2.2.4)
shifting the
omitted region of integration around the punctures from Di to D' amounts merely to
a shift in w( i) to w (i) + - fDS-D( O(z, T) dz A dz, and thereby a change of basis. So
the general deformation may be written,
6(EI = e-
27ri E-UiD
(E; zIl ) dz A d - E(EI
w(i) +
f
2i
i=1
O(z,)dz
D-D
Ad) .
(2.2.5)
Let us introduce a connection FI',(x) on the theory space bundle. Given sections
ai(x) (QI|
and IB(x)) = >i I(D)bi(x) on the bundle, one can define their
(A(x)I =
covariant derivatives by,
>E
D,(r)(A(I
where FM,
a Aai(Vi
0-,(AI
- airi
-
(A ,
(0(2.26)
(2.2.6)
Ej7-14i)I',(4V 1, and,
D,.(r) IB) - Iji) ,mb'+ IP,)r1b(2
- j IB)+ r,IB).
-27)
(2.2.7)
The above also defines the partial derivative of the sections. Greek indices are used for
theory space coordinates and Latin indices for symplectic coordinates. The covariant
derivatives of functions on the bundle are defined by,
i Cj
D,(P)F a 8,F - F'i FJ
FFa
talW/
(2.2.8)
where we introduce the notation analogous to (2.1.64),
-
i.
~(2.2.9)
Generalising, the covariant derivatives of a section of n-punctured surface states is,
n
Z-
D,()(I=
(•
Ti]),
(2.2.10)
i=1
The covariant derivative satisfies Leibnitz's rule,
D,(1)(AIB) = (D,(F)(AI)IB) + (AI(D,(F)IB)).
(2.2.11)
which means that the deformation,
6(E(x)l = 6x"D,(rF)(M(x)l,
(2.2.12)
is a valid CFT deformation. We will only consider connections for which there exist
sections of marginal operators Io,) and operators w, such that,
n
D,( (E=
2i
E-UDi
(E; zi,)
dz A d - (~I
i=
w( i ) .
(2.2.13)
Under a change in connection F, -+ F, + SF,, we have the corresponding change
WI -+
t) + 6rPf.
There is clearly a relationship between the range of integration,
the choice of connection and the choice of w( i ) so that, up to the choice of basis for 71,
the deformation is specified completely by the choice of marginal operator O0(z, f).
We will primarily be interested in symplectic connections for which the covariant
derivatives of the symplectic form and the sewing ket vanish,
D,(r) (W12 1= D,(F) IS1 2) = 0,
(2.2.14)
A special case is the canonical connection which is defined as the connection Fr for
which the excluded regions are the coordinate disks and for which the w,) vanish,
( )(
1
AMP
= 2ri
-,-UiDi
(E; zO1,,) dz A d.
(2.2.15)
This can be shown to be symplectic as a result of the vanishing covariant derivatives
of the ghosts and antighosts.
2.2.2
Special punctures and the operator /C
The above leads us to introduce the operator K: acting on moduli spaces. Given a
symmetric subspace A c P 9 ,,, we define )CA to be the subspace of P,n which includes
the set of surfaces obtained by adding a special puncture over the region E - UjiD
of each surface E C A. The puncture is called 'special' as it not symmetrised with
respect to the other punctures. The coordinate at the extra puncture is fixed by
first applying the minimal area metric to the original surface E and then applying
the general prescription outlined below to extract a family of local coordinates over
E - UiDi from a knowledge of the (conformal) metric on the surface. One thus
obtains a canonical set of local coordinates for each position of the added puncture
on --U Di. (Note that the choice of coordinates is essential for the proof of the ghostdilaton theorem, but it is irrelevant to the proof of quantum background independence
as a primary state is always inserted at the puncture). The orientation of KA is given
by [Vi, V2 , {A}] where V 1, V2 E TFPg,n+l are the tangent vectors associated with the
motion of the added puncture. The Schiffer variations corresponding to such motion
will also be found explicitly below.
The operator K satisfies the following properties,
K 2A = 0,
IC=A, A21=
(2.2.16)
A1 A2 +
A 2} ,
1
(2.2.17)
KAA = AKCA,
(2.2.18)
[0, IC]A = -(~, 3, A} ,
(2.2.19)
where A 1 , A, A E ? and V,3 is a three punctured sphere with punctures at 0, oo
with local coordinates z, 1/z respectively string vertex, and an (asymmetric) third
puncture at z = 1 which is special and whose coordinates are left arbitrary. It follows
from (2.2.15) that the covariant derivative of forms is given by,
D,(f)
Agn ([k]gn+l
)n+l,
(2.2.20)
where 1O,) = clIO,) is a dimension (0, 0) primary state (and hence does not care
about the coordinates at the special puncture at which it is inserted).
If we define a functional f, mapping from surfaces with a special puncture to
functions of the fields/antifields by,
f
()[[klgn+IQ)
k(A)
..
0) ,
(2.2.21)
g,n+1
then we find the following representation formulae,
fm({A, B}) = -{f (A), f,(B)} ,
f,(AB) = -Af,(A),
f, (A)
= -{SO,
2 , fj
(A)} ,
(2.2.22)
(2.2.23)
(2.2.24)
f,(KIA) = D,(P)f (A),
(2.2.25)
= Dt(r)So,2 ,
,3)
L(VL
(2.2.26)
I
(
i
Figure 2-7: Family of local coordinates. The map hp defines a local coordinate around
each point p E U parametrised by (AI, A2 ). The origin of w is mapped to p.
where A E P has no special punctures, B is a space of surfaces with a single special
puncture.
We now return to the problem of extracting a canonical family of local coordinates
over a punctured surface given a conformal metric. Consider a Riemann surface E
and some open neighbourhood U C E parametrised by two real parameters A1 and
A2 , and with local uniformizer z. Then each point p E U fixes (A1 , A2 ) and there
exists a function z(Ax, A2 ) = z(p). We may obtain a family of local coordinates by
assigning to each point p E U a mapping hp : Dw -+ D, from the complex unit disk
D,: IwI < 1 to some disk D, around p, where the origin is mapped to p [Fig. 2-7].
The coordinate zp(q) of any point q E Dp with respect to the local coordinates at p
is then given in its most general form by,
00
zp(q) =
a (Ai,
A2)Wn ,
(2.2.27)
n=O
where z(q) = hp(w). The family of local coordinates is then specified by the functions
an(A•, A2 ). Usually one sets zp(0) = ao(A 1 , A2 ) = z(A•, A2 ).
Given a conformal metric p(z, -) in a region U described by the local uniformizer
z, one can extract in a canonical way a family of local coordinates for U as follows. For
each point p E U, one demands that the pullback pw of the metric to the complex w
plane satisfy .,npw•w=o = od9p"Ww=o = 0, for all positive values of n. This prescription
uniquely fixes the coefficients an(A1 , A2 ) for n > 2, while ao and ac must be supplied
independently.
The pullback pW is related to pZ, the metric referred to the coordinates z,, as
follows,
pzIdzl = p'Idwl,
p = pI dzp'
(2.2.28)
We need to express pW as a power series in w and W. Performing a Taylor expansion
on pZ we find,
n=0
( Z. (n- m)!m! (z
-
zp(P))
0= +0
-=zP (l1+
E • ((Eaj)w
n=1
)) n-m
-(
m
zI )
( •)•)°) •,)
i=1
(2.2.29)
j=1
where we have used (2.2.27). Furthermore, we have for the remaining factor,
dzw
-
r=1
0
00
-1 1 E s
=I I(1a+
r=2
(2.2.30)
si
s=2
Combining these expansions the result is,
p = (pzIp - jai)I(1+
(E
,=1
p
S+
1+ znp
A((Eaiwi)9z
n=
p=2
W
)1
o
+
1Zq
a
zi
1
(2.2.31)
1) 2q•
q=2
The conditions on the pulled-back metric can now be applied by taking successive
derivatives of this formula. We notice that ao does not appear, and may be fixed as
was done earlier. The first condition states that ,wplw•=o = 1-Pvlw=o = 0, giving us,
c2
C2 =
1
z In pz
,
anda
2
=2
1
InpzP.
z
(2.2.32)
Since these equations determine the ratio a 2 /ca for each point p E R we can simply
write,
a2
(2.2.33)
-az In p .
2
Higher order coefficients are determined recursively from the higher order constraints.
Now, a, remains an arbitrary parameter. It is natural from (2.2.31) to set lail =
1/pZl . In this way the metric pW becomes of the form p" = 1 + - -.. The procedure
does not fix the phase of al over the region U, though we will mention later a way
to fix that phase in a natural way for the case of families of local coordinates over
curves.
We now calculate the explicit antighost insertions associated with the tangent
vectors that represent the motion of a puncture in a region U of some surface. We
will assume that as this motion takes place the rest of the surface data (including
the moduli, the position of the other punctures, and their local coordinates) does not
change. Therefore the Schiffer vector will be supported only on the moving puncture.
The associated antighost insertion will be used later to study the effect of insertions
of ghost-dilaton ID) and the state Ix) in correlators.
The local coordinate at the moving puncture is described by the family of local
coordinates zP (2.2.27) over a region U of a Riemann surface E. Associated with each
real parameter Ai there is a Schiffer vector v(Ai) given by [25],
vi (w) =
-Ai
-
(2.2.34)
E/3A,(A,,A2)
n=0
Expanding this as a power series in w we have,
vxi(w)
-
na
-(E
Wn-1) -1
k=0
n=l
- (a + + (+
1) an+
daE
_
k)
n)
(
00
00
00
m=0
k k
n
(n + 1) a
(rn
=
(2.2.35)
ak W
k=0
n=1
n=1
k=0
This is the form of the Schiffer vector for the most general choice of local coordinates.
The A•,n are now extracted by comparing coefficients of an in (2.2.34) and (2.2.35).
The first three coefficients are given by,
A,,o =
I3
=i
,,1 =
Sd
1 da 0
a, dAi
2a2 dao
(2.2.36)
,
1 dal
a d
(2.2.37)
1adAi ao dAi '
+ 2,2
2a2
dao
(2.2.38)
a1 dAs
a2 dA
\ a2
a1 dAi
These will be relevant when considering the ghost-dilaton theorem. The antighost
insertions corresponding to the Schiffer vectors (2.2.35) are,
O0
b(vA•)
= E(/Ai,n bn
+
i,, bn-l)
(2.2.39)
n=0
2.2.3
Quantum background independence
In the B-V formulation, a quantum theory is defined by the data (M, w, dp, S), where
M is the supermanifold with coordinates given by the fields/antifields, w is an odd
symplectic form on M, djp is a volume element on AM, and S is the master action. For
closed string field theory, M is just the subspace 7- of the state space of a conformal
field theory, while w, dp and S have all been given earlier. Consider two nearby
string backgrounds at points x and y in the theory space M, which are represented by
conformal field theories with data (-XI, w•, Idx, Sx) and (7,y, wy, dpy, Sy) respectively.
Establishing quantum background independence of closed string field theory then
amounts to finding a symplectic diffeomorphism carrying -x to Hy,,
Fy,z :
X
R,
-4 ,
(2.2.40)
in such a way that the pullback of wy is w",
(2.2.41)
WX = F~sW y ,
and the action weighted measure dps,,is pulled-back to dPsy,
dpxe 2S , /h
Fy,x(dpye 2SY/h).
(2.2.42)
The measure is given by dp = p Hi dg<'(where a basis is chosen for which p fieldindependent), so the latter condition translates into,
2S(VX,
2S(V,,
x)
)
exp
p(y)
exp((
sdet
(2.2.43)
If one considers an infinitesimal diffeomorphism relating a theory at x to one at
y = x + 6x we have,
x, x + 6x) = V4' + 6x" f (b,x) + O(6x 2 ) .
x+62= F i ('x,
(2.2.44)
Then (2.2.43) reduces to the condition,
h(O, Inp + str (0 f4)) = 0.
S + S~f ±
+
(2.2.45)
One can define the object B' by separating out a term proportional to the connection
from fr,
=f -F i - Bt.
(2.2.46)
The condition (2.2.41) that F i be a symplectic map requires that there exist an odd
Hamiltonian function B, such that
B
= {i,B,} ++ B = 2jj B..
Bi
(2.2.47)
(2.2.47)
Then (2.2.45) can be written,
D, (F)S - hAB, - {S, B,} = 2h(str F, - Oln p).
(2.2.48)
Establishing background independence amounts to finding the odd Hamiltonian B,
satisfying this condition. The solution is given by,
B, = B (2 ) - f(B13 1 ) ,
(2.2.49)
where B (2) is defined by,
(wl22J>)
O• 11
B(
2
,
(2.2.50)
and the sum of the 'B1-spaces' is defined by,
hgB
1
g,n
,
,
(2.2.51)
with B1,, for (n < 1) and B,,n for n = 0 being taken to vanish (the latter would
not contribute to background independence due to their lack of sewable punctures).
Each space B",, is a (6g - 3 + 3n)-dimensional subspace of Pg,n+1 with n ordinary
punctures which are symmetrised with respect with respect to change of labels, and
one special puncture (for inserting (0,O). These spaces are constructed explicitly by
first assuming the existence of B1,2 and then applying the following condition recursively (by taking B',, to be any homotopy interpolating between the two boundaries
suh that each surface remains symmetric with respect to the ordinary punctures) to
obtain all B'-spaces of higher dimension (note that dim /l = 6g - 3 + 2n),
aB
=V,
VV,'i
-
(2.2.52)
g,n+l,
g,n+l --
1,n
where V',,+1 is defined by,
g
n+l
,l+l-= v,n - E
, g{v,,n-m_+±2}
-
g,n±+2,
(2.2.53)
gj=O m=1
for n > 3 for g = 0 and n > 1 for g > 1, (since V,3 has already been defined as the
auxiliary three-string vertex). The operator I introduced in (2.2.52) simply changes
a (fixed-labelled) puncture of the space on which it acts into a special puncture. The
operator I has the following properties,
1 2 A = 0,
Z{A,,A 2} =
1{Z4,,A2}
(2.2.54)
+ {A17 A2} ,
(2.2.55)
[a, Z]A = 0,
(2.2.56)
(KIZ + IC)A = {A, , }.
(2.2.57)
and also satisfies the following useful representation identity,
f,(IV) = -{ f (V),B 2)} .
(2.2.58)
The space 70, in (2.2.57) is constructed as follows. Consider a three-punctured sphere
with an ordinary puncture at z = 0, and a special punctre at w = 0 where w - 1/z.
A second special puncture has a position interpolating form w = 0 to w = 1. The
resulting space is then antisymmetrised with respect to swapping of labels at the
special punctures.
Returning to the construction of the Bl-spaces, the space B1,2 itself must satisfy
the equation,
9,S1,o = 2(str F, -
a, In p) + fi(AL3, 2 ) + f,(EV1,1),
(2.2.59)
which holds if we choose BO,, to interpolate from IV 0 ,3 to V, 3 . However the existence
of singular tori AV•, 3 in the domain of integration in f,A(AB•, 2 ) means that the above
expression is not manifesly free of divergences. This is a pathalogical property of the
canonical connection F,. Choosing a different connection F, (by specifying a different
basis) as follows,
(w12 iJ7l)
=
(W12 1I~l) +(
o
t),
(2.2.60)
go,2
where Bg, 2 interpolates from V6, 3 to some new (nonsingular) auxiliary vertex Vo0, 3 , we
find that B 1,2 now interpolates from IV 0 ,3 to V0 ,3 . The integral is path independent
and may simply be deformed to avoid the singularity associated with AV 3 . :,This
concludes the construction of the Bl-spaces and establishes quantum background
independence of closed string field theory.
