Imperial College London
MSc Dissertation
T-duality, Double Field Theory and its
Geometry
Author:
Eduardo García-Valdecasas Tenreiro
Supervisor:
Professor Chris Hull FRS
September 18, 2015
Submitted in partial fulfilment of the requirements for the degree of Master of Science of
Imperial College London
Abstract
Double Field Theory (DFT) is an attempt to formulate a
T-duality invariant field theory incorporating stringy effects
coming from the winding of closed strings around compact
dimensions. The aim of this work is to review T-duality in
toroidally compactified string theory and how DFT naturally
arises in this context. We introduce string theory, Riemannian
geometry and supergravity as preliminaries. Then Kaluza-Klein
compactification is discussed and toroidally compactified string
theory is introduced. Then we discuss how T-duality arises
and we further study the T-duality transformations, both from
the group theory and Buscher points of view. Double Field
Theory is then naturally motivated by T-duality and presented
chronologically. Various actions are presented and their limits
studied, leading to the explicitly O(D, D) invariant action.
Finally, doubled geometry is studied, both in a coordinate
dependent and independent manner.
i
Dedicado a mis padres Miriam y Jesús por convertir el barro en porcelana y a Raquel por
pulirla. Sin vosotros nada sería posible.
ii
Acknowledgements
I would like to specially thank the generous support provided to me by the foundation "Fundación
Barrié". Without its help I would not have been able to come to Imperial College in the first
place. I would like to thank my supervisor, Chris Hull, for his guidance. I am also grateful to
Amihay Hanany for his support and help in taking decisions. I have also benefited from the whole
theory group at Imperial College and I would like to thank all lecturers for their knowledge and
my classmates for creating a very nice atmosphere. I would particularly like to thank Meera,
Eric, Joe and John for sharing their time with me while doing this project. Finally, I would like
to thank my parents, Miriam and Jesús for their unconditional support.
iii
Contents
1 Introduction
1
2 Preliminaries
4
Introduction to String Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1.1
Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1.2
Action and Equations of Motion . . . . . . . . . . . . . . . . . . . . . . .
4
2.1.3
Mode Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.1.4
Light-cone quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Introduction to Riemannian Geometry . . . . . . . . . . . . . . . . . . . . . .
13
2.3
Introduction to Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.1
3 T-Duality
18
3.1
Kaluza-Klein Compactification . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.2
Circular Compactification in String Theory . . . . . . . . . . . . . . . . . . .
19
3.3
Toroidal compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.3.1
Quantization and Physical constraints . . . . . . . . . . . . . . . . . . . .
22
3.3.2
Moduli Space and duality group . . . . . . . . . . . . . . . . . . . . . . .
26
3.3.3
The Buscher Approach - Abelian Duality . . . . . . . . . . . . . . . . . .
30
4 Double Field Theory
33
4.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
4.2
Doubling Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
4.3
Quadratic O(D, D) invariant action. . . . . . . . . . . . . . . . . . . . . . . . .
36
4.4
Physical constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
4.5
Cubic action and undoubled limit . . . . . . . . . . . . . . . . . . . . . . . . .
39
4.6
Full action and generalized Lie derivatives . . . . . . . . . . . . . . . . . . .
42
4.6.1
Background independent full action . . . . . . . . . . . . . . . . . . . . .
42
4.6.2
Generalized Lie derivatives, C-brackets and D-brackets . . . . . . . . . . .
44
4.6.3
The explicitly O(D, D) invariant action. . . . . . . . . . . . . . . . . . . .
47
Doubled Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.7.1
Generalized curvature and torsion tensors . . . . . . . . . . . . . . . . . .
47
4.7.2
Constraints in the connection . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.7.3
Generalized Ricci tensor and scalar . . . . . . . . . . . . . . . . . . . . . .
51
4.7.4
Coordinate independent formulation of DFT . . . . . . . . . . . . . . . .
52
4.7
5 Conclusions
54
Bibliography
56
iv
Chapter 1: Introduction
The aim of this work is to review T-duality and Double Field Theory. We will start by reviewing
closed bosonic string theory, as it will be needed later. String theory is a candidate for Theory
of Everything that replaces point particles with one-dimensional objects called strings. Different
vibrational states of the string account for different particles, producing an infinite variety of
them. String theory was initially developed in the 60’s as an attempt to describe the nuclear
force. It was soon realised that it naturally contained a spin-two particle, a graviton, making it
a natural candidate for a theory of quantum gravity. The first string theory to be developed was
bosonic string theory and it was soon realised that it was only consistent in 26D. Furthermore,
it incorporated a tachyon in its spectrum, making the theory unstable. This problem was fixed
with the introduction of supersymmetry and, consequently, fermions to the theory. This removed
the tachyon and fixed the dimension to be D = 10. By the 1980’s many string theories had been
developed and interest was decreasing. However, considerations about the consistency of the
quantum theory led to what is known as the first superstring revolution and only five theories
where found to be consistent. This five theories are known in the literature as Type I, Type
IIA, Type IIB and heterotic theories with gauge group E8 × E8 and SO(32). During the early
90’s these five theories were found to be related by a number of dualities, leading to the second
superstring revolution in 1995. In this year, Witten essentially showed that all theories are
equivalent and further introduced a sixth theory, living in eleven dimensions, that would also
be equivalent to them, M-theory. The formulation of this 11D theory is still unknown, but even
a full description of string theory is yet unknown.
String theory naturally leads us to the subject of supergravity that we will briefly introduce.
Supergravity is a field theory trying to incorporate both supersymmetry and general relativity.
This is achieved via setting supersymmetry to be a local symmetry, that is, gauging it. Ten
dimensional supergravity is easily found to arise as the massless tree-level approximation of
superstring theories. Analogously, 11D supergravity is believed to be the low-energy limit of
M-theory. All supergravity theories share an "universal" bosonic sector containing a metric field
g, a two-form field b and a scalar field called dilaton φ.
These preliminaries are treated in chapter 2. Chapter 3 then deals with T-duality. We start
this chapter by studying the concept of dimensional compactification in field theory through
Kaluza-Klein theory. The Kaluza-Klein theory is a field theory trying to unify electromagnetism
and gravity through a 5D pure gravity theory with one of its dimensions circularly compactified.
1
Chapter 1
Introduction
The electromagnetic field and its gauge symmetry naturally arise from the 5D graviton and its
diffeomorphism invariance.
Circular compactification is then discussed in the string theory framework. We see how Tduality naturally arises in this context as a symmetry interchanging R ↔ 1/R, being R the
radius of the compact dimension. As we mentioned above, a series of dualities were found to
relate the five string theories and, eventually, unify them. The existence of a duality between
two theories implies that both make the same physical predictions. They are, in fact, describing
the same physics. In particular, two string theories are dual if their partition functions are
the same. Dualities are not only present in string theory, being a well known example in field
theory the Montonen - Olive duality relating gauge bosons and monopoles. However, T-duality
is unique to string theory as it relies in the ability of strings to wrap in non-contractible circles
around compact dimensions. T-duality relates type IIA with type IIB and heterotic E8 ×E8 with
heterotic SO(32). As we will see, the implications of T-duality are extremely deep and shed light
in many stringy features. We will see how T-duality naturally hints at the geometry strings see
being radically different from the one we are used to for particles. Moreover, T-duality also hints
at a minimum scale in string theory, as compactifying in a small circle of radius R is equivalent
to compactifying in a big circle of radius 1/R.
The chapter on T-duality is finished with a section generalizing the previous procedure to arbitrary toroidal compactifiactions. We study the group structure of duality transformations
relating different solutions that describe the same physics. This transformations generalize the
R ↔ 1/R duality found in circular compactification. We finally introduce the Buscher approach
to T-duality and find the Buscher rules for an abelian isometry.
The final chapter is devoted to Double Field Theory (DFT). As we said, T-duality is a symmetry
distinctive of string theory that supergravity fails to incorporate. Double Field Theory is an
attempt to formulate a T-duality invariant field theory. This theory then incorporates some
stringy effects related to the winding of strings that are otherwise lost in supergravity. One
then seeks to incorporate some degrees of freedom describing the winding of strings in a field
theory. Even though this might seem contradictory, it is in fact achieved in DFT by doubling
the number of coordinates to accommodate those extra degrees of freedom. When toroidally
compactifying string theory both the classical momentum and the winding momentum 1 become
discrete. In fact, T-duality interchanges them, putting them on an equal footing. This leads to
the introduction of dual coordinates, the Fourier transformations of the winding momenta. This
is the starting point of Double Field Theory.
1
The winding momentum is the quantity counting the number of times a given string winds around each
compact dimension.
2
Chapter 1
Introduction
Previous work, trying to incorporate T-duality as a manifest symmetry of string theory was done
in [1] and [2]. In these papers the fundamentals of DFT are already present and, in particular,
coordinates are already doubled. The subsequent works by Siegel [3] and [4] further elucidated the
role of doubled coordinates and the structure of doubled space, with the development of covariant
derivatives for the doubled space. Finally, the foundational paper of Double Field Theory was
published in 2009 by Hull and Zwiebach [5]. In this paper the theory is presented and the action
to cubic order is constructed, as well as its gauge transformations. Soon afterwards, together