2.2.4
Ghost-dilaton theorem
The ghost-dilaton is present for any string background and is defined by,
1
D(z, )
2
(ca2e - eW ),
(2.2.61)
so that the corresponding Fock space state is,
ID) =
(c1 c-1 -_1F-1)00) .
(2.2.62)
This is BRST-closed, QID) = 0, and is a physical state which happens to be trivial
in the absolute BRST cohomology,
ID)= -QIX),
where IX)= - (c - 0o)0) .
(2.2.63)
However, it is not trivial in the semi-relative (physical) cohomology since bo IX) $ 10).
Neither the dilaton nor the state IX) is primary as they are not annihilated by either
L1 or L1, though they are annihilated by all other L,, L for non-negative n.
The goal of the ghost-dilaton theorem in closed string field theory is to show
that a background deformation in the direction of the dilaton changes the string field
measure dlas in the same way that a shift in the string coupling constant a would.
The form of the n-dependence of the action (2.1.67) tell us that schematically, the
proof requires one to show that for any correlator,
f D (zo
I)i(zI)
.
4n (Z))n
(2-
2g - n) (I)1 (zl) -'
n(Zn))r,
(2.2.64)
so that integrating a dilaton amounts to multiplication by the Euler constant of the
surface, which is considered to be a bordered Riemann surface of genus g with n
boundaries.
More specifically, analogous to the proof of quantum background independence,
there should exist a symplectic diffeomorphism F : W7, --+ W, relating two nearby
string backgrounds differing only by the value of the coupling constant such that the
measure at n is pulled back to the measure at r;',
F* (dp(K)e 2S (
)/ h )
= dp(W')el2S( ')/h.
(2.2.65)
For the case when the coupling constants differ infinitesimally by,
S= (1+ aE),
(2.2.66)
where a is a constant to be determined, one can show that the diffeomorphism must
be implemented by some odd Hamiltonian function BD which satisfies the condition,
{S, BD}+hABD = ahd.
d
1
(Sl,o+2 In p)+a -(2g-2+n)h
2g - 2 +n
f
f(Vg ,n ) . (2.2.67)
g,n
In analogy with the case of background independence, the solution for the fielddependent terms is simply given by [28],
BD = B(2 ) - fD(l),
(2.2.68)
where the subscript D corresponds to the insertion of a dilaton state at the special
puncture and B2) is defined as in (2.2.50). In this instance, it is found necessary to
extend the sum B' to include the higher genus spaces B1,o (g > 2) without ordinary
punctures. These may be added without affecting the proof of background independence in any way, precisely due to the lack of sewable punctures. One would like
also to define a space B 1 corresponding to a once-punctured torus, but the recursion
relations would require that BL, o = ABL, 2 -ZIB°,, which is inconsistent as it does not
satisfy OBlo = 0. If terms with g = 1, n = 1 are extracted by hand, the recursion
relations for all defined Bl-spaces may be expressed in a single equation,
031
AB•1 - {V, B'} -I
= V•,3 + KCV -
+ AB1,2 +-•V1,1 .
(2.2.69)
Although this suffices to prove the dilaton theorem for the field-dependent terms of
the action, the proof remains incomplete as it does not account for the vacuum terms,
which are somewhat more difficult to analyse. In particular, we are still left with the
requirement,
d
1
fD (-I,1)
- fX(A0VO, 3 ) =
- S1 ,0o +
In p ,
(2.2.70)
at genus one, and,
fo(ICVg,o) = (2g - 2) f (V,o),
(2.2.71)
for g > 2. The issue of the vacuum graphs and the ghost-dilaton will be tackled in
Chapter 3.
To understand why (2.2.68) is correct, we will apply the results of §2.2.2 to outline
how the insertions of ID) and IX) give the Gaussian curvature R (2 ) and the geodesic
curvature 0k ( ) respectively, so that integrating the insertions extracts the Euler characteristic of each surface. We are interested in evaluating integrals of the form,
fDCA)
([d+2]g,n+1j)
...
)nOD)n+,
(2.2.72)
where A C Pg,n is a d-dimensional space of surfaces of genus g with n > 1 punctures
(n > 3 for g = 0), and also integrals of the form,
f(,CA)
f,(z:A)--
-1
f
:.
(Q[d+l]g,n+1
1 T)
...
I')nIX)n+l,
(2.2.73)
where LA
-{V6,
I 3 , A}. We shall let (61,... ,d) represent a set of coordinates for
the space A and ((I,...,(d, A1, A2 ) coordinates for ICA, where A,, A2 are two real
parameters associated with the position of the special puncture. A family of local
coordinates for the special puncture has been derived in §2.2.2.
Let us consider the Schiffer vectors arising in (2.2.72) and (2.2.73). The Schiffer
vectors associated with the Ai cannot change any of the data on each E E A, and
need only be supported at the special puncture,
(2.2.74)
i = 1, 2,
)) ,
· 0, v
v \ = (0,
The remaining d vectors can be written in the form,
Vk
k
k = 1, ... , d,
+ Vk ,
(2.2.75)
where each Schiffer vector has been split into a vector V9k, supported only on the
original n punctures which in general can change the underlying moduli of E, and a
vector v'k, which is only supported at the special puncture and need only 'correct'
the local coordinate there,
S0) ( 1)
.
v•k = (0,..., 0,
,
(2.2.76)
The antighost insertion b(v k)10)n+1 corresponding to •k vanishes since only modes
bn with n > 0 appear for local coordinate changes. The forms appearing in (2.2.72)
and (2.2.73) are,
(Q[d+ 2]gn+llD)n+l = (-27i )2 -d-3d
1
A ... A dd A dA1 A dA2
-(E,,n+llb(v,)
...
b(vd)b(vl)jb(vx 2)jD),+1,
(Q[d+l]gn+llX)n+l= (-2ri ) 2-n-3gdf 1 A ... A dd A
•(Eg,n+l1b(v)cs,)
(2.2.77)
dAj
i=1,2
(2.2.78)
- -bb(d)b(v,) Ix)n+1
Now using (2.2.39), (2.2.62) and (2.2.63), one can show that,
IW()
dA1 AdA2 b(v ,1)b(vx,)jD) = 10) w(2)
w( 1) ) =
X = 0) w
dA2 b(va,)jx)
2)
,
(2.2.79)
(2.2.80)
i=1,2
R (2 )
where w(( ) and w ) are related respectively to the Gaussian curvature two-form
and the geodesic curvature one-form k(0) associated with the metric p on E,
(2) = 2dz A dz OOp = iR(2),
(2.2.81)
W) = i[dO - i(dz a In p - d9lnp)] = ik cl) .
(2.2.82)
Here 0 is the phase of a 1 which is set to be equal to the phase (with respect to the
uniformizer z) of the tangent to the coordinate curve y along which the IX)insertion
is integrated. Inserting these results into (2.2.77) and (2.2.78) we find that,
fD(ICA)
fx (A)
=
n!
1 /([dli
=
' j)
('
n! A"27"
...
27
2A
-UiDi
R (2),
(2.2.83)
k().
(2.2.84)
L). 1
_(r1-UDi)
The Gauss-Bonnet theorem then implies that the sum of these extracts the Euler
characteristic of the bordered surface E - UiDi (thus fixing a = 1),
fD(ICA) + fx(LA) = (2g - 2 + n)f(A).
(2.2.85)
In the particular case of the metric satisfying the minimal area condition, the geodesic
curvature k(1) and therefore fx(iLA) vanishes leaving precisely the equation required
for the ghost-dilaton theorem to hold for the field-dependent terms in the action. The
subtleties arise for the vacuum graphs as there are no ordinary punctures available
to accommodate the moduli-changing Schiffer vectors of (2.2.75).
The deformation in the direction of the ghost-dilaton is a valid CFT deformation which changes the string background by shifting the string coupling constant
it. However it turns out that the deformation does not give a new CFT, but rather
corresponds only to a trivial change of basis. This implies that the space of string
backgrounds includes an extra parameter not present in the space of CFTs and disproves the widely held assumption that a string background is simply a conformal
background. Moreover, it opens up the possibility that there exist string backgrounds
which are not conformal at all. The construction of string field theory around nonconformal backgrounds will be discussed next.
2.2.5
String field theory around non-conformal backgrounds
Let us recall some essential results from §4 of [11] which originally motivated the recent
investigations of string field theory around non-conformal backgrounds. It was shown
there that the algebraic structure of the classical closed string field theory corresponds
to a homotopy Lie algebra L,. This is defined by a set of graded commutative string
products m, = [B1 , ..., Bn]o which are multilinear maps from a tensor product 7 Oln
to W7,where 7W is the complete Hilbert space for a particular string background, and
the JBi) are annihilated by bo and L o .The products themselves are also annihilated
by bo and L o and must satisfy the identity,
E
o1(i, jk)[Bil, ..., Bi [Bjl, ..., Bjko]0
=
0.
(2.2.86)
{i1,jk};1,k > 0
1+k=n >0
where a(it, jk) is just a sign factor picked up from arranging the string fields B 1, ..., B,
into the required order. Roughly speaking, the string product mn is the element of the
string Hilbert space which is obtained when the string fields B 1 , ..., B, are inserted
into n punctures of the string vertex Vo0,+l. The products have ghost number given
by,
G([B1,..., B,]o) = 3 + Z(G(Bi) - 2),
(2.2.87)
i=1
so that the statistics of the product is intrinsically odd,
n
(2.2.88)
aB,.,ao = 1 + EZB,.
i=1
Defining the degree of Bi as d(Bi) = 2 - G(Bi), we find,
n
d([B, -... , B]o) = -1 + E d(Bi),
(2.2.89)
i=1
For n = 1, the product is defined to give the BRST operator acting on the state,
[B]o - QIB). For n = 0 the product [- ]Jo E 7 is just a special state and of degree +1
(or ghost number +3), which is taken to vanish for a conformal theory.
It was shown by Sen [35] that the above algebraic structure is stable under shifts of
the string field that correspond to classical solutions. However if we let IF-+ Io + 'I
with T0 a string field which does not satisfy the classical equations of motion, then the
new theory, whose vacuum corresponds to V' = 0, is a string field theory formulated
around a non-conformal background. This new theory has the same structure of the
original except that the product m 0 is now non-vanishing, [. ]0 = F where IF) is
Grassman odd and of ghost number +3. Also, we have [B]o = QJB) where Q is a
new BRST-like operator. It annihilates YF) and, unlike the original BRST operator,
is neither nilpotent nor a derivation of the product m 2,
Q[ ]== QIF) = 0,
(2.2.90)
Q2 1B) + [,S B]o = 0,
(2.2.91)
Q[BI, B 2 ]' + [QB 1 ,B 2]' + ()B1[B [ ,1 QB 2] + [7
1,
B 2].
(2.2.92)
It has been postulated in [24] that Q and JF) may be constructed from a generalized BRST charge Q and a string field F. The object Q is a contour-dependent
non-conserved charge which is taken to satisfy,
(2.2.93)
lim Q (,)F(0) = 0,
r-40
[Q(7)]2 =
Q(7 2) - Q(71) =
(2.2.94)
i
2-1
[2]
(2.2.95)
where the 'yi represent (homotopic) closed contours and F[1] and F [2] are respectively
the one-form and two-form associated to F. In symplectic language these conditions
take the form,
{Q, BF} = -f (0, 1),
(2.2.96)
f(V,),
(2.2.97)
{Q, f (E)} = f((KE),
(2.2.98)
S
=
where Q
1(w12 (2)I)1~
)2 = SO, 2 is the Hamiltonian function corresponding to
the BRST operator which is just the kinetic term of the string action.
By introducing some generalised moduli spaces whose existence will be derived
in Chapter 4, one can construct the SFT action around non-conformal backgrounds.
These generalised 'B-spaces' are moduli spaces of decorated Riemann surfaces of
genus g with n ordinary punctures and h special punctures, denoted Bf,,. They are
symmetric with respect to exchange of labels of ordinary punctures and antisymmetric
with respect to exchange of labels of special punctures.
It is necessary to extend the definition of the antibracket to take into account the
ordering of the special punctures. Let {B 1 , B 2 }' be the usual antibracket applied to
two spaces Bi having ni ordinary punctures and hi special punctures. On sewing,
the resulting surfaces will have ihl + i2 special punctures. Consider the inequivalent
splittings of the complete labelled set of special punctures into two groups (il,..., if,)
and (i' 1+1,... , i t 1+ 2 ) where order of labels in each set is irrelevant. For each splitting,
the ordered punctures on the B1 side of the sewn surface are labelled using the first
set of labels and for B 2 the second set. A sign factor is assigned to each splitting
according the permutations required to bring (il,..., i~h+1 2 ) to standard ascending
order. The distinct splittings are then summed over.
The final, generalised antibracket { -,7 } is now defined by,
{B 1, B 2}
(_)fn(1+(B2){B
1,
(2.2.99)
B2 }'.
The antibracket identities with respect to A and 0 are generalised as follows,
{B 1 ,B32
=
()(B1+hl+l)(B2+
(_)(B1+fl+1)(B2+22+L){
A{BI,22
{81, B2
32
=
{B, B }, Ba}
l){B 2 , B) 1 },
(2.2.100)
+ cyclic = 0.
(2.2.101)
{Al, B 2 } +-- (-)Bfil1l{
1 l,
A
2}
(aB1, 1 B 2} + (-)B1'1l{B+ 1 , BW
2} ,
,
(2.2.102)
(2.2.103)
The KC operator also needs to be generalised. If the original surface has f punctures, K will insert a puncture labelled i + 1 throughout the surface minus the unit
disks around the ordinary punctures and then subtract the n copies of this surface
where the label of the new special puncture is exchanged with each of the original
ones. The following new identities then hold with respect to the antibracket and d
(nilpotency and commutation with A still hold),
IC({BI1, B2 )
=
(-)"2+B2'1ICB
1
, B2 } + {B 1, CB2 },
(2.2.104)
(2.2.105)
[, K]B = (-)"+{( ,3 7 Bn} .
The operator I which changes an ordinary puncture into a special puncture is
generalised in precisely the same way as IC. Of the identities (2.2.54)-(2.2.57), only
the identity with respect to the antibracket is affected,
{B1, B2} =
(-)>
3
2+fl2+l{B11 ,
B2} + {,
zB 2} .
(2.2.106)
Similarly, the homomorphism from the B-V algebra of Riemann surfaces to the
B-V algebra of functions in W7is generalised to,
f(Bg
)
=
!L
(g,,
g,n
n,
I')n lF)>
...
...
IF)l .
(2.2.107)
The usual homomorphism identities still apply. The classical action for string field
theory around non-conformal backgrounds was found in [24]. The generalisation to
the quantum action will be stated in Chapter 5.
Chapter 3
Vacuum vertices and the
ghost-dilaton
In this chapter we complete the proof of the ghost-dilaton theorem by showing that
the coupling constant dependence of the vacuum vertices appearing in the closed
string action is given correctly by one-point functions of the ghost-dilaton. To prove
this at genus one we develop the formalism required to evaluate off-shell amplitudes
on tori.
3.1
Vacuum vertices of genus g > 2
In the following we assume some familiarity with the notation of [22], which should
be consulted for definitions. If a shift of the ghost-dilaton is to change the coupling
constant in the g > 2 field-independent terms of the string action the following
equation (2.2.70) should hold,
fD (CVg,o) = (2g - 2)f(Vg, 0 ) ,
g > 2.
(3.1.1)
The left hand side denotes a ghost-dilaton insertion over the vacuum vertex of genus
g, and the right hand side is (minus) the Euler number of a genus g surface times
the vacuum vertex. The purpose of the present section is to provide a proof of this
equation.
The right hand side involves the integration over Vg,o C Mg,o, of a (6g - 6)-form
defined in (2.1.31),
f(V,,0) -
"g
d(E
|b(~
Q)
... b(~"6 g- 6 )10) .
(3.1.2)
In this equation N= -27ri, and d - d= A- ..
Ad<6g_ 6 , where (61, ·
6g-6) is a set of
coordinates in Vg,,. Given a surface E E ,V,0, C E Pg,1 denotes the same surface, with
the addition of an extra puncture equipped with a local coordinate (defined up to a
phase). In addition, v% is the Schiffer vector associated to the tangent a/81i . More
precisely, it represents a tangent in T2P9 ,1 chosen to project down to O/O9i E Tr~Mg,
upon deletion of the extra puncture. While the ingredients used to build the form
integrated in (3.1.2) depend on position and coordinates around the puncture, as well
as a choice of representatives for the tangents, the resulting form is independent of
this data [30,31].
Let us elaborate on the ambiguity in the choice of Schiffer vectors representing the
tangents in (3.1.2). If the local coordinate at the puncture is denoted as z, Schiffer
vectors v'(z) = ao + a1z + -- -, regular at the puncture, can only change the position
and local coordinates at the puncture. Since the above form is independent of this
data, the change b(i',) -+ b('Y,) + b(v') should make no difference in (3.1.2). This
follows immediately from b(v')0O) = 0, which, in turn, holds since bn>- 1 and b.>_1
annihilate the vacuum.
Consider now the left hand side of (3.1.1). Here we must integrate over the position
of an extra puncture on each surface in the space Vg, 0 . Since the ghost-dilaton is not
primary we need a a family of local coordinates throughout each surface of V,,o. Such
a family is obtained by introducing, continuously over Vg,o, a conformal metric on
every surface. This metric is used to define a family of local coordinates via the
prescription of Ref. [36], as elaborated in §3.1 of [22]. We now write the left hand
side of (3.1.1) as follows,
d(A dA A dA2 - ( lIb(v,)
fD(KEV,o) -=.2-3g
. ..
b(v,
6
g
6
)b(vA,)b(v, 2)ID).
(3.1.3)
parameters
The
real
in
(3.1.2).
used
coordinates
same
,
,
the
The ý are coordinates M 9 0
(A1 , A2 ) parameterize the position of the dilaton puncture. The Schiffer vectors v\ are
chosen to alter only the position and coordinate data for the dilaton puncture, while
the vC, represent the tangents that change the moduli of the surface, and possibly the
data at the puncture. The vectors vE, differ from the vectors 'v appearing in (3.1.2)
by vectors v' regular at the puncture. They also may differ by irrelevant "Borel
vectors" i. Such vectors satisfy (Elb(3) = 0.
We now recall from Ref. [22] §5.2 that: dA1 A dA2 b(v\,)b(v\,2 ) D) = iR(2 )(p)I0),
where R (2) (p) is the curvature two-form associated to the conformal metric p used to
extract the coordinates for the ghost-dilaton insertion. Replacing this into (3.1.3) we
find,
fD(Kg,0)
K 3-39
j
d
b(,) ... b-b(Q'J 6g 6 )|0)
j/(
R(2)(p) ,
(3.1.4)
where we used the remarks below (3.1.2) to replace the vectors vc, by the vectors
v',. As emphasized earlier, the form d<(E Ib(vi,) . .. b(Q~ g- )j0) is independent of the
position and coordinates of the puncture placed on E. Therefore, the integral over
E in (3.1.4) can be readily evaluated to give - f, R (2 ) = 2 - 2g. Back in (3.1.4) we
find,
6
fD(CVg,o) = (2g - 2)
3- 3
6
0d (( Ib(vQ,) ..-. b(
6
Comparison with (3.1.2) immediately gives the desired result.
g-6 )10)
.
(3.1.5)
3.2
Off-shell amplitudes in tori
In this section, we use a representation of the punctured torus in terms of a sewn
three-punctured sphere to find the form of the Schiffer vectors which independently
generate modulus deformations and local coordinate changes. We then use these to
obtain explicit expressions for the canonical forms over the moduli space of tori and
in doing so set up for the formalism required to evaluate off-shell amplitudes in tori.
3.2.1
Once-punctured tori from three punctured spheres
Let (Ei; wlj and (E2; w2 1 denote the surface states corresponding to the punctured
Riemann surfaces E1 and E2. We single out a puncture on each surface, labelled
puncture one and puncture two, and having local coordinates wl and w2, respectively.
Let E(q) denote the surface produced by sewing together these surfaces with sewing
parameter q. The surface state corresponding to the sewn surface is given by,
(E(q)l
= ( 1 ; wi
|(E 2;
2
qL-)
q 0 R12)
R
,
_)E1
-2
Uq EI2 WW
1 2 = q .
(3.2.1)
Note that the second state space could equally well be used for the Virasoro operators.
This follows from the exchange symmetry of JR12 ).
Consider now the canonical two punctured sphere zi (z) = z, z2 (z) = 1/z, with an
additional puncture at z = 1 whose local coordinate w will be defined as,
w=
1
I In z.
(3.2.2)
This three punctured sphere will be denoted by R 123 , and the corresponding surface
state by (R 123 1. The sphere is described in the w-plane as the cylindrical region
determined by the identification: w - w + 1.
Suppose we now sew the coordinates z, and z 2 via the identification zIz 2 = q,
where q = exp(27rir). In terms of the z uniformizer this means identifying according
to z - z exp(27riT), which in w coordinates reads w -, w + T. It follows from the
identifications that we have obtained a torus with Teichmiiller parameter T. The
surface state (E(7); wI describing this once-punctured torus may therefore be written
as,
(E(T); WI = (R
123
1q l) q - 1(
R 12)
.
(3.2.3)
This is the once-punctured torus w ~ w + 1, w -, w + 7, with local coordinate w at
the puncture w = 0.
We now consider describing a general family of once-punctured tori. The local
coordinate at the puncture will depend on the modulus of the torus. Every torus will
be explicitly realized in the w-plane via the identifications w - w + 1 and w - w + T,
and the puncture will be at w = 0. As we change the modulus, the local coordinate at
this puncture is conveniently described by a function showing how the unit coordinate
disk I1<• 1 embeds into the torus,
1
w = h.,,(() = a(T, ) ( + -b(r,
2
t)
(2 + ...
(3.2.4)
Note that the function need not depend holomorphically on the modulus T-. The
above equation may be inverted to give the local coordinate as a function of the
uniformizer w of the torus, ( =
(7T,7, w). For brevity, we will write ( = & (w), and
the w dependence will be left implicit in some cases.
Following (3.2.3), the surface state for a torus of modulus T having coordinate &
at the puncture is given by,
( ();
q
=q-•o
- ( 23
(3.2.5)
|R12 ),
where the three punctured sphere RM 3 is the canonical two-punctured sphere R 12
equipped with an extra puncture at z = 1:
•
zl(z) = z,
R123 :
z 2 () =
/z,
3
(z)
=
(w(z)).
(3.2.6)
Here w is the coordinate defined in (3.2.2). The choice of coordinate around the third
puncture of the sphere ensures that the torus will have the desired local coordinate
ý (w) at the puncture.
3.2.2
Forms on the moduli space of punctured tori
Our aim here is to write the canonical string forms on the moduli space P•,1 of oncepunctured tori. The general case becomes clear once we write forms on the subspace
of P1,i parametrized by (3.2.4) which defines the local coordinate at the puncture as
a function of the modulus of the torus. This is the case because the most general
tangent to P 1 ,1 is a tangent that changes the modulus of the torus and adjusts the
local coordinate at the puncture.
We consider a torus of modulus T and write the one-form and two-forms as,
(OE ' (7=
([2]1,
(r)I =
1' (RC 1qL
N-K
3
- I (R
3
qq
q"LIR 12)
0
[dTb(l)
R 12 ) dr A d;
(
) +dTDb
b(3)
(3.2.7)
b
(3 ) (
where fN= -27ri is a normalization factor, and the surface state representing the once
punctured torus was built by sewing. As usual, the antighost insertions are located at
the free puncture. The vectors 0/07 and 0/d7 denote the Schiffer vectors representing
the deformation of the modulus together with the corresponding deformation of the
local coordinates. Since surface states for punctured spheres are well-known [25,37],
the construction of the above forms only requires the explicit expressions for the
Schiffer vectors for the family of tori described in (3.2.4), and the prescription to
build the corresponding antighost insertions.
Let us first consider the case where regardless of the modulus of the torus, the
local coordinate at w = 0 is always taken to be the coordinate w. In this case we say,
with a slight abuse of language, that the coordinate does not change as we change
the modulus. The Schiffer vectors must not deform the local coordinate at the free
puncture; in terms of the three-punctured sphere, the torus deformation only entails
a change of sewing parameter for the sewing of the first two punctures. It is therefore
convenient to place the antighost insertions on an interior puncture, say puncture
one. The two-form then reads,
([2]11
(
b~
- (R123 IqL )q)
) R12)d Adt .
)b()(
(3.2.8)
In this presentation, it is a standard calculation to show that,
b
= -2ribo,
b(
= 2ribo
0,
(3.2.9)
which can now be used to write,
([21,1
(R123 qL
bo)b)JR12) dr A d .
(3.2.10)
The Schiffer vectors on a torus. We can now discuss Schiffer vectors v(w) defined on
the neighborhood of the puncture in the once-punctured torus.' In the brief analysis
which follows, we shall consider the possible vectors according to their behavior near
the puncture at w = 0. Let us begin by considering those vectors fields which vanish
at the puncture,
v(w) = w" + O(w+') , n > 1.
(3.2.11)
Such objects do not extend holomorphically throughout the torus since no bounded
elliptic function exists. Since they extend all the way inwards to w = 0, where they
vanish, their effect is to change the local coordinate at the puncture. Corresponding
to (3.2.4), the explicit form of the Schiffer vectors that change the local coordinates
as T changes, can be read from Eqn.(6.10) of Ref. [25],
v,(w) = -
Oh
(A(w))
, vf(w) = -
Oh
((w)) .
(3.2.12)
These vectors are clearly of the type indicated in (3.2.11).
The constant vector v(w) = 1 is well defined throughout the torus and therefore
changes neither the modulus nor the local coordinate at the puncture. The case of a
vector with first order pole is more interesting:
1
v(w) = - + O(WO) .
(3.2.13)
This vector cannot be extended analytically throughout the torus as no elliptic function with a single first order pole exists. Such a vector field, which neither extends
inwards nor outwards, must change the modulus. We are especially interested in finding the vector field which changes the modulus without a corresponding change in
local coordinate at the puncture. To this end, it is useful to consider the logarithmic
derivative of the Jacobi theta function 01(wIT),
01(w -7) w1
'For a pedagogic discussion of general properties of Schiffer vectors and Schiffer variations see
Ref. [11].
where the prime denotes differentiation with respect to the first argument. (We follow
the conventions of Ref. [38]; note however that there q = e"'). This logarithmic
derivative has simple properties under translations,
u(w + ,rr I) = -2i + u(w I) .
u(w + 7r I()= u(w I) ,
(3.2.15)
For our purposes, it is convenient to choose the particular Schiffer vector,
vo(W) =
u(irw IT) ,
(3.2.16)
which, by virtue of (3.2.15) satisfies,
vo(w + 1) = vo(w),
vo(w + T) = 1 + vo(w).
(3.2.17)
As will be seen later, this choice of Schiffer vector is tailored precisely to change the
modulus without changing the local coordinate at the puncture (again, in the sense
that its dependence on the uniformizer is unaltered). Since 0-functions are entire, the
only pole of vo is at w = 0.
To conclude our analysis, consider vectors of the form,
v(w) =
±
+ O(w1-n),
Wn
n
2.
(3.2.18)
Vectors with such leading behavior can always be reduced to one of the cases already
discussed by subtracting a suitable linear combination of elliptic funtions. In particular the Weierstrass P function may be used to eliminate a second order pole, while
its derivatives can be used to eliminate higher order poles.
The Antighost Insertions. We now describe the antighost insertions corresponding to
the Schiffer vectors discussed above. We treat separately the contributions from vectors that change coordinates only and from vectors that only change moduli. Finally,
the results are combined to give the expression of a general form on the moduli space
of once-punctured tori.
The antighost insertions corresponding to the coordinate changing Schiffer vectors
in (3.2.12) are given by,
b(v,) =
b(v:) =f
b(w)v,(w) +
b(Qd)vt(w) ,
f2dw7ri b(w)v,(w) + f 2-xi bI()
(3.2.19)
v,(w) ,
where overbar on a vector denotes complex conjugation, and we integrate using
f dw/27riw = f df/2rif = 1. In these equations b(v,) and b(vf) are simply the
portions of b(ala7) and of b(al/i) representing the coordinate changes due to the
change in modulus. The explicit oscillator form of the antighost insertions follows
readily from the above equations once the Schiffer vectors in (3.2.12) are written as
power series in w. Note that the insertions are acting on the external puncture, but
use the w uniformizer. This is convenient when it comes to re-expressing them in
terms of the first and second state spaces, the spaces that are traced over. Since b
is primary, (3.2.19) is coordinate independent and the insertions can be expressed in
terms of oscillators b(W) by using the Schiffer vectors referred to the ( coordinates.
This may be useful since the external state is written in terms of such oscillators.
Let us now consider the antighost insertion corresponding to a change in modulus
only. The vector field we need is precisely the vector v,(w) introduced in (3.2.16). If
we define,
b(v)
27i b(w)v(w),
b()
27Tr b(
)o(W),
(3.2.20)
then, the correctness of our claim requires that,
6
(Rý 3 1qL q 1) 1R12) b(vo) - (R1
2(-2wi b )) qLq)1l1')lR
3
12)
.
(3.2.21)
This equation relates an antighost insertion on the puncture to an antighost operator
inside the trace and by virtue of (3.2.9) it justifies the assertion that the antighost
insertion b(vo) describes a change of modulus only, without affecting the local coordinate at the puncture. The proof of this identity is given in the appendix. In a similar
manner,
(RM j(1)•
qL
)
IR12 ) b(V) = (Ri 3f (2wi[• 1 )) q)q
(3.2.22)
R 12).