with Hohm, they derived the full background independent theory [6]. Finally, in [7] the theory
is presented in a fully T-duality covariant language,. The geometry of the theory was developed
by Hohm and Zwiebach in [8] and [9]. The geometry they found is strikingly different from the
usual Riemannian one. For instance, the curvature tensor is not fully determined by the fields,
even with vanishing torsion.
Double Field Theory unifies the metric and the two-form in a single T-duality covariant object
called "Generalized Metric", H. Then, the symmetries of this object are studied, leading to the
notion of generalized diffeomorphisms and, eventually, to the fully T-duality covariant action
in terms of H and T-duality covariant derivatives. In this work a constraint called "strong
constraint", not implied by string theory, is required. This constraint simplifies the treatment,
but implies that the theory developed here only depends on half of the total (normal and dual)
coordinates. Then, the theory we treat is not truly doubled, but only a restriction of the full
Double Field Theory, yet to formulate. The limit when the fields do not depend on the dual
coordinates is studied, leading to the familiar results from supergravity, as one would expect.
3
Chapter 2: Preliminaries
2.1
Introduction to String Theory
In this section we will introduce the basics of String Theory needed for understanding how
T-Duality is realised. We will closely follow [10, §. 2], [11, §. 3], [12, §. 2] and [13, §. 2].
2.1.1
Generalities
String theory was initially developed in the late 1960s as a description of the strong nuclear
force. However, a spin two particle, reminiscent of the graviton appeared insistently in the
theory. This led to considering string theory as a candidate for a quantum gravity theory. A
naive approach to quantum gravity reveals that it is non-renormalizable1 , as the mass dimension
of the coupling constant is [GN ] = −2. In order to remove the UV divergences, in the absence of
and UV fixed point, we need to smear the interaction out. In quantum field theory this turns out
to be complicated as smearing the interaction spreads time and breaks causality. String theory
deals with this issue by substituting point-like particles with one dimensional objects of length
Ls = 1/Ms called strings, where Ms is the string energy scale. At energies lower than Ms , strings
behave as point particles and a quantum field theory is recovered.
Different particles are obtained from different string vibrational states. The energy of these
particles increases with the excitation state of the string, creating an infinite tower of particles
in steps of Ms . Every string theory naturally incorporates a massless second rank symmetric
Lorenz tensor Gmn that behaves like a graviton. Hence string theory naturally includes quantum
gravity.
2.1.2
Action and Equations of Motion
A relativistic particle moving in space-time sweeps a trajectory known as a worldline. We know
that, in the classical theory, the trajectory will extremize its length. This can be cast as an
action principle:
S0 = −m
Z
ds = −m
Z q
1
−Gmn (X)Ẋ m Ẋ N dτ
(2.1)
At least superficially, as we still do not know whether the divergences cancel due to a non-trivial UV fixed
point [14].
4
2.1
Introduction to String Theory
Where X M are the space-time coordinates, ds the line element and X˙m = ∂X m /∂τ . Hence the
variational problem extremizes the length of the curve swept by the particle. The background
metric Gmn describes the space-time geometry. The X m (τ ) describes the trajectory of the particle
in space-time as a function of a parameter τ . These functions are said to provide an embedding
of the worldline in space-time. Similarly, strings sweep a two-dimensional surface Σ as they
move in space-time. This surface is called worldsheet. Points on the worldsheet are labelled
by two coordinates (σ 0 , σ 1 ) = (τ, σ), being τ timelike and σ spacelike. Given a D-dimensional
space-time MD , which we take to be Minkowski (R1,D−1 ) for convenience, the classical string
configuration is given by a set of functions X M (τ, σ) specifying the position in space-time of
the worldsheet point (τ, σ). These functions provide an embedding of the worldsheet σ in the
space-time MD .
The action (2.1) can be readily generalized. One expects the classical theory to extremize the
total area of the worldsheet and so the natural generalization of eq (2.1) for the classical string
action is:
1
dA.
(2.2)
SN G = −
2πα0 Σ
This is known as the Nambu-Goto action. The constant α0 is called Regge slope and is related
Z
to the tension of the string as,
1
.
(2.3)
2πα0
Now, the space-time metric induces a metric hab in the worldsheet. This object is just the
T ≡
pushforward of the background metric, which for Minkowski space-time is:
hab = Gmn (X)∂a X m ∂b X n = ηmn ∂a X m ∂b X n ≡ ∂a X m ∂b Xm .
(2.4)
Using the volume form for this induced metric, the Nambu-Goto action can be recast as
Z √
1
SN G = −
− det h dσdτ.
(2.5)
2πα0 Σ
The square root in the action makes it difficult to quantize. We introduce an independent worldsheet metric gab not to be confused with the induced metric mentioned above hab . Using this
metric we can write an alternative, easier to quantize but classically equivalent action:
1
SP [X, g] = −
4πα0
Z
Σ
dτ dσ − det g g ab (τ, σ)∂a X m ∂b Xm .
p
(2.6)
This action is known as Polyakov action. It can be easily seen that this action is equivalent to
SN G . We introduce the notation g ≡ detg and use δg = −ggab δg ab to eliminate gab from the
action using its equation of motion:
1
δg SP = −
4πα0
⇒
⇒
1
dσdτ (−g)−1/2 −ggcd δg cd g ab + (−g)1/2 δg ab ∂a X m ∂b Xm = 0
2
Σ
Z
1
−
4πα0
Z
Σ
dσdτ (−g)
1/2
1
hab − gab g cd hcd δg ab = 0
2
1
hab = gab g cd hcd .
2
5
(2.7)
Chapter 2
Preliminaries
Where we have used the definition of the induced metric hab in Eq. (2.4). Taking the determinant
in both sides of (2.7) one finds,
1
(−h)1/2 = (−g)1/2 g cd hcd ,
2
(2.8)
and substituting this into Eq. (2.6) we recover the Nambu-Goto action, proving that the two
actions are classically equivalent:
1
S[X, g] = −
2πα0
Z
dτ dσ(−h)1/2 .
Σ
(2.9)
The Polyakov action provides an abstract worldsheet point of view describing a string moving
in space through a 2d field theory coupled to 2d gravity. From this point of view the background
coordinates X m behave as D − 2 scalar fields in 2 dimensions with a graviton gab . The remaining
degrees of freedom are removed by gauge fixing. This scalar fields are just the positions of the
string in space-time and, upon quantization, they will describe the quantum fluctuations of the
string in space-time.
As is familiar from QFT, in order to quantize a theory, local symmetries need to be fixed. From
now on we denote the worldline coordinates as ξ ≡ (τ, σ). The Polyakov action has the following
symmetries:
• D-dimensional space-time Poincaré invariance: it is also a symmetry of the worldsheet
theory, as it is embedded in space-time. The coordinates ξ a are left invariant, while the 2d
fields transform as,
n
m
X 0m (ξ) = Λm
nX + a
,
0
gab
(ξ) = gab (ξ).
(2.10)
• Reparametrization (Diffeomorphism) invariance in the worldsheet: the usual diffeomorphisms from coordinate invariance in the worldsheet,
X 0m (ξ 0 ) = X m (ξ) ,
0
gab
(ξ 0 ) =
∂ξ c ∂ξ d
gcd (ξ).
∂ξ 0a ∂ξ 0b
(2.11)
• Weyl invariance: invariance under local rescaling of the 2d metric,
X 0m (ξ) = X m (ξ)
,
0
gab
= Ω(ξ)gab (ξ).
(2.12)
The Weyl invariance does not appear in the Nambu-Goto action, it is an extra redundancy of the
Polyakov formalism. Weyl-equivalent metrics correspond to the same embedding in space-time.
We can use diffeomorphism and Weyl invariance to gauge away the degrees of freedom in gab
and set the so-called conformal gauge, where the metric is flat: gab = ηab . This gauge leaves
a residual local symmetry known as conformal symmetry. Field theories invariant under this
symmetry are called conformal field theories (CFT’s) and are extremely important in modern
6
2.1
Introduction to String Theory
physics. Furthermore, conformal invariance ensures ultraviolet (UV) finiteness in string theory.
More details on CFT’s can be found in [15]. Our system is then flat with orthogonal coordinates
(τ, σ). We fix an arbitrary line in the τ direction to be σ = 0 and denote l as the length of the
string in the σ direction. The Polyakov action finally reads:
1
SP = −
4πα0
Z
Σ
d2 ξ∂ a X m ∂a Xm .
(2.13)
Varying the action we obtain the equations of motion for the scalar fields:
∂a ∂ X =
a
m
∂2
∂2
−
∂σ 2 ∂τ 2
!
X m = 0.
(2.14)
The general solution for this wave equation is given by:
m
X m (τ, σ) = XLm (τ + σ) + XR
(τ − σ).
(2.15)
m are left- and right-moving waves. Now, the equations of motion for g
Where XLm and XR
ab (Eq.
(2.7))are imposed as constraints for the system. Written in components they read:
a=b
⇔
∂τ X m ∂τ Xm + ∂σX m ∂σXm = 0,
(2.16)
a 6= b
⇔
∂τ X m ∂σXm = 0.
(2.17)
The constraint in Eq. (2.7) can be elegantly expressed in terms of the energy-momentum tensor.
The energy-momentum tensor of the Polyakov action in (2.13) is that of a theory of scalars
fields:
Tab =
1
(∂a X m ∂b Xm − ∂ c X m ∂c X n ηmn ηab ).
2πα0
(2.18)
And the Eq. (2.7) reads simply,
Tab = 0.
2.1.3
(2.19)
Mode Expansion
Introducing left- and right-moving worldsheet coordinates ξL,R ≡ τ ± σ , ∂L,R ≡ ∂/∂ξL,R we can
add and subtract the two equations above ((2.16) and (2.17))to obtain the so-called Virasoro
constraints,
∂L X m ∂L Xm = 0 ,
∂R X m ∂R Xm = 0.
(2.20)
There are three different boundary constraints in string theory; periodicity for closed strings and
Neumann and Dirichlet boundary conditions for open strings. We will only treat closed strings
in the present work. The boundary condition for closed strings is expressed mathematically as,
X m (τ, σ) = X m (τ, σ + l).
7
(2.21)
Chapter 2
Preliminaries
If we take −π < σ ≤ π, l = 2π the, condition for the closed string becomes,
X m (τ, σ) = X m (τ, σ + 2π).
(2.22)
And we can expand the fields in σ as,
X m (τ, σ) =
X
einσ Xnm (τ ).
(2.23)
n
The equations of motion (e.o.m) (2.14) for the normal modes become, using the linear independence of the exponentials to obtain n equations,
X
−n2 einσ Xnm (τ ) =
X
einσ Ẍnm (τ )
⇒
Ẍnm = −n2 Xnm .
(2.24)
n
n
Where Ẋ m ≡ ∂X m /∂τ and X 0m ≡ ∂X m /∂σ. In quantizing the harmonic oscillator one replaces
the phase space variables x(τ ) and p(τ ) with a(τ ) and a† (τ ) and then quantizes the theory
replacing Poisson brackets with commutators. We will proceed in a similar way with the closed
string, introducing infinitely many creation and destruction operators. Let us begin by defining,
P m (τ, σ) ≡ √
1 m
Ẋ (τ, σ) + X 0m (τ, σ) ,
2α0
P̄ (τ, σ) ≡ √
1 m
Ẋ (τ, σ) − X 0m (τ, σ) .
2α0
m
(2.25)
And further define the analogues of the creation and annihilation operators, αnm (τ ), ᾱnm (τ ),
Pm ≡
X
αnm (τ )e−inσ
,
P̄ m ≡
X
ᾱnm (τ )einσ .
(2.26)
n
n
We will now find the most general mode expansion for the X m fields. We start by isolating X 0m
and integrating it using Eqs. (2.25):
∂X m (τ, σ)
=
∂σ
s
α0 X m
αn (τ )e−inσ − ᾱnm (τ )einσ