Our results are now complete. The antighost insertions representing both changes
of moduli and local coordinates are given as
b
( ) = b(v)
+ b(v,),
b
) = b(v) + b(vt).
(3.2.23)
These expressions can now be substituted back in (3.2.7) and give the explicit expressions for the canonical string forms on the moduli space of once-punctured tori.
3.3
The case of genus one
In this section, we apply the earlier results to prove the ghost-dilaton theorem at
genus one. According to [22], the theorem is proven if the following equation holds,
fD (,)-
fX
,3 ) =
K
d (S
1 ,0 +
1In p
(3.3.1)
.
The expression in parenthesis in the right hand side corresponds to the elementary
contribution to the one-loop free energy appearing in the string action. It is expected
to be independent of the string coupling n, so ideally one would hope that the left
hand side of the equation vanishes. It does. In what follows, we verify that in fact
both terms appearing in this left hand side vanish independently.
Preliminary remark. We shall consider first fx(A~i 0 ,3 ), with iX) = -co 10),
fx (AVo, 3 ) -=
dO
(d1123
bo(1)eioLo(1)
C(3)10)3
.R12)
(3.3.2)
The geometrical interpretation is that we take our choice of three-string vertex (denoted here by (V123 1), and twist-sew two of the punctures, inserting the state JX) at
the third puncture. It happens that certain simplifications occur in the special case
where the three-string vertex is chosen to be the Witten vertex (W123 1. This example
gives an insight into how the expression may be evaluated for the general case.
The (W 12 3 1 vertex satisfies the conservation law: (W 123 I(c + c + 3) ) = 0,
which allows us to replace the co (3) in (3.3.2) by the factor (co(1) + Co(2 ) ) immediately
to the right of the surface state. This factor annihilates the reflector (R 12 ), and all
that remains is the term picked up by anticommutation with bo(1) ,
fx(A0,
3
(3.3.3)
IR12) 10)3.
2r (W1lei'Lo
23
Let us consider the integrand of this expression. We have a three-punctured sphere
with two punctures sewn together, and a state (in this case the vacuum) inserted in
the last puncture. Such an object is a once-punctured torus of some modulus T(0).
The calculation of T as a function of 0 is in general nontrivial, but will not be necessary
for our arguments. Referring back to our discussion of §3.2.1, any once-punctured
torus with arbitrary modulus r can be made by sewing together the punctures of
R•• 3 with an appropriately chosen sewing parameter, and local coordinate
may therefore rewrite (3.3.3) as,
d O(R 3 q(1)-()
fA
(
=
I
JAVo, 3
e2" 7().
q
2
R12) 10)3
fa
=
dO
Vo, 3
2-
',.
We
L,()_Z(1)
12
Iq(Rq 0 IR12)
(3.3.4)
It becomes clear that we are actually dealing with the supertrace
where q =
of the operator qLo-Lo 2 .
The supertrace of any operator vanishes unless the operator explicitly contains the
factor bocobo
0 o . The necessity of the factor b0 co follows by considering the holomorphic
The
state space splits into two sectors, identical except for the fact that they
sector.
are built upon the vacua 10) and coj0) respectively. The different fermion numbers
of these two sectors means that their contributions to the supertrace cancel. It is
therefore necessary to project out one of these sectors while still conserving ghost
number, the only operator suitable for this being boco. Similarly, the antiholomorphic
sector requires the factor bo0 0 .
Since neither Lo nor L 0 contain ghost zero modes, the integrand in (3.3.4) vanishes
identically. Note that such simple arguments could not be made for (3.3.3). Upon
deletion of the third puncture, the leftover two-punctured sphere does not coincide
with R 12 and the absence of ghost zero modes is not manifest.
The general case. We now consider to the case of the general three-string vertex.
Since fx(AVo, 3 ) is simply an integral of the canonical string one-form along a oneparameter subspace of 71,1, the considerations of the previous section indicate that
it can be written as,
fx(AVo, 3 ) = JrV2One
,
(R
IqL
IR1 2) [drb(")(i
+ d-b(
-Z(-)8 (48 IAI,) -
can readily show that (R 12 JA( 1)IR12) =
str(A)
)
)3, (3.3.5)
Here the tori are built by gluing two punctures of the T-dependent R(l3 spheres, where
the coordinates at the third puncture are determined by the desired coordinates on
the punctured tori AVo,3 . The antighost insertions represent both changes of moduli,
and changes in the local coordinates as outlined in detail in the previous section (see
(3.2.23)).
An important property of the R¶ 3 sphere is that the local coordinates at punctures
one and two reach puncture three. This means that we can, by way of conformal
transformations, map any operator acting on the third state space to an operator
acting on the first or on the second state space. Such mapping couples neither leftmovers with right-movers nor ghosts with antighosts. Exploiting this fact to our
advantage, we use conformal mapping to transfer all oscillators in the third puncture
to the first puncture. Once again, the vacuum state deletes the special puncture of
(R~ I,3 and what remains is an expression of the form,
fx(AVO, 3)
, (R 12 1(dT(b (1) +
-
1))
+ da(b(1) + (1))) (c(1)
-
(1))q )0 o) ioR 12)
(3.3.6)
where each ghost or antighost term indicated schematically above, is in general a
sum of many modes. For our present purposes we only need to know the holomorphic
or antiholomorphic characted of a given term. This is displayed by the absence or
presence of an overbar. It is now quite clear that at most two of the required four
ghost zero modes may be present in any one term. The integrand therefore vanishes
identically.
We can now apply the same ideas to fD(V,1j),
L
=
1,)
12) (3)
b(3)
D)3
I-1,1
l
dT^A dT(R
R1()
12I(b(z) +
(b'))()
+ V(l))(c(1)c(1)
Z(1)(1)) ql,0q,
1R12)
(3.3.7)
where once again we used conformal mapping to move all oscillators to the first
puncture (recall ID) = (clc- 1 - clc_1)O0)). This time the expression vanishes due to
the absence of the combination coco. We have thereby shown that the left hand side
of (3.3.1) vanishes. This concludes our proof of the ghost-dilaton theorem for the field
independent terms in the string action.
3.4
Appendix
We consider here the proof of (3.4.8). The starting point is,
(R
f
(w)b(W)Vo(w) qLo qL0R
12 )
(3.4.1)
where the integral is around the point w = 0. Note that the antighost insertion is
written in terms of the w-coordinate at the puncture. This expression is viewed as
an integral on the sphere R; 3 . (See Fig. 3-1).
=00
Z
Izl
= Iql_
Iz21=
Iql1
Figure 3-1: One-punctured tori from three-punctured spheres. The torus of modulus
7 is built by sewing two punctures in the sphere R 123 .
The periodicity v,(w) = vo(w + 1) in (3.2.17) implies that the vector field v, is well
defined on the annulus bounded by the curves C1 and C2 , corresponding to Izl I = Iq|1/2
2
and 1z
2 j = 1q11/ , respectively. Since v, is analytic on the annulus minus the puncture,
it follows by contour deformation that the integral around w = 0 is equal to the sum
of two integrals, one over C1 and the other over C2:
d
w
b(w)v
oW
c2 2j
(R6b(w)v(w) -
c 2ri[
i
q
(1)
1
o 0) IR12) .
(3.4.2)
Consider the antighost field b(w) in the second integral and express it in terms of z2
coordinates,
b(w) ... 4 q)
)1
12
=
dz2
(dw
2b(z2)
. .q-" o4-
R12 ).
(3.4.3)
The antighost, living in the second state space, can be taken all the way to the
reflector IR12) at which point it can be re-expressed in terms of the coordinate zl,
b(w)
q
..-
o R12
( dz' 2. . qL,
)b(zi)IR12) .
(3.4.4)
In order to bring the antighost back to the left we use qLob(z) = q2 b(qz),
b(w)Q
.(1L)
(
0 q IR•)12
=
()
2
w q2b(qzl) ...
L(1) -rM
q 0q
0 R 12 ) ,
and expressing the antighost back in w coordinates we obtain,
dzl 22q d w 2
L (1)()
po
q
o l2R1)
b(w
+
r)
...
q
dw
dzl qzl
(3.4.5)
(3.4.6)
The factor in front of the antighost field is precisely unity. Returning to (3.4.2), we
have,
(R1 3I
dw
2 .b(w)vo(w)
27
Z'
dw
db(w
1C2 27rz
-
)2
+ T)vo(w)1 q~otL(1)
0 - o0 1RR12).
I
(3.4.7)
Using v,(w + 7) = 1 + v,(w), and a change of variable to move the contour from C2
to -C 1 ,
dw
SdW b(wT+)v,(w) =
c25i
dw
2ri b(_(w+r) [v(w+r)-1] =
2c
-
dw
2ri b(w) [vo(w)-1].
(3.4.8)
The terms containing vo(w) in (3.4.9) then cancel, leaving only,
(R'
3
-
o q-o
(i IR12).
12)
db(w) q-L(1)
cJ 27ri J
(3.4.9)
This integral is evaluated in the zl coordinate and gives simply (27ri)bo1 ) . We therefore
have
1)
T
(R'1'23 q-LMO
o q,0o1
R12)
b(w)vo(w)
co•Sdw
2 7i
This was the desired result.
(-2i
= (R"l
12
b(1)) 0 qL(1)0 (1)JR )
12
(3.4.10)
Chapter 4
Consistency of quantum
background independence
In this chapter we will generalise the work [23] of Zwiebach from the classical to the
quantum case, showing that uniqueness and background independence of the master
action to all orders implies a set of cohomology vanishing theorems for the closed
string action, and postulated the existence of all punctured higher genus interpolating
moduli spaces BWn of positive dimension.
4.1
Review and notation
We will need to make use of the properties of connections on the space of conformal
field theories which have been outlined in §2.2.1 and need not be repeated here. We
will also make use of the string vertices V =
hi9 Vg,n (where the sum is over n > 3
for genus zero, n > 1 for genus one, and n > 0 for all higher genera) satisfy the
recursion relations (2.1.77),
V + hA
+
1
2
{, V} = 0.(4.1.1)
The string action may then be written (Eqn.(3.35) of [15]),
S = Q + hSj,o + f (V),
(4.1.2)
which satisfies the B-V master equation (2.1.47),
1{S, S} + hs = 0.
(4.1.3)
The interpolating 'B-spaces' have been introduced (using various notations) in the
papers [15,22-24,26], and are responsible for first and higher order infinitesimal background deformations of the theory via the canonical transformations implemented by
their associated Hamiltonians. They also play an important r6le in the formulation
of string field theory around non-conformal backgrounds.
We will define Bf, as the moduli space of decorated Riemann surfaces of genus g
with n ordinary punctures (each surrounded by a coordinate disk) and with T special
punctures. The B-spaces are symmetrised with respect to labellings of the ordinary
punctures and antisymmetrised with respect to labellings of the special punctures,
and we recall that the space B ,n has dimension 6g - 6 + 2n + 3M. The spaces Bo
with no special punctures will be identified with the usual string vertices V 9,,. In the
papers [15, 22, 26) the B-spaces having a single special puncture with n > 2 at genus
zero and n > 1 at higher genus were introduced, being responsible for implementing
first order background deformations. In [23, 24], higher order deformations of the
classical theory were considered and it was found necessary to extend this complex
to include the spaces f, for all n > 1 and A > 2 which implemented them.
Like the string vertices V, the B-spaces also satisfy recursion relations, and the
ones which will be needed for the purposes of this chapter are those (2.2.69) satisfied
by the spaces B',
B = (K - Z)V + V,0 + hAB3,2 + hV1,,,,
where B31
g, B1,, and Jv
0 + hA + {,
(4.1.4)
-}. There is no space B1• so the
terms involving objects of genus one and with one special puncture but no ordinary
punctures have been extracted by hand to leave an equation which holds for all (g, n).
Other useful formulae relating the various operators acting on moduli spaces and
functions are collected in the Appendix for reference.
4.2
Background independence revisited
In this section, we review the derivation of the condition for quantum background
independence [15]. This was originally carried out for the case where the B-V density
p, being dependent only on x, was a section on the theory space bundle. Here we shall
derive the background independence conditions for the more general case where p =
p(x, x) is allowed some field-dependence and hence is promoted to a function on the
1i d4 i p e 2S / ,
bundle. Given that the invariant B-V measure takes the form dis H
there should be no problem in transferring some of the field-dependence of the action
to the B-V density, and we verify this by analysing the transformation properties of
our background independence condition under field-redefinitions. Nevertheless, since
it is always possible to choose a frame in which p is field-independent, we choose
for simplicity to restrict our analysis to this case. After briefly reviewing the origin
of general symplectic connections [15], we write the explicit form of the Hamiltonian
B,(F) implementing background deformations for general symplectic connections, and
show that background independence to first order amounts to a 'vanishing' theorem
for As cohomology classes.
4.2.1
Field-independence of the density p
Having chosen field/antifield coordinates (Vi) on the symplectic manifold we can
define the string field 'IT) as usual by,
(4.2.1)
1 F>1•.
where the superscript x denotes the theory space dependence. We will make x implicit
in the following. Let us investigate the effect of allowing the density p = p(0, x) to
have an explicit dependence on the field/antifield coordinates (though we are always
able to choose coordinates such that the field-dependence vanishes). In the fieldindependent case, p drops out of the expression for the delta operation, but this
explicit p-dependence is reinstated on allowing the field-dependence, which may be
determined as follows,
1 (-In p) w"-ajF
AF =2pH)0(pw"
F)
1=()
+ a,--7--F))
(4.2.2)
= t{lnp, F} + AF,
where we have defined the p-independent hatted delta operation by,
.
-=S(
12)6,
F+1
(4.2.3)
and have used the notation,
ajIX)
= (C/I
(4.2.4)
.
We note that A, unlike A, is in general neither a nilpotent nor a scalar operator.
We may now reconsider the consistency condition for local quantum background
independence. This condition (2.2.48) was derived for the case of the canonical con-
nection F, of [39], and reads,
D (r)S = hAB, + {S, B,i + 2h(str r, -
, In p).
(4.2.5)
We would like to derive the general form where p = p(O, x) has a field/antifield
coordinate dependence. Given the measures dti = p(x, ,) IidJ d
and d/, =
p(y, y) fi do', we require the existence of a symplectic diffeomorphism F*,x such
that dlue 2 Sx /h = Fy*,(dp•e 2 Sy/h). Following Eqn.(4.9) of [15] we know that,
p(,OX7
) aPj.
Fy*,x(dpy) = p(,
sdet
-[
dy .
(4.2.6)
so that the background independence condition becomes,
exp
exp (2S(=x,X)
2S(PY),y)
) sdet [,b
p
(4.2.7)
.
If we consider infinitesimal diffeomorphisms, y = x + Jx, we may use (2.2.44),
z+', = F i(j=, x,x + 6x) =
~$ +
6x-'f (,,x) + 0(x 2),
(4.2.8)
to obtain,
y)
p(4V,YY)1
P(+,1 X
P(XI, X)
p(4'OOx)
1
_
, z) 5x" + P(
ap()
o)
P(_X)
p,0, '1 f. 6x .
(4.2.9)
So (2.2.45) is modified to,
as(S,XX)
+
rS(,x )
1
f + h
In p
,Inp f
+
f(4.2.10)
+ str
(4.2.10)
If we now separate from f the term proportional to the connection as in (2.2.46),
(4.2.11)
-F .- Bz,
and note that the condition that F' be a symplectic map reduces to the condition
(2.2.47) that there exists an odd Hamiltonian B, such that,
(4.2.12)
SB' atB
then it is clear that the new O-dependence results in a shift of the p-dependent term
of (4.2.5) as follows,
a, ln p
-+
O,lnp - (Inp)ýji
-+ Inp- (Inp)
(Inp)wiJj3•Bp
-
,I) - {Inp,B}
(4.2.13)
p)- {Inp, B} .
-+ D,(r) (In
where we have made use of Eqns.(4.11), (4.13) and (4.15) of [15]. The resulting
background independence condition is,
D,(F)(S+
1hln p)= hAB
1
+ {S + -hln p, B,} + hA
,
(4.2.14)
having used the fact that ½hstrF, = hAFLA, (where
_ - (12
P
1 2)While the form of the consistency conditions given in (4.2.14) is useful for demonstrating the relationship between p and the action, it is, for our purposes, also convenient
to write it inthe form,
D,(r)S = hAB, + {S, B,} + hAF, - -hD,()
S2
- -hD,,(r) Inp.
= AsB,(r) + hAr/,
Inp
(4.2.15)
where we have introduced As- = hAdps . = _{S, -}{S+llnp,}+
} + hA -- {S + 2n
- ~ hA".. .
In B-V quantisation the action is taken to transform as a scalar under string fieldredefinitions. It should therefore be possible to extract any scalar (and in general
field-dependent) component from the action and absorb it into the B-V density as an
additional factor. This procedure should leave the transformation properties of both
the action and p unchanged, so that the requirement that B-V measure dPLs be an
invariant remains satisfied. By analysing the transformation properties of (4.2.15) we
will now verify that it is indeed consistent for p to have such field-dependence.
Since the action is a scalar the LHS of (4.2.15), as well as the first two terms on
the RHS (in the first line of the equation) transform as scalars on the bundle. If the
equation is to be consistent, we must require that the remaining terms also transform
correctly. Let us see whether this is true.
Under a change of basis, 14i) -+ IJi)Nj(x), where Ný is an invertible matrix, we
i and a, are also invariants,
must have 'ii _+ (N-1')'j. We know also that Dj4i)0
and these facts allow us to derive the transformation properties
4i -+ F"' of the
connection (see [40]),
D'41(Ii)Vi
=
ooaoo
1i
+
,pii
p3
-j tD)NO,(N-1i) + Ij)NFAN-10
= kI)a,0 + Ij@)NaO,N-I
+ jD)NFIN-'N
(4.2.16)
,
where we have adopted matrix notation for brevity. The invariance of D,|• i) i then
implies that with our conventions,
r'i = (N- 1 ) Fk N' - (aN-1)i Nk.
(4.2.17)
Then the function F, = as,
IT ) 1 1 )2 associated to the connection transforms
(w12
F
-+
F,. - N(&o.N- 1 ).
(4.2.18)
The invariance of dps and the action S implies that the density must transform
according to,
p --+ p sdet
Also W --+
i
= psdet (N - 1) = pe- str InN-
N and therefore,
D, In p -2AF, = a, ln p - In pVF'
-
(4.2.19)
- 2AF,
,,lnp - O,,str In (N-') - In p-NN 1'F,NN'•0 + In pa-N(og,NN-1)
-2AF, + str (o,N-1)N - In p+ N(9,N-1)O
=
,, In p - In p
FQj
j
- 2AF,.
(4.2.20)
This verifies that the last two terms on the RHS of (4.2.15) transform as scalars, and
we conclude that it is indeed consistent to allow p some field-dependence. Nevertheless, given that frames always exist in which p may be chosen to be field-independent,
we will for simplicity restrict ourselves to this case in what follows. Choosing p to
be field-independent means that we may carry over directly the background independence condition (4.2.5), and also set A = A.
4.2.2
Origin of general symplectic connections
Until now, we have employed the canonical connection [F. There is no real reason
for restricting ourselves to this particular choice of connection and for the sake of
generality we would like to express the background independence condition in terms
of more general symplectic connections.
In order to understand how different connections are related to each other, we
will review §6.2 of [15] which demonstrates how a particular choice of connection is
related to a choice of three string vertex.
We are interested in the coupling constant-independent O(h) terms in the background independence consistency condition. For the case of field-independent p, the
O(h) condition is just Eqn.(6.2) of [15],
S,o = -0
1&
,ilnp+ A
+ f(AB, 2) + f (ZI,1) .
(4.2.21)
Now B, 2 interpolates from "V0,3 , which is the three string vertex with one special
puncture, to the auxiliary string vertex V•, 3, so the above equation seems to involve
singular tori AVX,3. The way this is avoided is to introduce a new vertex V0,3 with
one special puncture and two ordinary punctures such that AVo, 3 is not singular. We
can now introduce a new space B ,2 which interpolates between V, 3 and the vertex
V0, 3
and use this to define a new symplectic connection Fr (Vo,3) by,
(w12 1rl)
=
(wu
12 1r)±+f/(Q(1)0,l31O)t
(4.2.22)
Absorbing these changes into the canonical connection and LB,2, we make the replacement in (4.2.21),
=-/LF,,-+
AL. +/
/F~±fl(A/o, 2 )=
(a(°)l'lIO)
f,(A o,2 + AB1,2 ),
(4.2.23)
The integral is actually path independent, which allows us to define a moduli space
B0, 2 F) satisfying f,,(AB, 2 ()(r) = f ,(ABO, 2 + AB, 2 ) interpolating from 1V0,3 to Vo,3 in
such a way that it completely avoids the vertex V0,1,so that the resulting expression
avoids any singularities.
We see then that the supertrace of a choice of symplectic connection F, is determined by a choice of three-string vertex V0, 3 . Alternatively, a choice of symplectic
connection determines V)0,3 which in turn allows us to choose Ba,2 ().
4.2.3
Background independence and As-cohomology
Having recalled in some detail the origin of generalised connections, we may now
return to our covariant analysis of background independence in terms of arbitrary
connections. As noted in [23], the connection F, is a reference symplectic connection
and can be shifted as long as we preserve the symplectic nature of the connection.
Writing the canonical connection in terms of another symplectic connection, F =
F, - 6F, we find,
D, (F)S = D,(F + 6F)S + {S, 6FI,},
(4.2.24)
AF, = A(F, - j6,) ,
(4.2.25)
with a, ln p invariant. Rearranging terms, we may write the condition (4.2.5) for
background independence in terms of a general symplectic connection as,
D,.(r)S = AsB,(F) + hAF. -
1
hO, In p,
(4.2.26)
where the Hamiltonian B,(F) for deformations via general connections is,
B.(rF) = B,,(r) - 6r, = BP,(F) - (F, - ,,.).
(4.2.27)
Clearly B,(F) + F, is invariant under shifts of the connection, so this final expression
actually holds with respect to any reference connection F,, which may replace the
canonical connection P,. It follows from (4.2.26) that,
As(D,(r)S) = 0o,
(4.2.28)
since both AF, and 0, In p are field-independent. If we now require uniqueness of the
master action, we can use the condition (4.2.28) to derive a cohomology theorem as
follows.
Suppose we have a master action S satisfying the master equation. If we now
perturb this action slightly by APD,S, the new action will also satisfy the master
equation,
S{S + A"DS, S + A"D,S} + hA(S + APD,S) = A,"As(DS) = 0.
(4.2.29)
We already know from (4.2.28) that these these marginal deformations are As-closed.
The discussion in §6 of [15] still applies here, so that with an appropriate choice of
basis we have str F, = 0, In p, leaving the simplified condition,
D,.(r)S = AsB,,
(4.2.30)
which tells us that the marginal deformations are also exact. We are left to conclude
that the requirement of uniqueness of the master action reduces to a cohomology
theorem for the master action in that D,S, which being As-closed, must also be Asexact. We will come back to this point in more detail in §4.4. Let us now examine
the commutator of deformations.
4.3
The commutator of background deformations
In this section we will take a second covariant derivative of (4.2.26) and demonstrate
the existence of a As-closed 'field strength' H,,, which in its turn will imply the
existence of a Hamiltonian B,, from uniqueness of the master action.
4.3.1
The commutator conditions
We shall begin with (4.2.26) which expresses the background independence condition
in terms of a general symplectic connection,
D,(r)S = AsB,(r)+ hAF,. -
h0a, Inp.
(4.3.1)
Before proceeding, we derive the useful identity [D,, A]F = 0 (for arbitrary functions
F) which holds if the connection is symplectic. From their definitions we have,
AF
(-)
(4.3.2)
= OF - r•L. F,
D,F -8,F - F•r
F) =
(w
(-)J+
ij
F.
(4.3.3)
The commutator is calculated thus,
[D,, A]F = 9,( (-)i+ijwi -F)
-
k (()i+i wi•
Fk
• ()i+(,F -
'+(-,w' F
+-(-)
,FkdI k F)
)kI kl j
= (-)i+iq
19 F + ()j+kk
P kWkj
l(-)i+ij
F)
F -I
94F +
ij+i+kpkiiF
o9;,F
.
(-)i+l)(j+1)rjWki)
)i
)
0
(4.3.4)
as the vanishing of the expression in brackets is precisely the condition for the connection to be symplectic. (Note that the last two terms in the brackets are actually
equal, though we have chosen to separate the terms as above in order to make the
symplectic identity explicit).
Another result we will need is the following,
AR, =
=
I( ) p,
(0i 2-y(or
i-
+
+0
= alr,'- Oar.
,)
, rF
3 ,
(4.3.5)
where we have have used the fact that AF, is field-independent.
Additionally, the field-independence of p allows us to ignore its mixed covariant derivatives,
[D,,D,] In p = [O,, a,] In p = 0.
(4.3.6)
We are now ready to take a second covariant derivative of (4.3.1). Making the connection I, implicit we have,
[D,, D,]S = DlD,S - D,DS
= DAsB, - D,AsB, + h(DAF1, - Dv,AF,)
= D,{S, B,} - D,{S, B.} + hi(D,,AB, - DvAB,) + hAR.,,
= {DS, B,} - {D,S, B} + {S, D,B, - D',BS}
+hA(DBv, - D',B,) + h([DM, A]B, - [D,, A]B,) + hARt,.
(4.3.7)
where we have made use of (4.3.5) and (4.3.6). The result proven above allows us to
discard the commutators [D,, A] so that,
[D,, D,]S = {{S, B,} + hAB,, B,} - {{S, B,} + hiAB,, B,}
+As(DB,, - D,,B,) +
hAR,,
= {{S, B,},B,} - {{S, B,}, Bt,} + h{AB,,
+As(DB, - Dv,B, ) +
= {S, {B,, B,}}} + hA{B,,
B,} - h{AB,, B,
hAR1,,
(48)
(4.3.8)
B,,} + As(D,,B, - D,,B,) + hAR1,,
= As({B,., B,1 } + DB, - D,,B1 ) + hAR,,.
But we know that the action of the commutator is related to the antibracket with the
curvature,
[D,,D,,]S = -{S, R,}}(4.3.9)
= - AsR,, +hAR,,.
Combining (6.1.3) and (4.3.9) we find the consistency condition,
AsH/,, = 0,
(4.3.10)
where we have introduced the field strength,
H,=,
_ {B,, B,} + DB,, - D,B, + Rn,,.
(4.3.11)
Of course this equation must be satisfied since the condition results directly from
(4.2.26), which has already been solved explicitly. Eqn.(4.3.10) implies that a per-
turbed master action S' = S + Al"H,, also satisfies the master equation,
1
S {S', S'} + hAS'
2
=
1
{S + A"H,,, S + AX"H,,} + hA(S + A"vH,,)
2
= {S, S} + hAS + {S, A"H,,v} + hAA""H,~
AW{
2 S,H.} + h""AHg
(4.3.12)
= A" AsH,,
=0.
The hope that the master action be unique up to gauge transformations would require
that the perturbed action be merely a field redefined version of the original. This is
so if there exists a Hamiltonian B,, such that,
H,, = AsB1~.
(4.3.13)
So we see that the existence of B,, or alternatively, uniqueness of the string action,
implies a (higher) cohomology theorem for the string action which in turn implies
quantum background independence of the string action with respect to commutators
of deformations.
4.3.2
Analysis of gauge freedom of B,,
Having postulated the existence of the object B,,, let us now explore the extent to
which it is uniquely defined.
It was shown for the classical case in [23] that shifting the connection whilst
retaining the symplectic property does not alter H,. An identical argument which
we need not repeat here shows that this statement also holds in the quantum case,
so that B1, does not depend on the particular choice of symplectic connection.
From the nilpotency of As, any shift of B, by a As-trivial object will clearly also
satisfy the background independence condition (4.2.26),
Bi --+ Bp + As8 A, .
(4.3.14)
This results in a corresponding shift in H1, given by,
H/L
-
Hu + {AsA,,
-4H,, + As{AI,
B)} + {B,, AsAv} + D,(AsA,) - D,(AsAl)
BU} + {A, AsBv} + As{Bi,, Av} - {ASBI, A}
+D,({S, A,} + hAA,) - Do({S, A,} + hAA,)
-- H,, + As({B,, A)} - {Bv, AI}) - {AsB,, ) } + {AsB,, AM}
+h(D,,AAn
- DoAA, ) + {DS, A,} + {S, DA,} - {DS, A,} - {S, D,A/} .
(4.3.15)
We can now use (4.3.1) and the fact that D, and A commute,
Hpu -+ Ht + As({Bj,, )A}- {BV,, A,}) + {S, DIA,- - DAu} + hA(DA, - DvAM)
+{AsBt, A,} - {ASB,, Au} - {AsBu, A,} + {AsB,, A,}
-+ H,, + As({B,, Av} - {B,, A/,} + DA, - DA,,)
-Ht
+ As(DMA
- DvAt)
(4.3.16)
where DA,
{B, }} + D, is the 'gauge covariant derivative' introduced in [23]. Given
that we seek B,,~ such that Hj, = AsB,,, the shift B, -- B,+/AsAJ, must correspond
to a non-trivial gauge freedom,
BO
-+ B~, + DA - DvAo,.
There is also of course a trivial gauge freedom under B,, -+ B,, + AsA
8 ,.
give a detailed interpretation of these in the sequel.
4.3.3
(4.3.17)
We will
Consistency conditions and recursion relations for moduli spaces
We will now examine explicitly the consistency conditions derived in §4.3.1,
DB, - DB, + {B,, B,} + R1,
= AsB,,.
(4.3.18)
The aim of the present section is to show that the Hamiltonian B,, is the function
associated to some moduli spaces of Riemann surfaces of any genus and with two
special punctures. We will follow the analysis of [23] to derive the recursion relations
which must be satisfied by the higher genus moduli spaces that define the Hamiltonian
B,,,. We therefore take B,, to be a Hamiltonian of the form,
B,, = -fy,(B
2)
0121
l )O)l),
O
= - fZ2(
(4.3.19)
where B 2 is a sum of moduli spaces of surfaces with two special punctures, now
extended to include higher genus terms.