2 n
s
⇒
X m (τ, σ) = X0m (τ ) + i
⇒
α0 X 1 m
αn (τ )e−inσ + ᾱnm (τ )einσ .
2 n6=0 n
(2.27)
(2.28)
Now, taking the derivatives of P in Eq. (2.25) and using Eq. (2.26) together with the linear
independence of the exponentials one finds,
Ṗ m = P m0
⇒
X
n
α̇nm (τ )e−inσ =
X
−inαnm (τ )e−inσ
⇒
α̇nm (τ ) = −inαnm (τ ).
(2.29)
n
Proceeding similarly for ᾱnm (τ ) and solving the differential equations one finds,
αnm (τ ) = αnm (0)e−inτ ≡ αnm e−inτ
,
8
ᾱnm (τ ) = ᾱnm (0)e−inτ ≡ ᾱnm e−inτ .
(2.30)
2.1
Introduction to String Theory
Furthermore, one can find X0m (τ ) using the definition of the total momentum pm , the center of
mass momentum, of the string,
pm ≡
Z π
dσP m ≡
Z π
dσ
−π
−π
∂L
=
∂ Ẋm
Z π
dσ
−π
Ẋ m
1
=
0
2πα
2πα0
Z π
dσ
−π
X
einσ Ẋnm (τ ) =
n
Ẋ0m (τ )
. (2.31)
α0
Let us furthermore, perform the following integration and using the results in Eqs. (2.25), (2.26)
and (2.31) to find,
√
1
2α0 π
Z π
−π
dσP m =
1
=√
2α0 π
1
2α0 π
Z π
dσ
−π
X
Z π
−π
dσ Ẋ m + X m0 =
αnm (τ )e−inσ
=
r
n
1 m
2π
Ẋ
(τ
)
+
0
= pm =
0
2α0 π
(2.32)
√
2α0 α0m (τ ).
(2.33)
2 m
α
α0 0
⇒
Ẋ0m (τ ) =
And so it is easy to solve now for X0m (τ ) and find,
X0m (τ ) = X0m (0) +
Z
√
2α0
τ
0
α0m dτ
⇒
X0m (τ ) = X0m (0) + α0 τ P m .
(2.34)
Finally, one can use Eq. (2.30) and Eq. (2.34) to rewrite the mode expansion of the fields X M
in Eq. (2.28) as,
s
X0m (τ ) = X0m + α0 τ P m + i
i
α0 X 1 h m −in(τ +σ)
αn e
+ ᾱnm e−in(τ −σ) ≡
2 n6=0 n
(2.35)
m
≡ XLm (τ + σ) + XR
(τ − σ).
Where we have defined the right- and left-moving solutions,
s
α0 X 1 m −in(τ +σ)
α e
,
2 n6=0 n n
s
α0 X 1 m −in(τ −σ)
ᾱ e
.
2 n6=0 n n
X m α0 τ P m
XLm (τ + σ) = 0 +
(τ + σ) + i
2
2
X m α0 τ P m
m
XR
(τ − σ) = 0 +
(τ − σ) + i
2
2
2.1.4
(2.36)
Light-cone quantization
We now proceed to quantise the worldsheet theory described by Eq. (2.13). There are a number
of different ways of quantizing the string. We will use light-cone quantization, where the residual
gauge freedom is fixed in a non-covariant way but that allows the Virasoro constraints to be
solved and the spectrum of the theory to be described as a Fock space. Thus we gain simplicity
at the price of loosing manifest Lorentz invariance. More advanced formalisms, like covariant
BRST quantization, can be found in most of the texts cited above. We begin by introducing
space-time light-cone coordinates,
1
X ± = √ (X 0 ± X 1 ),
2
9
(2.37)
Chapter 2
Preliminaries
and using the index i = 2, ..., D − 1 for the remaining coordinates. In these coordinates the flat
metric is η+− = η−+ = −1 , ηij = δij and a vector V m becomes V− = −V + , V+ = −V − and
Vi = V i . The worldsheet metric is ds2 = −dξL dξR and so the residual gauge transformation
mentioned above is arbitrary reparametrizations
ξL → ξ˜L (ξL )
ξ˜R (ξR ),
,
(2.38)
and some Weyl rescalings. Now, the solution (2.15) to the wave equation is, for the coordinate
X + , X + (ξ) = X + (ξL )+X + (ξR ). One can then use (2.38) to set ξ˜L,R = 2X + and, together with
L
R
L,R
τ = (ξL + ξR )/2, one can fix gauge removing the conformal symmetries and breaking explicit
Lorentz invariance as follows,
X + (τ, σ) = τ.
(2.39)
Then the Virasoro constraints (2.20) are written as follows,
∂L XL− =
2
1
∂L XLi
2
,
−
∂R XR
=
2
1
i
∂R XR
.
2
(2.40)
And so X − can be solved in terms of the X i . Furthermore, the momentum can be seen to be
[16],
p− = −p+ = −
l
1
= − 0.
0
2πα
α
(2.41)
Now, using Eqs. (2.36) and (2.41) it is easy to see that we can rewrite the Laurent expansion of
the left- and right-movers as,
s
pi
xi
+ + (τ + σ) + i
XLi (τ + σ) =
2
2p
xi
pi
i
XR
(τ − σ) =
+ + (τ − σ) + i
2
2p
s
α0 X αni −in(τ +σ)
e
,
2 n∈Z−0 n
(2.42)
α0 X ᾱni −in(τ −σ)
e
.
2 n∈Z−0 n
(2.43)
Where xi and pi are the string center of mass position and momentum and the coefficients αni
and ᾱni are the amplitudes of the different string excitation modes for left and right movers,
respectively. Hence the final expansion of X i is,
s
pi
X i (τ, σ) = xi + + τ + i
p
"
#
α0 X αni −in(τ +σ) ᾱni −in(τ −σ)
e
+
e
.
2 n∈Z−0 n
n
(2.44)
The canonical conjugate momenta for the X i (τ, σ) are defined, as usual,
Πi (τ, σ) ≡
∂L
1
=
∂τ Xi (τ, σ).
i
∂(∂τ X )
2πα0
(2.45)
The theory can be now quantized by promoting the degrees of freedom to operators. Imposing
the following canonical commutation relations,
[X i , Πj ] = iδji
,
[x− , p− ] = −[x− , p+ ] = i
10
,
[xi , pj ] = iδji ,
(2.46)
2.1
Introduction to String Theory
one finds the following commutators,
i
i
[αm
, αnj ] = [ᾱm
, ᾱnj ] = mδ ij δm,−n
,
i
[αm
, ᾱnj ] = 0.
(2.47)
i
i
One thus obtains two infinite sets of harmonic oscillators with raising operators α−m
and ᾱ−m
for
m > 0. Hence the Hilbert space can be constructed by acting on a vacuum state |0i, annihilated
i and ᾱi , with the creation operators αi
i
by αm
m
−m and ᾱ−m . Now, after imposing conformal gauge
an arbitrary line was picked for σ = 0. The theory has to be invariant under redefinitions of this
line, meaning that it has to be invariant under translation in the σ direction, generated by,
Pσ =
Z l
0
dσΠi ∂σ X i =
XX
i
i
α−m
αm
−
XX
i m>0
i
i
ᾱ−m
ᾱm
≡ N − N̄ .
(2.48)
i m>0
Where the number operators for left and right moving oscillations have been defined:
N≡
XX
i
i
α−m
αm
,
N̄ ≡
i m>0
XX
i
i
ᾱ−m
ᾱm
.
(2.49)
i m>0
Since the translation generator must vanish we find the so-called "level matching condition":
N = N̄ .
(2.50)
This condition reflects the fact that the string does not have spacial points. Furthermore, it
turns out that the left- and right-moving sectors are independent and only related by the level
matching condition. Now, the Hamiltonian for the string vibrations is given by,
H=
1 N
+
N̄
+
E
+
Ē
.
0
0
α 0 p+
(2.51)
Where E0 and Ē0 are the zero point energies of the left and right movers. Furthermore, it can
be shown that,
E0 = Ē0 = −
D−2
.
24
(2.52)
From the mass-shell condition M 2 = −p2 , using Eqs. (2.51), (2.52) and for the critical dimension
D = 26, the mass-shell relation can be rewritten as,
2
D−2
M = 0 n + N̄ − 2
α
24
2
=
2
(N + N̄ − 2).
α0
(2.53)
As we mentioned before, the conformal symmetry is very important for the consistency of the theory. It turns out that it can be spoilt when quantizing due to the so-called "conformal anomaly".
Requiring that the conformal anomaly vanishes is equivalent to preserving Lorentz invariance
and only happens for D = 26 in the bosonic string theory, and D = 10 for superstring theory.
Hence the only dimension at which bosonic string theory is consistent is D = 26 and it is dubbed
"critical dimension". We will now study the spectrum of the theory. We will focus in the lightest
states, the states with smallest number of oscillators satisfying (2.50), according to (2.53). These
are:
11
Chapter 2
Preliminaries
Osc. Number
States
Mass
Mass for D = 26
Fields
N = N̄ = 0
|ki
M 2 = − 2(D−2)
12α0
α0 M 2 = −4
T
N = N̄ = 1
i ᾱj |ki
α−1
−1
α0 M 2 = 0
GM N ,BM N and φ
M2 =
2
α0
2−
D−2
12
At this stage an argument can be given for the critical dimension, although not a proof. In any
Lorentz invariant theory polarization states must transform in irreducible representations of the
little group. In the case of massless particles we can always write P = (E, 0, ..., 0, E) and so
the little group preserving this momentum is SO(D − 2), while for massive particles we can
write P = (M, 0, ..., 0) and the little group is SO(D − 1). Since i = 2, ..., D − 1, we see that the
i ᾱj |ki above transform as a second rank tensor of SO(D − 2) and must therefore be
states α−1
−1
massless. Requiring them to be massless implies D = 26.
As already advanced in the table above, the second rank tensor of SO(24) decomposes into three
irreducible representations that we name fields. These are the symmetric traceless second rank
tensor Gmn , or graviton, the antisymmetric part Bmn , or 2-form field, and the trace which is a
scalar field φ called dilaton. Besides, the theory also includes a tachyonic field T indicating and
instability in the theory vacuum. Luckily this field disappears in superstring theories. Superstring
theories include supersymmetry and, hence, fermions as opposed to bosonic string theory. The
fields Gmn , Bmn and φ are very important and will appear later in this work.
Let us now introduce a different formalism used for the Conformal Field Theory (CFT) living
in the WorldSheet, that of the Virasoro operators. We will just state some results following [16]
where the details can be consulted. Let us define,
P 2 = 2π(T00 + T01 ) ≡ 4πT++ ,
P̄ 2 = 2π(T00 − T01 ) ≡ 4πT−− .
(2.54)
In the quantum theory the Virasoro Operators are defined as follows,
Ln ≡
L̄n ≡
Z π
inσ
dσe
−π
Z π
−inσ
dσe
−π
T++
1
=
4π
T−−
Z π
1
=
4π
dσeinσ P 2 ≡
−π
Z π
−inσ
dσe
−π
1 X m n
:
α α
ηmn :,
2 m m n−m
1 X m n
P̄ ≡ :
ᾱ ᾱ
ηmn : .
2 m m n−m
(2.55)
2
Where ηmn is the flat metric in 26D and normal ordering has been introduced to avoid ordering
issues when treating operators. This prescription moves objects αnm with biggest n to the right.
In this language the classical Virasoro constraints in Eq. (2.20) can be rewritten as,
Ln = L̄n = 0,
12
∀n.
(2.56)
2.2
Introduction to Riemannian Geometry
And, upon quantization these conditions for a physical state become,
Ln |ψi = L̄n |ψi = 0,
∀n ≥ 1,
(2.57)
(L0 − L̄0 ) |ψi = 0,
(2.58)
(L0 + L̄0 − 2) |ψi = 0.
(2.59)
It is then, easy to see using Eq. (2.49) and Eq. (2.51) that for D = 26 Eq. (2.59) gives the massshell relation we already found in (2.53). This formulation of the mass-shell relation is easily
generalized to other set-ups by using Eq. (2.55) with the appropriate form of the oscillators. It is
easy to see that the condition in Eq. (2.58) gives just the level matching condition in Eq. (2.50).
2.2
Introduction to Riemannian Geometry
In this section we will do a brief reminder of Riemannian geometry. We will follow [17, §. 7] and
[18]. The basic element of Riemannian geometry is the metric. Given a differentiable manifold
M, it is a Riemannian manifold if it is equipped with a rank (0, 2) tensor g that is symmetric
and positive definite. This object is called a Riemannian metric. If the object is not positive
definite but g(U, V ) = 0 implies V = 0 for a non-zero vector U , it is called a pseudo-Riemannian
metric. The results that follow also apply to pseudo-Riemannian metrics.
The diffeomorphisms, or coordinate transformations, in this manifold are governed by the Lie
derivative, which measures how a given tensor transforms, when parallely transported an infinitesimal distance along a vector X. In a coordinate basis eµ ≡ ∂/∂X µ the Lie derivative is
written,
LX Y = (X µ ∂µ Y ν − Y µ ∂µ X ν )eν .
(2.60)
Now, we would like derivatives of tensors such as ∂µ gνδ to transform as tensors as well, but that
is not the case in general. We define a covariant derivative that transforms as a tensor. In a
coordinate basis,
∇µ Y ν ≡ ∂µ Y ν + Γλµν Y ν .
(2.61)
Where the Christoffel connection Γλµν measures the failure of the normal derivative to transform
as a tensor. A connection is said to be metric compatible, or simply called metric connection, if
it verifies,
∇µ gνδ = 0.
(2.62)
Solving this constraint for the Christoffel connection yields,
Γλµν =