In terms of moduli spaces the right hand side of (4.3.18) may be written,
AsB, , = -{S, f,.(B 2)}
-
-
hAf,2(B
2)
-{Q + f(V) + hS 1, o, ft, (L32)} + hf2,.(AB 2 )
= f,(oB2 + {,
2} + hAB 2)
(4.3.20)
= f,y((6B 2).
Recalling (from §3.2 of [23]) that the left hand side of (4.3.18) is independent of the
connection, we may simply apply the results derived in §6 of [23] for the canonical
connection (noting of course that B1 now includes the spaces of higher genus) which
tells us that,
D,B, - D,B, + {B,, B,} + R,1 = f,,(~,o + (K - -)B13'
-
1{B',
B1}).
(4.3.21)
Putting these together, (5.4.7) becomes,
fv(7o0, + (K:
-
1)1
(4.3.22)
BB'}) = f,(01VB2).
-2
This equation will be satisfied if,
v132
=
7, + (KI- 1Z)B1
{B' , 8 } .
-
(4.3.23)
This has the same form as the classical formula, except that 6 v now contains in its
definition the additional operator hA.
Let us verify the consistency of (4.3.23) by checking that 6 v acting on the RHS
vanishes. Consider each term separately we find,
bv~1= a7,
6v
+ {V, ',-
+ A2%1 = I
, + {V,
1} ,
(4.3.24)
(KC-IZ)B1 = [6v,KIC- 2]B1 + (K1 - Z)6vB3'
= {VO, 3 + (K - z)v, B'} + (K - z)(v;,3 + (K - z)v + n
= {VO, 3 + (K - Z)V, 1' -
1(-2{B1 B'1}) = -{SvB
VO,3 - {V, 70,} + hcal3
,}
=
-{V,
3
+(K
-
2
,2 + h
+ hAiV,21,1
I)V, 3}
1,1)
(4.3.25)
(4.3.26)
We remind ourselves from the discussion of §4.2.3 that 13B,interpolates between 1V 0,3
and some vertex Vo, 3 determined by the choice of connection, so that there is no longer
the unwanted singularity associated with AVV, 3 . So (4.3.23) is consistent if,
K(AB1, 2 -+
IV,1) = 0 .
(4.3.27)
Both terms in this expression consists of the operator K acting on a torus or tori with
a single special puncture. It is fairly simple to see why each term must vanish.
Consider a torus Tio with a single special puncture. The operator K: adds another
special puncture over the remainder of the surface of the torus, antisymmetrising with
respect to the two punctures. The translational symmetry of the torus means that for
any relative position of the two punctures of any torus in ITT1,o there will be another
torus with the two punctures with positions reversed. The antisymmetrising property
of K ensures that this pair of twice-punctured tori will occur with the opposite sign
and cancel. This pairwise cancellation means that each term in (4.3.27) vanishes,
thus verifying the consistency of (4.3.23) and thereby (4.3.19).
4.4
As-Cohomology
ometry
classes and theory space ge-
Having introduced B-spaces with two special punctures in the previous section, we will
outline in this section how uniqueness of the master action implies the existence of Bspaces with more than two punctures. We will show that the quantum generalisation
of the analysis [24] has an efficient description in terms of differential forms on the
theory space manifold which is parametrised by the marginal operators.
Let us consider a basis of marginal states, {1O1), ... , ION)}, where N may be
infinite. For a given point S in the theory space manifold, these form a basis of the
tangent space T,Ms. This N-dimensional theory space manifold Ms may therefore be
parametrised locally by coordinates {xl,... ,xN}. We will use the following notation
for differential forms on T*Ms,
1
(4.4.1)
A ... A dx" .
A-A,,..dx.,,Ad
A(,)
In this language the B, are components of a one-form 'gauge field' B(1) = Bdx ' ,
and the H,, are components of a two-form field strength H(2 ) = 1HAdx"
A dxz such
that H( 2) = /AsB( 2) = AsBA,dxA A dxz.
It is convenient also to introduce a kind of 'gauge-covariant exterior derivative' D =
D+ {B, -} (not to be confused with the actual exterior derivative d on T*Ms), which
is defined by,
DA(n) - DA(n) + {B(1), A(n)}
1
= •-DoAV-...,dxAo A ... A dxn"
1
- (DAooAA1...
An+
{ Bo0, A, ... , })dx'O
(4.4.2)
AA... A dx".
If we define the antibracket of two forms by,
{A(n), C(m)} = n!m! {A,,..,
C,+,',...An+m}d&
I
A-... A dx" Ad
1 A... A dx
""
+dx
+m,
(4.4.3)
then D has the property,
Df{A(n), C(m)} = { A(), C(m)I} + (-)n{A(n), DC(m)}I.
(4.4.4)
As a useful identity, we show that D commutes with As,
[As, 7D]A(n) = A(DA(n) + {B(1), A(N)}) + (S, DA(n) + {B(1), A()}} = DAA(A)
)DAsA(n)
+ {AB(1), A(n)} + {B(i), AA(,)} + D{S, A(0)}
-(DS, A(n)} + {B(1), (S, A(n)}} + {{S, B(1)}, A(n)} - V)AsA(n)
= VAsA(n) - {DS, A(n)} + {AsB(1), A(n)} - DAsA(W)
=0.
(4.4.5)
Another useful property is the following,
1
DDAA(,)~ = -D),
=
2n!
,
bA,,·...Adx
[Dp,,
S21 {H
2n!
"
A dx
b A dx" '
A ... A dx"
,76b]Al...,n dxza A dxzb A dx"' A ... A dx An
(4.4.6)
a7b,
Al,... n}dx~a A dxAb A dx•M A ... A dZlX,
= {H(2), A(n)} ,
where we have made use of the identity [•,
TD,] .
= {H,,, • } (note the sign correction
to Refs. [23] and [24]).
Let us now proceed to show how to recursively construct in a simple manner the
n-form field strengths H(n) and gauge fields B(n) for all n > 2. Treating the modified
action S -S + 1 h ln p as a zero-form and BA and I, as components of one-forms, we
may write the background independence conditions (4.2.26) as,
VS = A(B(1) + r(l)).
(4.4.7)
Acting once again with the gauge-covariant exterior derivative,
DDS = DA(B, + r,)dxl A dx"(4.4.8)
= AH(2) -
(4.4.8)
But we know from (4.4.6) that DDS = {H( 2), S}, from which immediately follows
the result we derived earlier (now written in terms of forms),
AsH(2 ) = 0.
(4.4.9)
Now, by the same argument which was used in (4.2.29), we know that we can add
to the action any As-closed function to get a new action also satisfying the master
equation. The hope that the master action be unique implies no non-trivial Ascohomology, which leads us naturally to the requirement that H( 2 ) be As-exact,
H( 2) = AsB( 2 )
(4.4.10)
We have already shown that the Hamiltonian B,, may be obtained from moduli
spaces of surfaces with two special punctures.
The procedure to construct higher forms goes as follows. Let us define an auxiliary
three-form,
H 3) -=DB( 2 ).
(4.4.11)
Acting on this with As, we find,
AsH( 3) = AsDB(2) = )AsB(2) = DH( 2) = 0,
(4.4.12)
where we have used (4.4.5), (4.4.10) and finally the Bianchi identity for H,,,, Eqn.(2.26)
of [24]. This means we can simply choose our As-closed three-form field strength to
be,
H(3) - H=3 ) = DB(2 ),
(4.4.13)
which is a condensed way of expressing the analogous classical result Eqn.(2.27) of [24].
Once again, uniqueness of the master action requires the existence of a corresponding
three-form gauge field such that,
H(3 ) = AsB(3 ).
(4.4.14)
Finding the four-form field strength is still simple, albeit not quite as trivial. We first
define an auxiliary four-form,
H(4) =
(4.4.15)
)B(3 ).
Acting with As gives a long chain of identities,
AsH' 4) = AsDB(3 ) = DAsB( 3) = DH 3) =
= {H( 2 ), B( 2 )}= {AsB(2),B(2)=
2)
B( 2)(4.4.16)
B 2 ) ( ,B(2)(
From this we can extract the As-closed four-form field strength,
H(4) - DB 3) - ~(B(2 ), B(2 ).
(4.4.17)
Note that this simplified expression agrees with Eqn.(2.33) of [24]. By repeating
and
-DB(p,1),
the same procedure that is, defining an auxiliary p-form by H) =
then acting upon it with As to eventually extract a As-closed p-form H(p), we may
construct all higher n-forms H(,) = AsB(.) ad nauseam. We will refer to this last
equality as the n-th vanishing theorem for As cohomology classes. Indeed one can
shown by induction that the general formula is,
n-2
H() = DB(n-1) +
(-)m+1{B(m), B(n-m)}
(n > 2).
(4.4.18)
m=2
Generalising the results which have already been demonstrated for B1 and B ,,
we can assume that the antisymmetric coefficients of the n-form Hamiltonians B(n)
are given by functions (with the appropriate marginal state insertions), of moduli
spaces with n special punctures,
gIn)f
(f·..ft,
1!.•
-
.
(4.4.19)
We note that B n = Eg,k>0 gn,k extends over a complete set of positive-dimensional
moduli spaces of punctured Riemann surfaces for all genera, and all numbers of
ordinary punctures compatible with the dimensionality requirement.
Just as before, these Ba-spaces may be explicitly constructed using their recursion
relations, but we should defer this task until Chapter ch:geomact, when we have before
us the complete B-complex and corresponding recursion relations.
In summary, what he have shown is that the requirement of unique physics implies
the need for background independence. The statement of uniqueness and background
independence at the p-th order of deformations implies the p-th vanishing theorem
for the As cohomology class of the master action, so that uniqueness and background
independence to all orders implies a set of As cohomology vanishing theorems for the
master action S. In addition, the space of equivalent theories related by marginal
deformations is simply the equivalence class (contained in the manifold Ms) of the
As-cohomology of which the action S is a representative.
4.5
Appendix
We collect here for reference some useful formulae used in this chapter. We assume
that the states 10I)
are BRST-closed.
{A, B} = -(-_)(tA
+i)(B+
{n+1)(B,
A},
(4.5.1)
(_)(A+1iA+1)(c+c+l){{{A, B}, C} + cyclic = 0,
(4.5.2)
(-)(A+fa+1)(c+nc+1l){A, {B, C}} + cyclic = 0,
(4.5.3)
A{A, B} = {AA, B} + ()A+iiA+lA AB} ,
(4.5.4)
A = 0,
(4.5.5)
As{A, B} = {AsA, B} + (-)A+A4+1{A, AsB} ,
(4.5.6)
D,{A, B} = {DA, 1} + {A , DB},
(4.5.7)
f,u({V,A}) = -{f(V), fv (A)},
(4.5.8)
6 vSvA
= 0,
[6 v, IC]A = (-)A+fa{(~V,
(4.5.9)
+ KY, A},
[6v,Z1]A = (_)A+tA {jy, A}.
(4.5.10)
(4.5.11)
ZKA = 0,
(4.5.12)
(ICZ + ZIC)A = {A, 7 } .
(4.5.13)
ZZA = 0.
(4.5.14)
aT1 =
0v',3 ,
(4.5.15)
Chapter 5
String vertices and inner
derivations
In this chapter, we shall introduce a new moduli space B°,2 associated with the BRST
Hamiltonian Q, and use it to attempt to identify the action of the operators a, IC and
I with the antibracket sewing of elements of the B-V algebra of string vertices. The
fact that we succeed shows that these operators are expressible as inner derivations
if the spaces B30, 2 and Bo", satisfy certain unusual properties. We then show that
the recursional relations for the B-spaces takes the form of a master equation for the
B-spaces. Finally, we show that the B-V delta operation cannot be expressed as an
inner derivation of the B-V algebra. Unless explicitly stated otherwise, we will use
units in which h = K = 1 throughout.
5.1
The BRST Hamiltonian Q and the moduli space
B0,2
The BRST Hamiltonian Q = 1(w12IQ(2 )I'T)1'1I'
2,
which is also the kinetic term in the
string action, may be represented by the standard twice-punctured sphere with both
a co ghost and a BRST operator insertion as well as the two string field insertions. In
the spirit of [24], just as BC2)= (w121F))|I) 2 was identified with -f(B0, 1), it is natural
to introduce a moduli space B°,2 of dimension -2 which will be the standard sphere
with two ordinary punctures which we have just mentioned, and then to identify
Q with f(B°,2). If we include BO, 2 in the sum' B = g B-,n,
B,n, this will allow us
to absorb the BRST Hamiltonian into the term f(B) so that the expression for the
quantum action simply becomes,
S = Si,o + f(B),
(5.1.1)
where S1,o is the one-loop vacuum term, not present in the classical theory.
'Note for the quantum case that B also includes the higher genus spaces of positive dimension
introduced in Chapter 4.
We state an identity which we will make use of in the next section. Let A be any
moduli space. Then from Eqn.(3.25) of [24],
{Q, f(A)} = -f(aA - (-)A..ICA),
where have introduced for convenience the notation A
Given that Q = f(B°,2 ) this may be written,
(5.1.2)
A + nA for the grading.
f({IB3,2, A}) = f(OA - (-)ACKA).
(5.1.3)
This will be useful in finding moduli space identifications for the operators 1 and K,
to which we now turn.
5.2
0, K: and I - Explicit operator identifications
We will now suppose that the operators 9, K and I can be written as inner derivations
on the B-V algebra and shall attempt to derive the general expressions for these
operators in terms of elements of the B-V algebra of string vertices, using the standard
operator identities satisfied by them as constraints. We recall the following list of
operator identities,
&{A, B} = {aA, B} + (-)A+1 {A, B} ,
IK{A, B} = (-)+1{CA, L3} + {A, IB},
(5.2.1)
Z{A, 1} = (-)+'1{ZA, } + {A, zB}.
a2 = /C2 = _2 = 0.
(KI + ZIK)A = JA,I
(5.2.2)
},
[a,z]A = 0,
[0, IC]A = (.-).4)V ,3, A}.
(5.2.3)
We also have the following properties, the first being (5.1.3), and the second being
Eqn.(5.17) of [24],
f(OaA -(-) KA) = -{Q, f(A)} = f({ 2, A})(5.2.4)
f(IA) = {f (A), B (2} = f({A, 3Boj}).
(5.2.4)
The three conditions which needs be satisfied to write string theory around nonconformal backgrounds (2.2.96)-(2.2.98), are as follows,
) } = -f(7(0,),
{Q, BF
2,
Q}
= f(v1,),
(5.2.5)
(5.2.6)
{Q, f(E)} = f(KCE).
(5.2.7)
The third of these is already satisfied as it was used to derive the first of Eqns.(5.2.4).
The remaining identities not included in the list above are as follows,
021 zV,,0
(5.2.8)
70, = KBCo,1,
(5.2.9)
Vo, 3 = KB, 2 .
(5.2.10)
The first is Eqn.(4.18) of [23], the second is Eqn.(5.18) of [24], and the third was
mentioned as a postulate in a footnote in §5.3 of [24]. Our aim will be to find operator
identifications and conditions on BL,2 and B31, which satisfy all of these equations.
Let us first attempt to satisfy Eqns.(5.2.1). We shall consider sums of operators
of the form {X,A} = -(_)(A+i)(X+l){A, X}, and {A,Y} = -(-_)()(YA+)(+•y,A}
where X and y are elements of the B-V algebra. So we shall try the general forms,
aA = {L, A} + {A, R} ,
ICA = -{£,A} - {A, R},
(5.2.11)
ZIA = {r, A} + {A, )},
where L, R,7,
1, I and r are elements of the B-V algebra to be found. The reason
for the unusual notational and sign choices will be evident soon.
Let us consider the action of the general operator OA = {X, A} + {A, y} on the
antibracket, and use it to deduce conditions on our unknown elements. We find,
O{A, B} = {X, {A, B}} + {{A , B}, y}
S_(_)(B+1)(+1)((-)(A4+1)(+1){B,
-(-)(A+1)(+1)()(8+1)(
{X, A}} + (_)(A+1)(X+1){A, {B, X}})
9l+)(+){{y, A}, B} + (-)(+)(B++){{B,y}, A})
= {(-)(8+i)(A)(X, A} + (-)(8+1)(y+1)({A, y}, B}
+{A, (-)(A+1)(++1){X, B} + (-))()({B , y}} .
(5.2.12)
We apply this to the operators d, KI and I whose postulated forms are written in
(5.2.11), and compare with (5.2.1). We come to the following conclusions. For a, we
find that L is graded-even, and R vanishes. For KC, we find that R is graded-even and
£ vanishes. Finally, for I we find that 1 is graded-even, and r vanishes.
To summarise, we have £ = R = r = 0 and,
4A= {L, A} ,
KA = -{A, R} ,
IA = {A, } ,
(5.2.13)
where L, R and I are each graded-even elements of the B-V algebra.
Consider now the nilpotency conditions. These imply the following constraints on L,
R and 1,
{L, L} = {R, R} = {1, 1} = 0.
(5.2.14)
Let us now tackle the third set of conditions, Eqns.(5.2.3). For KIC + ICK we find,
(ICT + ZI)A = -{{fA, }, R} - {{A, R}, l}
= {{l, R}, A} + (-)A{{R, A},1} - {{,A, R},1}
(5.2.15)
= -{A, {1,R}},
which is of the form we seek. We may make the operator identification,
T = -{l, R} ,
(5.2.16)
which must be non-vanishing. Turning now to the vanishing of [0,Z], we find the
condition,
{L,l} = 0.
(5.2.17)
A similar calculation for [0, K] implies the operator identification,
V0, 3 = -{L, R} ,
(5.2.18)
which must also be non-vanishing. The fourth set of conditions, Eqn.(5.2.4) helps us
to identify the vertices which L, R and 1 are associated with. In particular we obtain
the conditions,
L + R = B° ,2
(5.2.19)
1= B0,1'.
(5.2.20)
The latter condition fixes the form of 1 to be that stated in [24]. Consider now the
conditions for string theory around non-conformal backgrounds. Eqn.(5.2.5) requires
that,
{B0, 2; 0,1} = {l, R},
(5.2.21)
Similarly, Eqn.(5.2.6) requires that,
2{B,2,
0,2
2}
= {L, R}.
(5.2.22)
Finally, we consider the remaining identities, Eqns.(5.2.8)-(5.2.10). These imply the
following set of constraints,
{L, {l, R}} = {I{L, R},1},
(5.2.23)
{l, R} = {Bo,1 , R}I,
(5.2.24)
{L, R}= {Bo,
(5.2.25)
2
R},
It is a remarkable fact that this entire gamut of constraints is soluble, and that the
unique solution is given by the following two requirements,
8°,2 = L + R,
(5.2.26)
130,1 = 1,
supplemented with the following conditions,
{L,L} = R,R}= {1,1} = {L,1})
£=R
= r = 0.
(5.2.27)
We shall suggest an explanation for these conditions in Chapter 7. These imply the
set of operator identifications Eqns.(5.2.13),
MA = {L, A} ,
ICA = -{A, R},
(5.2.28)
IA = {A, I} ,
and the special vertex identifications,
3= -{L, R},
S= -{, R}.
(5.2.29)
The success we have had in reproducing the usual operator identities verifies that the
operators o and K: can, as was shown to be the case in [24] for I, be described as
inner derivations of the B-V algebra.
5.3
Recursion relations and the string action
Given all the preparatory work of the last section, we will now simply state the form
of the recursion relations and the string action around general backgrounds and then
check that they are correct. The recursion relations are given by a 'geometrical'
quantum B-V master equation for the B-space complex,
-{B, B} + AB = 0,
2
(5.3.1)
and the string action is given by Eqn.(5.1.1),
S = Si,o + f(B).
(5.3.2)
Here we have B =
g,,3,, where the sum is over B-spaces with all values of
(g, n, f) except B, 0o, B,o0 , Bgo,o and those with g = 0 and n + i < 1. That this action
satisfies the master equation is clear in view of the form of the recursion relations and
the field-independence of S1,o. For a background which is conformal, this reduces to
the correct form S = Q + S1,o + f (V) where the kinetic term is Q = f (B ,2).
In order to gain some insight into how the recursion relations of (5.3.1) work, we
will re-express them in such a way as to make them look a little more familiar. Let
us rewrite the negative-dimensional B-spaces in their operator form and let us also
introduce the notation B= B, 2 + 0, 1 + B = L + R + 1 + B, where the object B
is the restriction of B to the non-negative dimensional B-spaces. Then the recursion
relations may be written as follows,
O= {B, B} +AB
= {L,R}
= -,3
+ {l, R} + {L+ R,B} + {1,B} ~~,}
+ (8- I)B +
-
B +{
+a
(5.3.3)
+ AB,
or perhaps more provocatively as,
a3=
, +,,-fo,1
+ L3-
z-
-2{,1L}- AB,
(5.3.4)
where in the first line we have expanded B in terms of its components and used the
fact that vertices are all graded-even to extract the operators 0, K and I. Eqn.(5.3.4)
should be compared with Eqn.(5.10) of [24] 2. The equivalence between our form of
the recursion relations and (the quantum generalisation of) those claimed by Zwiebach
is now transparent.
We might mention here that we have tacitly used the fact that A(L + R) vanishes
in deriving (5.3.3), which is in accord with AQ = 0 [11]. A geometrical explanation
for this will also be suggested in Chapter 7.
Before we end, we shall discuss briefly the possibility of expressing the B-V delta
operator as an inner derivation.
5.4
The B-V delta operator - an inner derivation?
The B-V delta operator satisfies several identities similar to those of 0, K: and Z. We
have shown in §2 that these three operators may consistently be expressed as inner
derivations in such a way that the usual identities are satisfied. We now ask ourselves
the question of whether the same might be possibly with the A operator. This seems
an unlikely proposal, but it turns out to be surprisingly close to working, the failure
being a somewhat indirect implication of the B-V antibracket being a derivation of
the graded-commutative and associative dot product on moduli spaces.
Let us list the usual identities known to be satisfied by the A operator,
2
A{A, B} = {AA, B} + (-)A+{A, AB},
(5.4.1)
(OA + AO) = 0,
(5.4.2)
[A, K] = 0,
(5.4.3)
Recall that M E IC - I and that the A-term would be expected in the naive quantum generalisation of Eqn.(5.10).
A 2 A = 0.
(5.4.4)
Assuming the identifications derived earlier for 0, /Cand 1, the second identity actually follows from the first which we demonstrate as follows,
AdA = A{L, A} = {AL, A} - {L, AA} = -{L, AA} = -&AA,
(5.4.5)
while a similar calculation verifies the third identity. We might also use this construction to 'predict' the identity [A,Z] = 0.
Now (5.4.1) is very reminiscent of an analogous identity, Eqn.(5.2.1) which is
satisfied by 0, and this strongly suggests an identification,
AA = {X, A}.
(5.4.6)
where X is an element of the B-V algebra to be found. The fourth condition of
nilpotency gives the following condition on X,
(5.4.7)
{X, {X,A}}.
This vanishes identically only if X is graded-even and satisfies,
{X,X} = 0.
(5.4.8)
This all seems very encouraging as the logic so far has been identical to the cases of
and I. However, the B-V delta operator satisfies one more identity, Eqn.(3.14)
a, :C
of [27], which relates it to (and can be used to define) the B-V antibracket,
{A, B} = (-)A (A -B) + (-)A+l(AA)
.
B - A- (AB),
(5.4.9)
where the moduli spaces form an algebra under the graded-commutative and assoApplying (5.4.6) to (5.4.9), we find the following
ciative product denoted by the '-'.
condition,
{A, B} = (-)A{X,A
+(-)+1X
} -B - A -{X, B}.
(5.4.10)
But the antibracket satisfies the following property with respect to the dot product,
{X, A- B} = {X, A} . B + (-)(+
A- {X, B}.
(5.4.11)
Our space X is graded-even, so applying this to (5.4.10) we find that this would imply
that the antibracket identically vanishes! This shows that the B-V delta operator
cannot be described as an inner derivation of the algebra, despite initial promise.
Chapter 6
Geometrising the string action
In this chapter, we will extend the B-space complex by introducing the positivedimensional spaces U1,0 (being the moduli space of tori with a single special puncture),
BL,0 (the space of unpunctured tori), and 8B 0 where n > 2 (the moduli spaces of genus
zero surfaces with no ordinary punctures and at least two special punctures), and
explain why it is consistent to do so without spoiling earlier results. Armed with the
extended B-complex, we shall generalise the results of [24] and state the form of the
quantum string field action around arbitrary backgrounds. We find that the string
action then takes the elegant form S = f(B) while the recursion relations continue
to be described by a quantum B-V master equation for the sum of string vertices.
Unless explicitly stated otherwise, we will use units in which h = , = 1 throughout.
6.1
The moduli spaces B,
and BL3
All of the spaces mentioned above are distinguished by the fact that they are moduli
spaces of surfaces containing no ordinary punctures. This means that they cannot
be sewn, and hence can take an active part neither in background deformations (in
particular they should not affect the proofs of background independence or the ghostdilaton theorem), nor in the recursion relations, so one would expect that there should
be no problem in including them into the B-space complex. Let us then reconsider
the reasoning for leaving out these spaces, and readmit them if we find a satisfactory
excuse for doing so. We first discuss briefly why the space Bl o was excluded.
When the spaces B, n were first introduced for the proof of quantum background
independence, they were defined only for n > 2 at genus zero and for n > 1 at higher
genus (Eqn.(4.19) of [15]), the remaining spaces BL,o (for all genera g 2 1) being set
to vanish as they could not take part in background deformations and were therefore
irrelevant to the discussion.
However, their irrelevance to background independence was not a reason for dismissing them entirely, and this was acknowledged in proving the ghost-dilaton theorem (see [22] and Chapter 3), where all these spaces were reinstated, with the exception of the space Blo. The reason for this omission was that the general equation
derived in [22] implied that its boundary was given by OB ,o = -AB1, 2 - B0,1 , which
inconsistent as it did not satisfy &&B', o = 0.
This problem disappears if we assume the existence of the one loop vacuum vertex
B0,o satisfying the recursion relations (5.3.4) as we can then show that OB1', o = 0 is
in fact satisfied. In particular, we find the following,
B
o
= KB°,o - ZB,
1-
AB,2(6.1.1)
Taking the boundary once again and using the usual operator identities,
,MBI,o = acB°,o0 - azB0, 1 - BAB,2
0-
=
KC9B,
=
-Kcao,
1,1 +(6.1.2)
IBB, + aB,
,2 2 +zAB0,3 + AnK3
AzBo,3
=-0.
where we have applied the recursion relations to B °0, and B,, 2 to obtain the expressions
,o = -AB0, 2 and 9,2 = IC,-Z
,2
, 3 for their boundaries. The term KCB, 2 above
should be identified with V0,3 of §6.2 of [15], the usual V0,3 being a special case. We
shall discuss this in more detail momentarily.
Having assumed the existence of B°,0 satisfying the recursion relations, let us now
see further evidence to suggest why this is consistent. In particular, we note that
the proof of quantum background independence (§6 of [15]) for the field-independent
O(h) terms was particularly tricky, and left us with the unusual condition OB?,0 =
-7rFAVO, 3 on the boundary of B °,o. We will now show how our expectations for the
one loop vacuum vertex do away with these unwanted features.
We recall the field-independent O(h) condition for background independence Eqn. (6.9)
of [15],
y,S1,o
( (°)1,11O),
=
(6.1.3)
1,1+AB0,2
Having assumed the existence of the vertex V1,0 we can now write this as,
S,,o = ,f(V 1,0) = f,(IV1,0) = Lf
()1•1I,).
(6.1.4)
This allows us to rewrite the condition (6.1.3) as follows,
=
(V,0-(0)1,1I,-AB1)
0.
(6.1.5)
If we glance at (6.1.1) we see that integration region is simply B1,o0. A simple application of Stokes' theorem then explains why the background independence condition
is satisfied,
13,0
((O))1 '1
O
"
L•L IQIO
(
1
1(0)1,
1)- 0.
(6.1.6)
There is no need for any further analysis or application of auxiliary constraints.
This shows that it is algebraically consistent to introduce Bo, and we now give
a geometrical description of what this algebra implies. It was mentioned in [11] that
BO1, was 'not constrained' by the recursion relations. On assuming the existence of
B0 ,2, which is a twice-punctured sphere representing the kinetic term, this is no longer
true and, as we have mentioned, the recursion relations imply that B13,o = -AB°, 2.
This is as we would naively expect - as we increase the height of the internal foliation
of the vacuum graph to 27r, the diagram should split into the twice-punctured sphere
whose punctures are glued together by a propagator.
Usually we consider the string vertices to have stubs of length 7r, so that the propagator can have any length from 0 to oo. In general it is also possible to use shorter
stubs of length a say, where 0 < a < 7, in combination with 'cutoff propagators' [11]
which would be of length > 2(7r - a). The latter condition would ensure that no
nontrivial loops have length less than 27, as is required by the minimal area metric
prescription. If we are to apply the same conditions to the vertex L300, 2 as are applied
to the other vertices, we would expect that it be conformally equivalent to a cylinder
of length 2a. In this case, the general expression for the kinetic term would be,
Q = 2(R12C(2)2aLo
j)1j)2
(6.1.7)
For the usual case a = 7r, we require the insertion e20Lo+
This description of B O,2 elucidates some points regarding the choice of connection
F. In §4.2.2 we reviewed the origin of symplectic connections. In particular, a choice
of auxiliary three string vertex 1V
0 ,3 determines the vertex Ba,2 (which interpolates
between IB0,3 and V0,3 ), and this in turn determines the choice of connection through
(4.2.23). The particular case of the canonical connection r follows from choosing V',3
(introduced in §3.3 of [26]) as the auxiliary three-string vertex.