λ










µν 

+
1 λ
Tν µ + Tµλν + T λµν .
2
13
(2.63)
Chapter 2
Where
Preliminaries
 



λ

are the Christoffel symbols, defined as,



µν 





λ










µν 

1
≡ g λκ (∂µ gνκ + ∂ν gµκ − ∂κ gµν ).
2
(2.64)
And T λµν is the torsion tensor, defined as,
T λµν ≡ 2Γλ[µν] .
(2.65)
Hence, the torsion corresponds to the antisymmetric part of the connection. If the torsion tensor
vanishes on the manifold M, the metric connection ∇ is called a Levi-Civitta connection and is
completely fixed by the metric as follows,
1
Γλµν = g λκ (∂µ gνκ + ∂ν gµκ − ∂κ gµν ).
2
(2.66)
Now, one may study how a tensor is parallelly transported along a closed loop. This gives an
intrinsic notion of curvature. The failure to remain the same upon this translation is described by
the Riemann curvature tensor, that can also be described as the failure of covariant derivatives
to commute,
λ
[∇µ , ∇ν ]V λ = Rµνκ
V κ − Tµνκ ∇κ V λ .
(2.67)
Rκλµν ≡ ∂µ Γκνλ − ∂ν Γκµλ + Γηνλ Γκµη − Γηµλ Γκνη .
(2.68)
With,
For a Levi-Civitta connection, the Riemann tensor satisfies the following identities,
Rκλµν = −Rκλνµ ,
Rκλµν = −Rλκµν ,
Rκλµν = Rµνκλ .
(2.69)
Rκ[λµν] = 0,
(2.70)
∇κ Rξλµν + ∇µ Rξλνκ + ∇ν Rξλκµ = 0.
(2.71)
It also satisfies the following Bianchi identities,
Rκλµν + Rκµνλ + Rκνλµ = 0
−→
Tracing the Riemann tensor one obtains the Ricci tensor and, further tracing, one arrives at the
Ricci scalar or curvature scalar,
Rµν ≡ Rµκνκ = Rνµ ,
R ≡ Rµµ .
(2.72)
(2.73)
The action of general Relativity is, in vacuum,
Z
√
dx −gR.
14
(2.74)
2.2
Introduction to Riemannian Geometry
Where R is the scalar of curvature. And so the equations of motions read,
Rµν = 0.
(2.75)
The solutions to the equation above are called Ricci flat. Note that all the curvature tensors are
completely specified by the metric for a Levi-Civitta connection. All this geometry can also be
described in a non-coordinate fashion using vielbeins,
gµν = eᾱµ eβ̄ν ηᾱβ̄ .
(2.76)
Where ᾱ, β̄ are "flat" indices. Hence, the vielbein eᾱµ provides us with a flat frame at every point
in space-time. In this formulation one also introduces an Spin Connection Wµᾱβ̄ so that,
∇eᾱν = ∂µ eᾱν + Γµκν eᾱκ − Wµᾱβ̄ eβ̄ν .
(2.77)
And metric compatibility ∇µ eᾱν = 0 implies
Wµᾱβ̄ = Ωγ̄ ᾱβ̄ eγ̄µ + Γµνκ eᾱν eβ̄κ .
(2.78)
Where the Weitzenbock connection has been defined,
Ωᾱβ̄γ̄ ≡ eᾱµ ∂µ eβ̄ν eγ̄ν .
(2.79)
The formulation of Riemannian geometry presented above is coordinate dependent. A more
general formulation can be done in a coordinate independent way. The coordinate dependent
formulation is useful for making calculations and it can be derived from the more fundamental
coordinate independent formulation. Let us briefly review the coordinate independent formulation of Riemannian geometry following [17].
The Lie derivative can be defined as,
i
1h
(σ− )∗ Y |σ (X) − Y |X .
→0 LX Y ≡ lim
(2.80)
Where is a parameter and (σ− )∗ is the pushforward along the flow σ generated by the vector
field X µ a parameter distance −. The connection is then defined as a map taking two vector
fields and producing a third one with the following properties,
∇ : (X, Y ) 7→ ∇X Y,
(2.81)
∇X (Y + Z) = ∇X Y + ∇X Z,
(2.82)
∇(X+Y ) Z = ∇X Z + ∇Y Z,
(2.83)
∇X (f Y ) = f ∇X Y,
(2.84)
∇X (f Y ) = X[f ]Y + f ∇X Y,
(2.85)
∇X f = X[f ].
(2.86)
15
Chapter 2
Preliminaries
Where X, Y, Z are vector fields and f is a function on the manifold M . The directional derivative
of f along the flow of X µ is designed X[f ]. Metric compatibility is then written as,
∇V [g(X, Y )] = 0
∀ V ∈ Υ(M ).
(2.87)
Where Υ(M ) is the space of all vector fields defined over the manifold M and the map g(X, Y )
is a metric map, that is, symmetric and positive definite (for Riemannian metrics). Then, the
curvature and the torsion tensors can be defined through the following maps,
T : Υ(M ) ⊗ Υ(M ) → Υ(M ),
(2.88)
T (X, Y ) ≡ ∇X Y − ∇Y X − [X, Y ];
(2.89)
R : Υ(M ) ⊗ Υ(M ) ⊗ Υ(M ) → Υ(M ),
(2.90)
(2.91)
R(X, Y, Z) ≡ ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z.
Where the Lie bracket has been denoted as [·, ·],
[X, Y ]f = X[Y [f ]] − Y [X[f ]],
LX Y = [X, Y ].
(2.92)
Let us finally introduce an equivalent formulation of the torsion and curvature tensors that will
be useful later. They can be formulated through the following maps,
D
E
R(X, Y, Z, W ) ≡ (∇X ∇Y − ∇Y ∇X − ∇[X,Y ] )Z, W ,
(2.93)
T (X, Y, Z) ≡ hT (X, Y ), Zi = h∇X Y − ∇Y X − [X, Y ], Zi .
(2.94)
Where the product h·, ·i is the inner product in the tangent bundle, hX, Y i = ηmn X m Y n
2.3
Introduction to Supergravity
In this section we will discuss some aspects of the action of the bosonic NS-NS sector of supergravity, also corresponding to the low energy limit of bosonic string theory. In the previous
section we found that the low energy effective theory for 26D bosonic string theory has a tachyon
T , a metric gmn , a two-form field bmn and a dilaton field φ. These fields depend on the space-time
coordinates X m . Ignoring the tachyon we can introduce the action for this effective theory [11,
§. 3.2.1],
SSG =
Z
√
1
d X −ge−2φ R − Hmnp H mnp + 4(∂φ)2 .
12
26
(2.95)
Where some constants have been absorbed by the fields, R is the 26D Ricci scalar and Hmnp
is the three-form field strength tensor corresponding to the two-form field. The three form field
16
2.3
Introduction to Supergravity
definition and its Bianchi Identity are as follows,
1
H = db = (∂m bnp )dX m ∧ dX n ∧ dX p
2
dH = 0
⇒
Hmnp = 3∂[m bnp] ,
⇒
(2.96)
(2.97)
∂[m Hmnp] .
Being the second identity implied by the nilpotency of the d operator, d2 = 0, and where we
have used both differential form notation and explicit tensor notation. The square brackets imply
antisymmetrized indices. The action has the following local symmetries [18]:
• General Diffeomorphisms: the action is explicitly covariant and thus is invariant under
general coordinate transformations parametrized by the infinitesimal vector ξ m :
δgmn = Lξ gmn ,
δbmn = Lξ bmn ,
δφ = Lξ φ.
(2.98)
Where Lξ stands for the Lie derivative, defined as usual (see [17, §. 5.3.2] or Eq. (2.80))
and, for a vector is defined as,
Lξ V m = (ξ n ∂n V m − V n ∂n ξ m ) = [ξ, V ]m .
(2.99)
Where [ξ, V ]m is the Lie bracket. Using the Leibniz rule, the Lie derivative of the fields in
our theory is found to be,
Lξ gmn = ξ p ∂p gmn + gmp ∂n ξ p + gpn ∂m ξ p ,
Lξ bmn = ξ ∂p bmn + bmp ∂n ξ + bpn ∂m ξ ,
p
p
p
Lξ φ = ξ ∂m φ.
(2.100)
m
• Gauge principle of the two-form: as we have seen, the three-form H is closed and the
action will hence enjoy the following gauge symmetry,
δb = dΛ
⇒
δbmn = ∂[m Λn] .
(2.101)
Where Λ is some 1-form. Since the field strength three-form H is invariant under this
transformation and the other fields do not transform, the action remains invariant.
17
Chapter 3: T-Duality
In this section we will discuss the toroidal compactification of String Theory and how T-Duality
arises. We will start by illustrating the idea of compactification by discussing compactification
in field theory. We will the continue by compactifying string theory on a circle and showing how
T-duality arises in this set-up. Finally, string theory will be compactified in a general torus and
T-duality in this background will be discussed.
3.1
Kaluza-Klein Compactification
Compactification in field theory, as first proposed by Nordström [19], Kaluza [20] and Klein
[21], tried to unify electromagnetism and gravity by suggesting that we live in a compactified
4D space-time embedded in a 5D one with one compact dimension, being this compactified
dimension small enough as not to be detected in current experiments. We know that the four
dimensions we live in were once highly curved, so it is natural to conceive compact extra dimensions. In this theory, as we will show, the gauge symmetry of electromagnetism and the 4D
diffeomorphism invariance arise from 5D diffeomorphism invariance in a natural manner. We
will follow the modern treatment in [13, §. 8.1].
Let us consider a D-dimensional space-time with one toroidally compact coordinate, X d ,
∼ X d + 2πR.
Xd =
(3.1)
Where R is the radius of the compactified dimension and d = D − 1. We use indices M, N =
0, 1...D − 1 and µ, ν = 0, 1...d − 1 The D-dimensional Gmn metric will decompose into Gµν , Gµd
and Gdd , corresponding to the d-dimensional metric, a d-dimensional vector (or 1-form) and a
d-dimensional scalar, respectively. Explicitly, the metric can be written as,
ds2 = Gmn dX m dX n = G0µν dX µ dX ν + Gµd dX µ dX d + Gdµ dX d dX µ + Gdd dX d dX d ≡
≡ Gµν dX µ dX ν + Gdd (dX d + Aµ dX µ )2 .
(3.2)
Where we have defined Gµd ≡ Gdd Aµ . Note that the 4D fields are only allowed to depend on the
non-compact coordinates, as the physics should not depend on the compact ones. It is easy to
check that this parametrization has d-dimensional diffeomorphism invariance X 0µ (X ν ). There
also has to be the residual invariance under X 0d = X d + λ(X µ ). For the metric to be invariant,
the vector field Aµ must transform as
A0µ = Aµ − ∂µ λ(X µ ).
18
(3.3)
3.2
Circular Compactification in String Theory
Hence, gauge invariance naturally arises from the higher dimension diffeomorphism invariance.
Let us now consider how does the periodicity in the X d coordinate affect an scalar massless
field φ(X m ). Invariance under translations of 2πR in the X d direction imply quantization of
momentum pd = n/R so one can expand the field as,
φ(X m ) =
X
φn (X µ )exp inX d /R .
(3.4)
n
And, using the linear independence of the exponentials, the equation for the D-dimensional
scalar field decomposes in n equations,
"
∂ ∂m φ(X ) = 0
m
m
⇒
X
n
⇒
#
inX d
n2
∂ ∂µ φn (X ) − φn (X ) 2 e R = 0
R
µ
µ
∂ µ ∂µ φn (X µ ) = φn (X µ )
µ
n2
.
R2
⇒
(3.5)
(3.6)
Thus, from the d-dimensional point of view, the massless higher dimensional scalar field decomposes in a massless mode and and infinite tower of massive Kaluza-Klein (KK) modes. The
mass of the KK modes scales with R−1 and at energies much smaller than this scale the theory
will be that of a massless scalar field in d-dimensions. If we were to carry out experiments at a
higher energy we would produce heavier particles with conserved quantum number n. Hence a
particle moving in the compact dimension would not disappear from the D-dim point of view.
It would just have non-zero n [22]. This compactification procedure has been generalized for
higher dimensions. More fields are obtained but the basics are still unchanged.
3.2
Circular Compactification in String Theory
As we have mentioned string theory is only consistent in either 26 or 10 dimensions for bosonic
string theory and superstring theory respectively. In order to connect with the 4 dimensions we
live in, some of these dimensions should be compact enough for us not to detect them. In this
section we consider the simplest of such scenarios, compactifying 26D bosonic string theory in
a circle. We will follow[23] and [16] but also [11], [24], [25], [22] and [26]. The main difference
when compactifying dimensions between field theory and string theory is that closed strings may
wrap around the compact dimensions. We consider X 25 compactified on a circle and length of
the worldsheet σ coordinate l = 2π, with range −π < σ ≤ π. The boundary conditions for the
uncompactified dimensions remain unchanged and are given by (2.21). However, the periodicity
in the X 25 coordinate implies modifying the boundary conditions for that coordinate as follows,
X 25 (τ, σ + 2π) = X 25 (τ, σ) + 2πRω,
ω ∈ Z.
(3.7)
Where ω is the winding number and R is the radius of the compact dimension. Hence 0 ≤ X 25 ≤