Now the recursion relations determine that the boundary of B1, 2 is given by,
,2 =
C13,
2
-
0,3
,
(6.1.8)
1
which means that 0V,
3 is identified with C30 ,2 . Moreover, the choice of auxiliary
three-string vertex determining the connection is defined by the choice of B3,2 which as
we saw above encodes the choice of stub length and cutoff propagator. Now, B13is just
a cylinder of length 2a. The operator /C adds a puncture over the entire surface of this
cylinder, resulting in the vertex )0 ,3 . The vertex V1, 3 stems from the particular case
a -+ 0 corresponding to a degenerate cylinder (being the overlap surface corresponding
to the standard twice-punctured sphere). In this case the special puncture may only
be added over the boundary of the coordinate disks. Each of the resulting threepunctured spheres are identical (being related simply by a rotation), and we thereby
recover the usual description [26] of V',3 . It was recognised in §6.2 of [15] that the
canonical connection F has some rather singular properties. We are now able to
identify the source of these with the degeneration of the cylinder described by B0,2 .
'Note that in (5.2.10), we identified LB00,2 = - {L, R} with V2, 3 . The identification with Vo, 3 is
more general, and it is V0 ,3 which should appear in the recursion relations (5.3.3).
The considerations above suffice to show that it is consistent to include the moduli
spaces3
B", and Bl,o into the string vertices.
6.2
The moduli spaces Bnon
The moduli spaces of spheres with at least two special punctures and no ordinary
punctures were not considered solely because they contribute ineffectual constants to
the action, and were therefore not essential to the usual discussions. Consequently
we see no harm in including them and they will be reinstated in what follows.
6.3
The geometrised string action
Having successfully introduced into B the missing string vertices of non-negative
dimension, we can now state the resulting expression for the action. It takes the
elegant 'geometrised' form,
(6.3.1)
S = f(B3),
where B = Eg,n,, L3,n, where (g, n, h) may take all values except when g = 0 and
n + h < 1. That this action satisfies the master equation is clear in view of the form
of the recursion relations, which are still given by a quantum master equation (5.3.1)
for the string vertices,
2
{B, B} + AB = 0,
(6.3.2)
The boundaries of the newly introduced moduli spaces are summarised as follows,
B,, = KBo - zB°, 1 - AB,,2
°,° = -A130°, 2,
(6.3.3)
OB2,o =0,
f , - zB~,-1,
ofB,o = ICB
( > 2).
So we have seen how the moduli space 30,2 introduced in has provided the key to
incorporating into the sum of string vertices B the spaces B1, 0 and B0,0 which, together
with the spaces Bf,o, complete the set of string vertices of non-negative dimension.
The entire theory has been encoded elegantly in terms of the set of string vertices,
and the string fields IT) and IF), which define the function mapping the string vertices
to the action.
301 and B3,
One might note that the three moduli spaces B1, 0, B
0 remain to be
defined. These will be discussed in Chapter 7 which will be concerned mainly with
new insights related to background independence stemming from our geometrised
formulation.
Chapter 7
Path towards manifest background
independence
In Chapter 6, the theory was encoded elegantly into the set of string vertices satisfying
a quantum master equation, as well as the string field lI) and the state IF) defining
the mapping from surfaces to functions. The set of string vertices are the background
independent component of the formulation while the string fields are the backgrounddependent component. In this chapter we argue that the choice of background (and
hence the string fields) and the delta operation may be associated with the zero genus
moduli spaces with only a single ordinary puncture. We also explain why, should this
hypothesis be correct, the manifest background independent formulation of the theory
is given simply by the sum of the completed set of string vertices (this being the
geometrical analogue of the usual field theory action), whose recursion relations are
expressed via a classical master equation. Except where explicitly stated otherwise,
we will use units in which h = n = 1 throughout.
7.1
Completing the set of string vertices
Let us summarise the current state of our theory. It has been simplified to such
an extent that it is now completely described around arbitrary backgrounds by the
complex B, satisfying a quantum B-V master equation,
{B, B}+ AB = 0,
(7.1.1)
as well as the function mapping the set of B-spaces to the action S which is a function
of the fields/antifields,
S = f(B).
(7.1.2)
To construct this function we need a knowledge of the background which will determine the string field IT) of ghost number two, and the fermionic state IF) of ghost
number three. If we are somehow able to express this information in a background
independent way, we will have achieved our main ambition of attaining a manifest
background independent formulation of the theory. A few suggestive observations
will now lead us to postulate the way in which we expect this goal to be realised.
The object B is the formal sum of moduli spaces of decorated Riemann surfaces
B ,n for all positive values of (g, n, h) with the exception of BoO,o, B, and B°,,. It
would be satisfying if there could be some way of incorporating them into the theory
and thereby completing the sum contained in B.
Adding the spaces B°,0 and B0,o0 should pose no problems. They are represented
by an unpunctured sphere and a sphere with a single special puncture respectively,
and their lack of ordinary punctures implies that they do not couple in any way to
the remainder of the theory, contributing at most a harmless constant to the action.
The space B. 1, remains, and we shall discuss this object now.
We recall from §2.2.5 that the algebraic structure of the classical closed string
field theory corresponds to a homotopy Lie algebra L, defined by a set of multilinear
graded commutative string products mn = [B1, ..., Bn]o. For a conformal background,
it was found that the product mo must vanish, [ -]0 = 0, while for string theory around
a nonconformal background it was associated to some non-vanishing ghost number
three state IF) E R7, [ -]o = F. Now, the product mn is associated to the n-punctured
vertices 1B,n+1 with fields inserted at n of the punctures. So we expect the product
mo, which determines the string background, to be associated with a once-punctured
sphere B, i1 with no insertions. Let us suppose that this is indeed the case, so that
the background independent object BO, 1 actually encodes the information about each
particular string background.
Having introduced the spaces B°,0 1,B, 1 and Ba,0 , let us also assume the master
equation for the B-spaces is still satisfied. We need to ask what the effect would be
on the recursion relations. The spaces B°,0 and Ba,0 have no sewable punctures, and
can be safely ignored in this regard. However, BO,, is not limited by this restriction.
Let us then define the operator U as follows,
UA = {B0,, A}.
(7.1.3)
Existence of 8,0, then implies that there would be a contribution of UB to our master
equation. The B,,1 has naive dimension -4, whereas the boundary operator 0 is
represented by the sewing of B0, 2 which is of dimension -2. This would signal the
loss of the usual form of our usual recursion relations. If the contribution from BO, 1 is
nilpotent, one may be able to recover recursion relations of some kind, but they will
not agree with the geometrical identity for the moduli spaces of Riemann surfaces.
Furthermore we would also have to explain the existence (or vanishing) of new objects
{B0,, BO, 1}, {B,, 1, B, 1} and {B00, 1, B0,2 } which do not seem to have any interpretation
in the theory as yet.
With a view to resolving this issue, let us consider the surface which BOO describes.
It is a sphere with a single coordinate disk whose boundary corresponds a Hilbert
space. This is topologically equivalent to an infinite complex plane, the point at
infinity representing the puncture and the boundary being some contour around the
origin. This description is fine when there is no additional structure such as endowed
by a metric. Suppose now that we try to find a metric on the surface which satisfies
the minimal area conditions. In the first place, we note that attempting to put a
metric on the sphere will always result in two singular points. One of these may be
associated with the puncture, but the other one has no such identification. Adding to
this the requirement that nontrivial closed loops have length > 2r, the geometrical
description of B ° is as a semi-infinite cylinder, completely foliated by saturating
geodesics of length 21 [41], with a Hilbert space associated to the puncture at the
finite end. This description will prove useful later when we try to identify the sewing
of B°,j with the delta operation.
Before we do this we need to postulate sewing rules which will serve as a useful
heuristic to both explain the constraints (5.2.26) and (5.2.27) on 80,2 and B30,1, as well
as motivate our eventual description of the sewing of B0,1.
7.2
A postulate about sewing
We begin with a few observations regarding B 1 , and B °,2 . The former has only one
sewable puncture while the latter has two. Also, (5.2.26) tells us that B1, has one
° comes in two forms, denoted 'L' and 'R'. This suggests
form, denoted '1', while B0,2
that there is a relationship between the number of forms and the number of sewable
punctures which these negative-dimensional (overlap) vertices have.
Also, we note that the Hamiltonian B(2) = -(R' 2 Co-(1)F)1i|) 2 , (where we have
chosen arbitrarily to label the ordinary puncture '1', and the special puncture '2'),
may be written in many different ways by using (2.1.19) to move portions of the co
insertion between the two punctures. The same is true of the BRST Hamiltonian
Q = f(B°,2) = -½(R'2Q(1)co(1X)))2. We would like some way of extracting
from these possibilities, one particular way of writing B(F ) and two particular ways
of writing Q. We therefore make the following observation.
If we sew Bo1,, there is only one unique way to write B ) in such a way that the
ghost insertion is associated with the unsewn puncture,
LB
2) =
f (l) = -(R(21Co(1) IF)l)2 .
(7.2.1)
The reason for the notation will be clear momentarily. Similarly, if we sew B ,° 2, there
are only two ways (depending on which of the two punctures are sewn) to write Q in
such a way that the insertions are at the unsewn puncture,
RQ =
f(R)
= (R2 C_-(2)Q(2)I1)
2
LQ = f(L) = -~\(R2'('1)-(1e)
)2 ,(7.2.2)
1 )2,
(7.2.3)
corresponding respectively to the sewing of the first or second puncture.
This lead us to our first sewing postulate which simply states that it is 'illegal'
to sew any puncture whilst there are operators inserted at the Hilbert space at that
puncture. While we do not have a proof of this, the intuitive reasoning behind it is
that the information at the Hilbert space corresponding to the sewn puncture would
be lost when the coordinate disk (and hence the Hilbert space) is removed. What this
implies is that when a puncture is sewn, it is essential to ensure before sewing that
any operators at the corresponding Hilbert space are transferred to Hilbert spaces
at other punctures. This provide a useful heuristic which explains why on sewing
B1,g 'collapses' into the object 1. Also, B0,2 can collapse into either L or R, with the
condition that L + R = B00, 2. Together, these offer a possible explanation of the origin
of (5.2.26).
The process of rewriting operators around one Hilbert space in terms of operators
at all other Hilbert spaces can always be carried out if other Hilbert spaces are
available. However, when we wish to sew the only puncture of once-punctured objects,
it is necessary to use the common technique of introducing an auxiliary Hilbert space
by adding an extra ordinary puncture arbitrarily to the surface and inserting a vacuum
state at the Hilbert space around the auxiliary puncture. The point is that the sewing
postulate above is acceptable as it does not affect any earlier results.
We are not yet able to explain the origin of (5.2.27), so we shall simply state
them as our second sewing postulate. To this end shall assign a 'handed-ness' to
these vertices on sewing according to which puncture is sewn, so that 1 (which has
been chosen by convention to sew puncture '2') and L (which is that component
of B0°,2 which sews the same puncture as 1) are designated to be left-handed, while
R is designated 'right-handed'. Our second sewing postulate then states that the
antibracket of any two spaces with the same 'handed-ness' vanishes, and this then
correctly reproduces (5.2.27). These rules are admittedly rather ad hoc and we hope
insha'Allah that derivations of these rules (or plausible alternatives) are forthcoming
in future.
These rules have some useful consequences. They can be used to explain the
vanishing of AL and AR which was used in deriving the recursion relations (5.3.4)
of Chapter 5. The reasoning is that we are attempting here to sew both punctures of
the spaces L and R at the same time. More specifically, sewing one of the punctures
means that we must transfer the operator insertions co Q to the Hilbert space around
the remaining puncture, which renders this remaining puncture unsewable, and this
explains why the expressions vanish.
7.3
The sewing of Bo,, and the B-V delta operator
Let us reconsider our identification of BO, 1 as a semi-infinite cylinder in light of the
sewing postulates. According to our sewing rules, when we try to sew the ordinary
puncture we would need to map our insertions to other Hilbert spaces before the
puncture could be sewn. But B ,1 has only a single puncture and we might try to resort
to using an auxiliary puncture as was described in the previous section. However even
this does not work for B00, 1 , as the insertions would still 'leak' away down the semiinfinite cylinder, where they have nowhere to go. Clearly, if our expression is not
to vanish, the boundless end of the cylinder must somehow lead to another Hilbert
space on sewing, and the only Hilbert spaces available are those which remain on
the surface to which the space B0, 1 is being sewn. What this argument suggests is
100
that sewing LB,,, to a surface is equivalent to sewing together two punctures of that
surface! In other words, we argue that the operator we denoted as U is none other
than the B-V delta operator. As an immediate corollary, the original quantum master
equation for the complex B reduces to the classical master equation, {B, B} = 0.
It was shown at the end of Chapter 6 that the B-V delta operator cannot be
an inner derivation of the B-V algebra, yet it may seem that we are attempting
to do precisely that here by identifying it with the sewing of B°,1. However the
sewing here does not coincide with antibracket sewing as is clear from the description
we have given above, and so {B°,1, - } is not an inner derivation on the B-V algebra
despite appearances. That the identities (5.4.1)-(5.4.4) follow from an inner derivation
interpretation will be considered as merely a happy coincidence. Note that the objects
{B°,1 , BI,1 } and {Bo,o, B01, } vanish from lack of sewable punctures, while {B0,, B0, 2 }
vanishes for the reason described at the end of the previous section.
7.4
Manifest background independent formulation
If we suppose that the identifications we have suggested above are correct, then we no
longer need to retain explicitly the action S, as all the information we need to define
the theory is already contained in the full set of B-spaces. Indeed our final 'manifest
background independent' formulation of the full quantum closed string field theory
would be given by the following 'geometrical master action',
B
(where B -=
(7.4.1)
g,n,i>O B ,n), satisfying the 'geometrical classical master equation',
B,B} = 0
(7.4.2)
We shall conclude our analysis at this point.
7.5
Conclusion
If the arguments which have been been presented in this chapter can be made rigorous,
we will have succeeded in finding a fundamental geometrical representation of string
field theory which underlies the usual formulation. Furthermore, the unexpected
appearance of the Batalin-Vilkovisky master equation, which is satisfied by all gauge
field theories strongly suggests that this is a phenomenon which is not restricted to
string theory, and may be of much more general applicability, where our string vertices
B would be replaced by more general algebraic objects.
It is of interest to generalise our ideas to the case of open-closed string theory [42],
and superstring field theories as formulated in the language of [43], in the hope of
eventually clarifying important issues such as the nature of the space of backgrounds
and the existence of dualities. If such a program is successful, one would expect that
the results would also be of direct relevance to field theory as a whole. Of course
much more work needs to be done before such ideas can be properly established.
101
There are still many gaps in our understanding which need to be filled before we
can be satisfied that we have found the manifestly background independent formulation of string field theory. The arguments we have brought forward merely suggest
that our identifications are consistent, and certainly do not represent proof of correctness. Besides this we have yet to show how to extract the states in theory and neither
do we have explicit expressions for the BRST operator Q, the fermionic state IF), the
one-form F[11 or the two-form F [2]. Also, some of the issues which remained open at
the end of [24] still remain to be tackled. There are probably many other difficulties
which we have not mentioned or of which we are not yet aware. Nevertheless, we hope
that the results of this thesis will constitute solid progress towards our understanding
of string theory.
The problems which remain unsolved are left as an exercise for the reader.
102
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