2πR. The winding number counts the number of times the string winds around the compact
19
Chapter 3
T-Duality
Figure 3.1: Strings winding around the S1 compactified dimension. String 1 has ω = 0 as it does
not wind. Strings 2, 3 and 4 have winding numbers ω = 1, ω = −1 and ω = 2, respectively.
Image taken from [11]
direction, as seen in Fig. 3.1. The winding number is conserved in interactions [11]. From the
world-sheet point of view, states with non-zero winding number are topological solitons, states
with a non-trivial topology [13]. Let us now redefine the X 25 coordinate in order to have the
same boundary condition in both the compact direction and the non-compact ones,
X̃ 25 (τ, σ) = X 25 (τ, σ) − ωRσ.
(3.8)
The boundary condition then becomes X̃(τ, σ + 2π) = X̃ 25 (τ, σ). We can then use results from
section 2.1.3. Thus X̃ 25 admits a Fourier expansion,
X̃ 25 =
X
Xn25 (τ )einσ ,
(3.9)
n
and oscillators can be defined using Eqs. (2.26) and (2.25) for n = 25. Hence Eq. (2.28) is also
valid for the compact coordinate. However, the zero modes are changed as follows,
√
1
2α0 π
Z π
−π
1
2α0 π
Z π
1
= 0
2α π
Z π
dσP 25 =
−π
dσ Ẋ 25 + X 250 =
dσ
−π
X
e
inσ
X̃˙ n25 (τ ) + ωR
!
=
(3.10)
n
1
1
= 0 X̃˙ 0m (τ ) + ωR = 0 Ẋ0m (τ ) + ωR =
α
α
r
2 m
α .
α0 0
And we find, proceeding analogously with P̄ 25 one finds,
α025 =
1 25
Ẋ
(τ
)
+
ωR
,
0
2α0
ᾱ025 =
1 25
Ẋ
(τ
)
−
ωR
.
0
2α0
(3.11)
Since the equations of motion remain unchanged, Eq. (2.34) still holds. Using it together with
p25 = Ẋ025 /α0 and (3.11) we find,
s
α0 25
(α + ᾱ025 )τ.
2 0
(3.12)
2 25
ωR
ᾱ0 = p25 − 0 .
0
α
α
(3.13)
X025 (τ ) = X025 (0) + Ẋ025 (τ )τ = X025 (0) +
The left- and right-handed momenta are identified as,
pL =
r
2 25
ωR
α0 = p25 + 0 ,
0
α
α
pR =
20
r
3.2
Circular Compactification in String Theory
So that the total momentum verifies,
p25
0 ≡p≡
Ẋ025
= pL + pR .
α0
(3.14)
Putting (2.28), (3.12) and (3.13) together one finally finds the mode expansions,
s
α0 X αn25 −in(τ +σ)
e
,
2 n6=0 n
s
α0 X ᾱn25 −in(τ −σ)
e
.
2 n6=0 n
α0
25
XL25 (τ + σ) = X0L
+ pL (τ + σ) + i
2
α0
25
25
XR
(τ − σ) = X0R
+ pR (τ − σ) + i
2
(3.15)
25 +X 25 and X 25 (τ, σ) = X 25 (τ +σ)+X 25 (τ −σ). When the theory is quantized
Where X025 = X0L
0R
L
R
the momentum in the compact direction becomes discrete, just as it did for the Kaluza-Klein
compactification,
p25 =
m
,
R
(3.16)
m ∈ Z,
The left- and right-handed momenta become,
m
1
+ 0 Rω,
R
α
pL =
pR =
m
1
− 0 Rω.
R
α
(3.17)
In the uncompactified dimensions the quantisation is unchanged. However, the Virasoro operators for the compactified dimensions are now modified, as they are given in terms of the new
α025 , ᾱ025 in Eq. (3.11). Hence the mass-shell relation given in Eq. (2.53) becomes,
(L0 + L̄0 − 2) |ψi = 0
⇒
h
i
: 2N + 2N̄ + α0m α0n ηmn + ᾱ0m ᾱ0n ηmn − 4 : |ψi = 0
⇒
N + N̄ +
⇒
α0 2 α0 2
p + (pL + p2R ) − 2 |ψi = 0.
2
4
(3.18)
(3.19)
Where p2 = pµ pµ , being µ = 0, 1...24 and N, N̄ are given in Eq. (2.49). Now, using Eq. (3.17)
and Eq. (3.19) one finds the mass spectrum of the theory,
M 2 = −p2 =
m2 R 2 ω 2
2
+ 02 + 0 (N + N̄ − 2).
2
R
α
α
(3.20)
We then see four contributions to the mass: that coming from the momentum (m2/R2 ), potential
energy from the string winding (R2 ω2/α0 ), the oscillators energy (N + N̄ ) and the zero-point
energy (−2). Furthermore, the condition for a physical state given in Eq. (2.58), that gave the
level matching condition also gets modified as follows,
α0
α0
(L0 − L̄0 ) |ψi = 2N − 2N̄ + p2L − p2R |ψi = 0
2
2
⇒
h
i
N − N̄ + mω |ψi = 0.
(3.21)
So, for non-zero winding numbers the number of left- and right-movers is no longer equal,
ω 6= 0
⇒
21
N 6= N̄ .
(3.22)
Chapter 3
T-Duality
The spectrum can be easily found as we did in section 2.1.4, see [11] or [16] for further details.
One would then find a 2-form, a graviton, two vectors and one scalar. However, the mass depends
√
on the radius and if R = α0 additional modes become massless. Then the massless spectrum is
formed by a two-form, a graviton, six vectors and 10 scalars. Moreover, the gauge symmetry is
enhanced from U(1)L ×U(1)R to SU(2)L ×SU(2)R (see [16], [24]. The breaking of SU(2)L ×SU(2)R
down to U(1)L ×U(1)R is actually a Higgs mechanism.
If one looks at the mass-shell relation in Eq. (3.20) is is easy to see that the mass spectrum of
the theory is left invariant under the following transformation [23], [16], [24], [11],
√
R
α0
√ ←→
,
m ←→ n.
(3.23)
R
α0
Hence, the complete spectrum of the theory at radius R is the same as the one at radius α0/R,
provided the winding and momentum modes are interchanged. This is the simplest case of Target
Space Duality Symmetry (T-duality) and it is an stringy phenomenon coming from strings being
able to wrap around compactified dimensions. Thus the spectrum is the same for very big and
very small circles. This hints at strings not being able to probe arbitrarily small distances, this
is, that there is a minimum distance scale. However, although strings do indeed see a peculiar
geometry at small scales, there is non-perturbative structure below this scale [13].
From Eq. (3.17) we see that the T-duality transformation acts on the momenta as,
pL ←→ pL ,
pR ←→ −pR .
(3.24)
Thus, T-duality can be seen as a parity operation in the right-moving sector of the theory. The
invariance of the spectrum proves that the free theory, and hence the partition function at one
loop order, is invariant, but it does not ensure that the whole interacting theory is invariant
under T-duality transformations. To show that the full theory is indeed invariant under Tduality one must prove that the full partition function (to arbitrary genus contributions) is
also left invariant. A proof can be found in [23]. Finally, considering invariance of the couplings
under T-duality transformations one can see that it acts non-trivially in the dilaton field. See
for instance [13, §. 8.3] or [11, §. 3.2]. The resulting transformation is,
α0 /2 Φ
e .
=
R
1
Φ0
e
3.3
3.3.1
(3.25)
Toroidal compactification
Quantization and Physical constraints
In this section we will generalize to an arbitrary number of toroidally compactified dimensions
the procedure followed in the previous section for the circular compactification. Let us consider
22
3.3
Toroidal compactification
n toroidally compactified dimensions. The space-time will be the product of a compact n-torus
and a non-compact d-dimensional manifold M d × T n , d = 26 − n. We will mainly follow [16, §.
10.2], [12, §. 7.3], [23] and [13, §. 8.4]. The space-time metric will then have the following form,
ds2 = ηµν dX µ dX ν + Gij dY i dY j .
(3.26)
Where X µ are the non-compact coordinates and the Y i the compact ones, with µ, ν = 0, 1...d−1,
i, j = 1, 2...n and Gij is the internal metric describing the geometry of the compact space T d .
This metric will generally be non-diagonal for tori with non orthogonal circles. For a rectangular
torus, with all n circles perpendicular to each other, the metric is Gij = Ri2 δij , where Ri are the
radii of the circles in the different dimensions.
The action S = S U + S C has a contribution coming from the non-compact coordinates
S U and a contribution from the compactified coordinates,
1
S =−
4πα0
C
Z
d2 σ
h√
i
−gg ab ∂a Y i ∂b Y j Gij − ab Bij ∂a Y i ∂b Y j .
(3.27)
Where, as we discussed in 2.1.2, for the theories we are considering (Weyl anomaly free), the
metric in the worldsheet is conformally flat. In this action the 2-form field Bij has been allowed
to take non-trivial background values in the compact dimensions. This is usually refereed in
the bibliography as "Compactification with background B-field". The data of this CFT are d2
couplings encoded in a symmetric (Gij ) and antisymmetric (Bij ) matrices. This data can be
arranged for convenience in one matrix E with symmetric part G and antisymmetric part B,
Eij ≡ Gij + Bij .
(3.28)
The E matrix is called "Background Matrix". The boundary conditions for closed strings compactified on a torus are then, in analogy with Eqs. (2.21) and (3.7),
X µ (τ, σ + 2π) = X µ (τ, σ),
Y i (τ, σ + 2π) = Y i (τ, σ) + 2πω i ,
(3.29)
ω i ∈ Z.
(3.30)
Where the ω i are the winding numbers counting the times the string winds around each of the
torus cycles. Let us now introduce oscillators αni and ᾱni using Eqs. (2.26) and (2.49),
Pi ≡ √
i
P̄ ≡ √
X
1 i
Ẏ + Y 0i ≡
αni e−inσ ,
2α0
n
X
1 i
Ẏ − Y 0i ≡
ᾱni e−inσ .
0
2α
n
(3.31)
The generalization of the procedure followed in section 3.2 to find the mode expansion is straightforward. Essentially one just needs to rewrite everything adding the index labelling the different
23
Chapter 3
T-Duality
compact coordinates. For the uncompactified coordinates one finds, as usual, the mode expansion in Eq. (2.36). For the compact coordinates the mode expansion is the generalization of the
one found for the circular compactification (3.15):
α0
i
YLi (τ + σ) = Y0L
+ piL (τ + σ) + i
2
s
α0 X αni −in(τ +σ)
e
,
2 n6=0 n
s
α0
i
YRi (τ − σ) = Y0R
+ piR (τ − σ) + i
2
(3.32)
α0 X ᾱn25 −in(τ −σ)
e
,
2 n6=0 n
Y i (σ, τ ) = YLi (τ + σ) + YRi (τ − σ).
Where we have naturally defined,
piL
r
≡
2 i
α ,
α0 0
piR
r
≡
2 i
ᾱ .
α0 0
(3.33)
Substituting the modes expansions (3.15) in the boundary condition in Eq. (3.30) one finds the
following,
α0 i
pL − piR = ω i .
(3.34)
2
And, as before, to ensure the periodicity in the coordinates Y i we need to require that the
i
translation operator eiPi Y in single valued. This then implies,
ePi (Y
i +2πmi )
= e Pi Y
i
⇒
Pi = mi ,
(3.35)
mi ∈ Z.
Where Pi is the total momentum of the string, defined to be the zero mode of the canonical
momentum density pi . The canonical momentum density of action (3.27) is given by,
pi =
i
1 h j
δ Lc
0j
=
Ẏ
G
+
Y
B
.
ij
ij
2πα0
δ Ẏ i
(3.36)
Thus, the total momentum Pi , given by the zero mode of pi can be obtained upon integration,
1
Pi =
dσpi =
2πα0
−π
Z π
⇒ Pi =
Z π
dσ
−π
i
α0 h i
pL + piR Gij + piL − piR Bij
2
⇒
i
1 h i
pL + piR Gij + piL − piR Bij = mi .
2
(3.37)
(3.38)
Where we have used Eqs. (3.35) and (3.32). Now, substituting Eq. (3.30) into Eq. (3.38) and
using Eq. (3.34) one finds,
2piR Gij = 2mj −
2
(Gij + Bij )ω i
α0
⇒
2piR = 2Gij mj −
2 i
ik
k
δ
+
G
B
ω
.
kj
α0 j
(3.39)
And proceeding similarly for piL one finds the left- and right-handed momenta,
piR = −
piL
1 i
1
ω + Gij mj − 0 Bjk ω k ,
0
α
α
1
1
= 0 ω i + Gij mj − 0 Bjk ω k .
α
α
24
(3.40)
3.3
Toroidal compactification
Imposing canonical commutation relations, as we did in section 2.1.4, but with the mode expansion in (3.32), one can find the following commutation relations for the oscillators [23],
i
i
[αm
, αnj ] = [ᾱm
, ᾱnj ] = mGij δm,−n .
(3.41)
Also following section 2.1.4, the number operators are found to be [23],
N=
X
i
i
α−m
(E)Gij αm
(E) ,
N̄ =
m>0
X
i
i
ᾱ−m
(E)Gij ᾱm
(E).
(3.42)
m>0
Then the spectrum of the theory can be found in the usual way,
(L0 + L̄0 − 2) |ψi = 0
⇒ : N + N̄ +
α0 2 α0 2
pL + p2R − 2 |ψi .
p +
2
4
(3.43)
Where p2 = pµ pµ . Hence,
M 2 = −p2 =
1
2
2
2
p
+
p
.
N
+
N̄
−
2
+
L
R
α0
2
(3.44)
Where N , N̄ are, as usual, given by Eq. (2.49). Analogously, the level matching condition
becomes,
L0 − L̄0 |ψi = 0
N − N̄ =
⇒
⇒
α0 2
pR − p2L
4
N − N̄ = mi ω i .
⇒
(3.45)
(3.46)
This expression generalizes (3.21). The number of left- and right-movers is only equal when the
string does not wind around any compact dimension. Let us now study the expression for the
mass-shell relation (3.44),
α0 2ω 2
1
α0 2
pL + p2R =
+ 2 mj − 0 Bjk ω k
02
2
2 α
α
"
=
1
mi − 0 Bis ω s Gij =
α
ωk Gks + Bjk Gij Bis ω s + α0 mi Gij mj − mi Gir Brj ω j + ω j Bir Grj mj =
0
α
(3.47)


=
#
mi
 α0 Gij

i
ω 

Bir Grj
−Gir Brj
1
α0 (Gij
− Bir Grs Bsj )

 mj 
 
 .
 
ωj
Where the antisymmetry of Bij has been used. Let us now make the following field redefinitions,
suppressing indices, G → α0 G and B → α0 B. Then we can rewrite the mass-shell relation in Eq.
(3.44), using the result above, as follows,
α0 M 2 = 2(N + N̄ − 2) + Z T M Z.
(3.48)
Where we have defined the following quantities,
 
m
 
Z =  ,
 
ω

 G−1

M =

BG−1
25

−G−1 B
G−
BG−1 B


.

(3.49)
Chapter 3
T-Duality
Furthermore, it is easy to see that the Level Matching condition in eq. (3.46) can be rewritten
as follows,



 0 1  m 
n  j 

N − N̄ = mi ω i 
   = Z T JZ.
 

ωj
1n 0
(3.50)
Where 1n is the n × n identity matrix and we have defined,


 0 1n 


J ≡
.


1n 0
3.3.2
(3.51)
Moduli Space and duality group
In this section we will find the Moduli Space of the string theories compactified in Tori as
detailed in the previous section. We will also find that the T-Duality group from the circular
compactification is enlarged to the discrete group O(n, n; Z). We will mainly follow [23], [13, §.
8.4], [12, §. 7.3] and [16, §. 10.4].
Let us introduce some concepts from Lattice Theory that will be useful in the following. A
D-dimensional lattice is defined as,
Γ=
(N
X
)
(3.52)
ni ∈ Z .
ni e i ,
i=1
Where ei , i = 1...N are a basis of a vector space, V , of dim D. The lattice inherits the scalar
product from the vector space, and so its metric is the same one the vector space has, gij . We
will be interested in V = RD and V = RN−M,N . A lattice Γ is euclidean (lorentzian) if V is
euclidean (lorentzian). A lattice is called integral iff {x · y ∈ Z ∀ x, y ∈ Γ}. An integral lattice is
even if {x · x ∈ 2Z ∀ x ∈ Γ} and odd otherwise. Using this language a D-Torus can be defined
as the following coset space,
TD =
RD
.
2πΓD
(3.53)
Being Γ the lattice associated to V = R. The lattice dual to Γ is defined as,
Γ∗ : {y | y · x ∈ Γ ∀ x ∈ Γ}.
And a lattice is self dual iff Γ = Γ∗ . A lattice is unimodular if vol(Γ) =
(3.54)
p
|g| = 1. It can be
shown that a lattice is self-dual iff it is integral and unimodular [16]. Let us now define,
s
ωL ≡
s
α0
pL ,
2
ωR ≡
26
α0
pR .
2
(3.55)
3.3
Toroidal compactification
Then, the vector ΩT ≡ (ωL , ωR ) spans a lorentzian lattice Γ(d,d) with inner product given by
(3.46),
2
ΩT JΩ = ωL2 − ωR
= 2mi ω i ∈ 2Z.
(3.56)
And so the lattice is even with metric J and associated vector space Rd,d . Moreover, since
|det(J)| = 1, the lattice is unimodular. Since the lattice is even and unimodular, it is selfdual. Now, all even self-dual lorentzian lattices with signature (n, n) are known to be related by
O(n, n; R) transformations [23]. These rotations preserve the Lorentz product but do, generally,
2 ), hence modifying the spectrum of the theory (see Eq.
change the euclidean product (ωL2 + ωR
(3.44)). Thus, each lattice Γ(n,n) corresponds to a different theory, this is, a different toroidal
compactification.
The space that parametrizes all the possible inequivalent theories is called the Moduli Space. Let
us now study the space parametrizing the string theories compactified in a toroidal background
as described in the previous section, this is, its Moduli Space. Since all Lorentzian lattices are
related by O(n, n; R) transformations, the Moduli Space will be the coset space of this group
with the isotropy group, which is the group leaving the Lattice invariant. The lattice is invariant
2 are independently preserved. This preserves both physical conditions (3.44)
if both ωL2 and ωR
and (3.46). Thus, the isotropy group is O(n; R)L × O(n; R)R and the Moduli Space is, up to a
discrete symmetry that will be described in the following,
0
M(n,n)
=
O(n, n; R)
.
O(n; R)L × O(n; R)R
(3.57)
But, there is a discrete group of O(n, n; R) that permutes the individual points of the lattice
but takes the lattice Γ(n,n) to itself. This discrete group is O(n, n; Z), since it only relabels the
basis vectors of the lattice. Hence, upon this identification, the space of inequivalent theories is,
M(n,n) =
O(n, n; R)
.
O(n; R)L × O(n; R)R × O(n, n; Z)
(3.58)
We will now study how the background matrix in (3.28) transforms under the action of
O(n, n; R). The most general element of O(n, n; R) takes the matrix form,


a


b

O ∈ O(n, n; R) : 
c d


| O T JO = J.
(3.59)
Where a, b, c and d are n × n matrices. Thus, O(n, n; R) is the group of orthogonal matrices
preserving the metric J. This results in the following conditions for the n × n matrices,
aT c = −cT a,
bT d = −dT b,
aT d + cT b = 1,
bT c + dT a = 1.
Let us now define the vielbein for the metric Gij , e ≡ eαi , so that Gij =
(3.60)
P α α
ei ej or, removing
α
the indices, G = eT e, where the indices α, β... are the "flat indices". Let us now consider the
27
Chapter 3
T-Duality
following matrix,


e

F ≡

B(eT )−1 

(eT )−1
0
(3.61)
.

Making use of the fact that B T = −B, as B is antisymmetric, it is easy to observe that this
matrix satisfies F T JF = J. Hence F is an element of O(n, n; R). Consider now the following
map for some g ∈ O(n, n; R),
g(K) ≡ (aK + b)(cK + d)−1 .
g : GL(n, R) 7→ GL(n, R),
(3.62)
Where K is an n × n matrix and a, b, c, d are n × n matrices satisfying (3.60) for some g =
a b
c d
and thus, g ∈ O(n, n; R). Using this map for F we now consider its action on the identity,
F (1) = e + B(eT )−1 ((eT )−1 )−1 = eeT + B = G + B = E.
(3.63)
Furthermore, it is easy to see that,

FF
T

G − BG−1 B

=

−G−1 B
BG−1 

G−1
 = M −1 .

(3.64)
Where M is defined in (3.49). Thus F is a vielbein for the matrix M −1 . Hence, we are tempted
to treat M −1 as a metric, as some sort of "Generalized Metric". Indeed, a reader familiar with
generalized geometry will notice that this matrix has the same form as the "Generalized Metric"
in the context of generalized geometry [27]. However, they are not exactly the same. This "Generalized Metric" will appear again when we discuss Double Field Theory (DFT). Now, under an
O(n, n; R) transformation g, M −1 (E) becomes Mg−1 ≡ gM −1 (E)g T . Then one finds,
F 0 F 0T = M −1 (E 0 ) ≡ M −10 ≡ Mg−1 = gM −1 g T = gF F T g T
⇒
F 0 = g(F ).
(3.65)
So, using (3.63) we finally find the transformation of the background metric under the action of
O(n, n; R),
E 0 = F 0 (1) = g(F (1)) = g(E) = (aE + b)(cE + d)−1 .
(3.66)
This transformation details how all different toroidal backgrounds are related by O(n, n; R)
transformations. The transformation for the metric is more complicated, hinting that the background matrix E will be a better object to consider when formulating a T-duality invariant
theory. It can be shown, using the fact that B T = −B, that G0 = (E 0 + E 0T )/2. Using this
together with Eq. (3.66) and the analogous transformation for E 0 one can show that G0 satisfies
[23],
G = (d + cE)T G0 (d + cE),
G = (d − cRT )T G0 (d − cE T ).
28
(3.67)
3.3
Toroidal compactification
For a transformation to be a symmetry of the string, the canonical commutation relations in
Eq. (3.41) must be preserved. This fixes the transformation of the oscillators to be [23],
αm (E 0 ) = (d − cE T )αm (E),
ᾱm (E 0 ) = (d + cE)ᾱm (E).
(3.68)
It is easy to see that Eqs. (3.66) and (3.68) imply that the number operators N and N̄ in Eq.
(3.42) are left invariant under duality operations.
One may now study if there are equivalent backgrounds, this is, if there is any residual symmetry. As we have already explained, the transformations under the O(n, n; Z) group will produce
physically equivalent theories and generalize the T-duality symmetry we found when compactifiying in a circle. To proof that two theories are equivalent (dual), one needs to show that all
correlation functions are the same in both theories. However, we will only consider the necessary
condition (and sufficient at one loop order) that the spectrum (3.44) and the level-matching condition (3.46) are left invariant by the duality transformations. We will now study more deeply
the action of the group O(n, n; Z) on the background matrix E, given by Eq. (3.66) with a, b, c, d
integer-valued matrices, [28], [23], [16, §. 17.1]. Let us consider the transformations that leave
the spectrum and the level-matching condition invariant.
• B-shift: Let us first consider the following shift,
1
Bij → Bij + Θij ,
2
i
Θij = −Θji ∈ Z,
(3.69)
mi → mi + Θij ω .
i
j
ω →ω,
In matrix notation this corresponds to taking a = d = 1 and b = Θ,


1 Θij 


.


gB = 
0
(3.70)
1
It is easy to see that both the mass-shell relation and the level-matching condition are left
invariant,
αM 2
→
2(N + N̄ − 2) + Z T g T g T −1 M g −1 gZ = 2(N + N̄ − 2) + Z T M Z,
N − N̄
Z T g T JgZ = Z T JZ.
→
(3.71)
(3.72)
• Basis Change: Let us now consider transformations with b = c = 0, a ∈ GL(n, Z). Then,
using (3.60), d = (aT )−1 and the transformation matrix is then just,


a
0 


gC = 
.


T
−1
0 (a )
29
(3.73)
Chapter 3
T-Duality
Showing that the physical conditions are invariant is straightforward. We also see, using
(3.66), that this transformations acts on the background matrix as E 0 = aEaT . This is
just a basis change on the compactification lattice Γ(n,n) , that is, the large diffeomorphisms
preserved by the torus.
• Factorized Duality: Let us finally consider a transformation such that b = c = DK , where
DK is a n×n matrix with only one non-zero entry, the KK component, which is DK(K,K) =
1. Then, using (3.60) one finds the following transformation matrix,


1 − DK

gD = 

DK
DK 

1 − DK
(3.74)
.

This transformation generalizes the R → α0 /R we found in section 3.2 for the X K direction.
This is easy to see by setting B = 0 and setting the metric to be diagonal. Then the metric
transforms as follows,
G0 = (1 − DK + DK G)T −1 G(1 − DK + DK G)−1
⇒
G0KK =
1
,
GKK
G0ii = Gii ;
⇒
i 6= K.
(3.75)
(3.76)
It is easy to see that, again, the mass-shell and level-matching conditions are left invariant.
It can be shown that any transformation in O(n, n; Z) can be decomposed as a product of these
transformations [16], [23], so we have fully decomposed the group. There is an extra symmetry of
the theory not contained in O(n, n; Z) coming from the parity symmetry in the worldsheet [28].
This symmetry is realized in the action (3.27) as σ → −σ. Using the antisymmetry of ab we see
that this transformation is equivalent to B → −B and, thus, E → E T . This transformation does
not belong to O(n, n; Z) as it flips the sign of the lorentzian norm preserved by the orthogonal
group.
3.3.3
The Buscher Approach - Abelian Duality
In this section we will consider Buscher’s approach to T-Duality. We will consider the case of an
abelian duality. In his papers [29] and [30] Buscher studied T-duality transformations from the
world-sheet point of view with background fields. We will mainly follow Buscher original work,
as well as [31].
Let us consider a non-linear σ-model with B-form and dilaton φ fields present,
1
S=−
4πα0
Z
d2 σ
h√
i
√
−gg ab Gmn ∂a X m ∂b X n − ab Bmn ∂a X m ∂b X n + α0 −gR(2) φ(X) .
30
(3.77)
3.3
Toroidal compactification
Where R(2) is the world-sheet Ricci scalar. We now assume that there is an abelian isometry in
the θ direction, restricting ourselves to θ spacelike,
α = 1, 2, ...d − 1.
{X m } ≡ {θ, X α },
(3.78)
We can choose Gmn , Bmn and φ to be independent of θ, as we can always locally choose a killing
vector to be K = ∂/∂θ [32]. Let us now rewrite the action in an equivalent form substituting
∂a X 0 by Va and introducing the Lagrange multiplier θ̃,
S0 = −
1
4πα0
Z
d2 σ
h√
−gg ab (G00 Va Vb + 2G0m Va ∂b X m + Gmn ∂a X m ∂b X n )−
− (2B0m Va ∂b X + Bmn ∂a X ∂b X ) − 2 θ̃∂a Vb + α
ab
m
m
ab
n
0√
−gR
(2)
(3.79)
i
φ(X) .
If this action is extremized with respect to θ̃ one finds,
1
δS 0
=
4πα0
δ θ̃
Z
h
i
d2 σ 2ab ∂a Vb = 0
⇒
∂[a Vb] = 0.
(3.80)
This can be rewritten in form notation as dV = 0, being V a 1-form. Hence, V is closed and
using the Poincaré Lemma, in a contractible patch we may write V = dw for some 0-form w.
Relabelling w = θ one finds the original action. Hence we recover the original action, so the
two are indeed equivalent. Now, if one instead extremizes with respect to Va , one finds, up to
boundary terms,
δS 0
−1
=
δVa
4πα0
Z
d2 σ
h√
−gg ab (2G00 Vb + 2G0m ∂b X m ) − 2ab B0m ∂b X m − ∂b θ̃
i
=0
#
"
−1
b Va =
B0m ∂b X m − ∂b θ̃ .
G0m ∂a X m − √a
G00
−g
⇒
⇒ (3.81)
(3.82)
And substituting (3.82) in (3.79) one finds,
1
S̃ = −
4πα0
Z
"
2
d σ
√
−gg ab
−B0m B0n + G0m G0n
∂a X n ∂b X m +
G00
Gmn −
B0m
1
+2
∂a X m ∂b θ̃ +
∂a θ̃∂b θ̃ −
G00
G00
ab
−
#
√
G0m
B0m G0n − G0m B0n
∂a X n ∂b X m + 2
∂a X m ∂b θ̃ + α0 −gR(2) φ .
Bmn −
G00
G00
(3.83)
One can further do the following identifications, known as Buscher Rules [29] [30], to find a dual
action,
G̃mn = Gmn −
B̃mn
G0m G0n − B0m B0n
,
G00
B0m G0n − G0m B0n
= Bmn −
,
G00
G̃0m =
B̃0m
B0m
,
G00
G0m
=
.
G00
G̃00 =
1
,
G00
(3.84)
These identifications lead to a dual theory with action S̃. The fact that the two actions are
equivalent guarantees classical equivalence, however, quantum equivalence is more subtle as
31
Chapter 3
T-Duality
the conformal invariance might be spoilt, as we have already mentioned. For instance, in this
case, the equivalence of both theories at one-loop level requires the dilaton field to transform
non-trivially[30],
φ̃ = φ −
1
log G00 .
2
(3.85)
This ensures conformal invariance at one-loop level. In order to prove that both theories are completely equivalent (dual) one needs to show that they are equivalent on all-genera worldsheets,
leading to consider global issues.
32
Chapter 4: Double Field Theory
4.1
Motivation
In section 3.2 we saw that string theory compactified on a circle remains invariant under the
following transformation,
√
α0
R
√ ←→
,
m ←→ n.
(4.1)
R
α0
This means that string theory compactified on a big circle is equivalent to string theory compactified on a small circle, provided that we change the winding and momentum modes. This
duality might seem very strange at first as momentum modes (discrete momentum along the
compact direction) are very different from winding modes since the former is a fundamental
attribute of the string and the latter is a topological property of the state. However, there are
dualities even in Yang-Mills theories casting a similar doubt on what we mean as fundamental.
As an example, Montonen-Olive duality, first developed in [33], relates gauge bosons and magnetic monopoles. Hence, the concept of fundamental particle is blurred. Moreover, this T-duality
found in the circular compactification puts the momentum modes and the winding modes on the
same footing as they are interchanged by its action. In section 3.3 this T-Duality symmetry was
generalized to compactifications in an arbitrary torus T n . In this case, we saw that T-Duality
acts as an O(n, n; Z) transformation in a new object M −1 that groups all degrees of freedom
which we called generalized metric. In this case even the 2-form field and the metric get mixed
by the action of T-duality. We thus see that the way strings behave, and the geometry they see,
is extremely different from that of point particles.
However, some of these degrees of freedom are lost when one studies the low energy limit we
introduced in section 2.3. Indeed, the winding modes are very heavy and are discarded in the
low energy dynamics. Supergravity describes the low energy particle limit of string theory and,
since particles cannot wind, all phenomena related to the winding of strings is lost. At this point
it becomes clear that a field theory incorporating winding modes would allow us to study new
properties of string theory. This is exactly the aim of Double Field Theory (DFT). However,
it is non-trivial how to incorporate winding modes to a particle theory, since particles cannot
wind. It is clear, however, that one needs to add new degrees of freedom to those particles to
account for the winding modes. In DFT this is achieved by doubling the number of coordinates
of the space the particles live in.
As we have seen, string theory in this set-up is invariant under T-duality O(n, n; Z) transforma33
Chapter 4
Double Field Theory
tions and it is precisely this symmetry what implies that winding and momentum modes are to
be treated together. Hence, a good way of keeping the winding modes in a field theory would be
to enforce an O(n, n) T-duality symmetry in the field theory. This is precisely what DFT does.
Double Field Theory is a T-duality invariant field theory generalizing supergravity to include
some stringy effects.
4.2
Doubling Coordinates
We consider string theory with space-time Rd × T n , d = 26 − n, as we did in section 3.3. Again,
the coordinates X m , M = 0, 1...25, will be decomposed into non-compact directions X µ , µ, ν =
0, 1...d−1, and compact directions Y i , i, j = 1, 2...n. We saw that, in the compactified dimensions
one finds two discrete momenta, the normal one Pi = mi and the winding momentum P̃ i . In
Double Field Theory we want to keep the dependence in the winding momentum that was lost
in supergravity. Moreover, the fields will also depend on the momentum in the uncompactified
direction Kµ . Thus a particular field will have the dependences φ(Kµ , Pi , P̃ i ). If one Fourier
transforms the field, one obtains φ(X µ , Y i Ỹi ), where each coordinate is dual to a momentum as
follows,
Kµ → X µ ,
Pi → Y i ,
P̃ i → Ỹi .
(4.2)
One then finds the usual "physical" coordinates (X µ , Y i ) and some extra coordinates Ỹi dual
to the winding momentum. This extra coordinates will be compact, as they are the Fourier
transforms of discrete momenta. The (Y i , Ỹi ) are then the coordinates of a doubled torus T 2n .
Since T-duality relates winding and momentum modes, it will also relate Y i and Ỹi , so they are
dual to each other. It proves useful to introduce coordinates dual to the non compact coordinates
X µ , X̃µ . These coordinates do not enter the dynamics as the fields do not depend on them, but
they will ease the notation. In a similar way, the momentum dual to Kµ (and Fourier transform
of X̃µ ) can be introduced: K̃ µ . It is in this sense that DFT is a Doubled Theory, it doubles the
number of coordinates to account for the winding degrees of freedom. We can then group the
compact coordinates of T 2n in O(d, d) objects,


 Ỹi 
 
Y =  .
 
Yi
(4.3)
If all dimensions are compact (d = 0) this object has 2D elements and it behaves as an O(D, D)
vector as it transforms in the fundamental representation and the theory has an O(D, D; Z)
symmetry. However, if there are d non-compact dimensions, there are 2d periodic coordinates
and only O(d, d; Z) ⊂ O(D, D; Z) will be a symmetry of the theory [5]. It proves useful to
34
4.2
Doubling Coordinates
introduce 2D coordinates,




P̃ i
 Ỹi 
 






 X̃µ   X̃m 

 

Xi =   =   .
 


Y i
Xm








 




 µ
m
K̃  P̃ 

 

Pi =   ≡   ,
 


P 
Pm
 i




(4.4)
Xµ
Kµ
Where the O(D, D) covariant indices I, J = 1, 2...2D have been introduced. There is a natural
action of O(D, D) on the Xi coordinates,





a b   X̃m  aX̃ + bX 

 


X → gX = 
.
=


 


m
cX̃ + dX
X
c d
(4.5)
Where, as usual, we restrict to the discrete subgroup g ∈ O(D, D; Z). Furthermore, since there
are non-compact coordinates, we must only consider elements g ∈ O(d, d; Z) ⊂ O(D, D; Z)
leaving X µ invariant. Thus, the D × D matrices a, b, c, d are constrained to be,


â

a=

0

0 1d
,




0

 b̂

b=

0 0

 ĉ

c=

,


dˆ

d=

0

0 0
,


0

0 1d
.

(4.6)
Where â, b̂, ĉ, dˆ are d × d matrices such that,


â b̂ 


ĝ ≡ 
 ∈ O(d, d; Z).


ĉ dˆ
(4.7)
So that,

â 0



 0 1d

X 0 = gX = 

 ĉ 0



0
0

b̂
0
dˆ


0   Ỹi  aỸ + b̂Y 









 X̃µ  

0
X̃





.



i




0   Y  ĉỸ + dY 







0 1d
Xµ
(4.8)
X
Which is indeed the correct transformation for both compact and non-compact coordinates.
Using this notation the background fields are written as follows,


Ĝij


Gmn = 
0

0 

ηµν

B̂ij


Bmn = 
,

35
0
0

.

0
(4.9)
Chapter 4
Double Field Theory
Where Ĝij and B̂ij are the metric and the two-form in T n and Rd has a flat metric ηµν . The
background matrix defined in Eq. (3.28) becomes:


Êij

Emn = Gmn + Bmn = 

0
0 

ηµν
,

Êij ≡ Ĝij + B̂ij .
(4.10)
We will also use the O(D, D) object M −1 defined in Eq. (3.64) that looked like a "Generalized
Metric". For notational purposes let us define,


 G−1
−G−1 B 


Hij (E) ≡ M (E) = 
.


−1
−1
BG
G − BG B
(4.11)
And let us also recover the invariant O(D, D) metric,


 0 1D 


J =
,


1D 0
g ∈ O(D, D)
iff
g T Jg = J.
(4.12)
Being an invariant tensor, J can be used to raise and lower indices,
Hij = J ik J jl Hkl ,
J ik Jkj = δji .
(4.13)
Finally, we have seen that the physical conditions can be written, using this formalism, as follows,
2
αMD
= 2(N + N̄ − 2),
N − N̄ = P T JP.
(4.14)
2 ≡ M 2 − α0−1 P T M P . In this work we will
Where MD is the D-dimensional mass, defined as MD
be interested in the massless fields and so, we will restrict ourselves to N = Ñ = 1 which is the
only possibility fulfilling both physical conditions.
4.3
Quadratic O(D, D) invariant action.
In this section we will further develop our treatment of string theory in toroidal compactifications
and we will find the quadratic action from String Field Theory. We will follow the work by Hull
and Zwiebach in [5].
Let us start by remembering the field redefinitions G → α0 G and B → α0 B. Then, putting Eqs.
(3.17) and (3.40) together one finds the zero oscillators to be,
α0m
1
∂
∂
= −i √
Gmn
− Enp
m
0
∂Y
∂ Ỹ p
2α
1
≡ −i √ Dm ,
2
ᾱ0m
1
∂
∂
T
= −i √
Gmn
+ Enp
m
∂Y
∂ Ỹ p
2α0
1
≡ −i √ D̄m .
2
36
(4.15)
4.4
Physical constraints
Where we have already used Emn = Gmn +Bmn and (Emn )T = Gmn −Bmn due to the antisymmetry
of B. We have also used pm = −i∂m and ω m = −i∂˜m , where ∂m = ∂/∂Y m and ∂˜m = ∂/∂ Ỹm .
Finally, we have introduced the following derivatives,
∂
∂
1
− Enp
,
Dm = √
m
∂ Ỹ p
α0 ∂Y
(4.16)
∂
1
∂
T
+ Enp
D̄m = √
.
m
0
∂ Ỹ p
α ∂Y
It is easy to see that these objects, D and D̄, are derivatives with respect to right- and left-moving
coordinates. Le us now introduce the following operators,
≡
1 2
D + D̄2 ,
2
∆≡
1 2
D − D̄2 .
2
(4.17)
It can be shown [5] that is a Laplacian for the generalized metric H(E), while ∆ is a Laplacian
for the invariant metric J. Now, closed string field theory with N = Ñ = 1 gives us the following
string state [5],
m ᾱn c c̄ + e(p)c̄ c̄
|Ψi = [dp] − 12 emn (p)α−1
1 −1 + i
−1 1 1
R
q
α0
2
+
m
m
¯
fm (p)c+
0 c1 α−1 + fm (p)c0 c̄1 ᾱ−1
|pi .
(4.18)
Where [dp] is denoting the integral over continuous momenta Pµ and sum over discrete momenta
and winding number Pi , ω i . The c operators are ghost operators and c±
0 = 1/2(c0 ± c̄0 ). Then,
the quadratic action is given by [5],
(2κ2 )S (2) = −
2
hΨ| c−
0 Q |Ψi .
α0
(4.19)
Where Q is the BRST operator. The quadratic action can be eventually found to be [5],
(2κ )S
2
(2)
=
Z
2
1 m
1
1 n
2
mn
m n
[dXdX̃] emn e + D̄ emn + (D emn ) − 2dD D̄ emn − 4dd . (4.20)
4
4
4
Where we have the fields emn and d and the integrations measure is,
[dXdX̄] ≡ dd X µ dn Y i dn Ỹi .
4.4
(4.21)
Physical constraints
The level matching condition in Eq. (4.14) can be rewritten, for N = N̄ = 1, as follows,

P T JP = 0
⇒
 
 0 1 ∂˜

 
˜
 φ = 0
∂ ∂ 

 
1 0
∂
⇒
∂m ∂˜m φ = 0.
(4.22)
For any field φ. It is straightforward to see that this constraint can also be written as ∆φ = 0.
We will refer to this condition as weak constraint, in contrast with a stronger condition that we
37
Chapter 4
Double Field Theory
will shortly introduce. A more conventional way of writing the weak condition is in its O(D, D)
covariant form. Taking into account that Jij is used to raise and lower indices in the O(D, D)
formalism, we may write the weak constraint as follows,
∂i ∂ i φ = 0.
(4.23)
Let us now impose an stronger condition that we will shortly motivate. Let us assume that all
fields and their products are annihilated by ∆. This constraint can be written in the following
equivalent forms,
∆(φα φβ ) = 0,
∂m ∂˜m (φα φβ ) = 0,
∂i ∂ i (φα φβ ) = 0.
(4.24)
For any two fields φα , φβ . This constraint, also called the section constraint, goes beyond the
level matching condition and it implies that fields only depend on half of the coordinates of the
doubled torus. This requirement was introduced in [34] and is equivalent to requiring that the
fields are restricted to an n-dim toroidal subspace of the doubled torus T 2n that is null with
respect to J, or, in other words, that fields are restricted to a general isotropic subspace of
dimension n. In the mentioned work this is ensured by requiring the following condition,
∂
φ = 0.
∂ ũ
(4.25)
Where ũ are the coordinates of a subspace Ñ which is an n-dim torus such that N × Ñ = T 2n .
Furthermore, this constraint was found to be equivalent to the strong constraint (4.24) in the
publication [6]. We will now follow this work to see how the strong constraint does indeed
restrict the fields in the required manner. We have already seen that the strong constraint might
be rewritten as,
P T JP = 0
⇒
P · P = Pi Pj J ij = 0.
(4.26)
This implies that any two momenta Pα , Pβ associated with two Fourier modes of some two fields
must be orthogonal,
Pα · Pβ = 0.
(4.27)
This means that the momentum vectors Pα belong to a subspace of R2D with all vectors null
and mutually orthogonal. That is, they must belong to an isotropic subspace N ⊂ R2n . The
maximum dimension of such a subspace is D and an example would be the space spanned by
the following generalized vector,
PiT
= 0 Pm .
(4.28)
This is indeed the case ∂˜ = 0, that is, the supergravity limit. Now, any isotropic space is a
subspace of a maximal isotropic subspace (dim D). Hence, the strong constraint implies that
the momenta must lie in a D-dim isotropic subspace, as we wanted to show. It can be further
shown [27] that any maximal isotropic subspace of R2D is related to any other by an O(D, D)
transformation. Indeed, this means that there is an O(D, D) transformation that brings the
38
4.5
Cubic action and undoubled limit
space to the one described by (4.28). If there are n dimensions that are compact, the momenta
will be discrete and, as we discussed previously, only the subgroup O(n, n; Z) ⊂ O(D, D) will
leave the theory invariant.
To sum-up, the strong constraint implies that there is a T-duality transformation that reduces
the theory to usual supergravity. In this sense, the theory we have described, with the strong
constraint, is not really doubled. Then, the theory with strong constraint can be seen as a
reduction of the complete double field theory restricting it to a totally null subspace. However,
even though this version is not truly doubled, we expect it to exhibit some new structure and
properties that should be also present in the, yet to be constructed, full double field theory.
In the following sections we will present the full action of this restricted version of double field
theory and some of its properties studied.
4.5
Cubic action and undoubled limit
In this section we study the cubic action and the gauge transformations of double field theory
coming from string field theory. Later we compare the gauge transformations with the usual
ones from supergravity when the dependence in the dual coordinates is removed. We will follow
[5] and state some results without derivation as some calculations are beyond the scope of this
work. The cubic action derived from string field theory is [5],
(2κ2 )S (3) =
Z
"
[dXdX̃]
h
1
1
1
emn emn + (D̄n emn )2 + emn (Dm enp )(D̄n epq )−
4
4
4
i 1 h
− (Dm epq )(D̄q epn ) − (Dp emp )(D̄n epq ) + d (Dm emn )2 + (D̄n emn )2 +
2
i
1
1
+ (Dp emn )2 + (D̄p emn )2 + 2emn (Dmn DP epn + D̄n D̄p emp ) +
2
2
(4.29)
#
+ 4emn dD
mn
D̄ d + 4d d .
n
2
One can show that this action enjoys the following Z2 symmetry [5],
emn → enm ,
Dm → D̄m ,
D̄m → Dm ,
d → d.
(4.30)
And the gauge transformations are as follows [5],
1
δλ emn =D̄n λm + [(Dm λp )epn − (Dp λm )epn + λp Dp emn ],
2
1
1
δλ d = − D · λ − (λ · D)d.
4
2
39
(4.31)
Chapter 4
Double Field Theory
Now, the Z2 symmetry in (4.30) implies that the following doubled gauge transformations are
also a symmetry of the action,
i
1h
(D̄m λ̄p )epn − (D̄p λ̄m )epn + λ̄p D̄p emn ,
2
1
1
δλ̄ d = − D̄ · λ̄ − (λ̄ · D̄)d.
4
2
δλ̄ emn =Dn λ̄m +
(4.32)
All these gauge transformations have an implicit projection to the kernel of ∆ for the term linear
in the fields, in order tu fulfil the level-matching condition, as it is not preserved straightforwardly, see [5]. We can combine this transformations ,
δemn = δλ emn + δλ̄ emn = D̄n λm + Dm λ̄n + (Dm λp )epn + (D̄n λ̄p )emp +
1
1
1
+ (λ̄p D̄p + λp Dp )emn − (Dm λp + Dp λm )epn − (D̄n λ̄p + D̄p λ̄n )emp
2
2
2
1
1
δd = − (D · λ + D̄ · λ̄) + (λ · D + λ̄ · D̄)d.
4
2
(4.33)
We want to compare these gauge transformations with the ones supergravity provides. From
Eqs. (2.98) and (2.101) we see that the gauge transformations of supergravity can be written as,
δgmn = Lξ gmn ,
δbmn = Lξ bmn + ∂[m Λn] ,
δφ = Lξ φ.
(4.34)
Let us now define, for convenience, the following field,
e−2d ≡
√
−ge−2φ .
(4.35)
If we study how this quantity transforms we find the following,
1
δ(e−2d ) = e−2d ∂ m ξm − ξ p ∂ m gpm + g mn ξ p ∂p gmn − 2ξ m ∂m φ = ∂m (e−2d ξ m ).
2
(4.36)
And so e−2d transforms as a density,
δ(e−2d ) = ∂m (e−2d ξ m ).
(4.37)
Let us now calculate the infinitesimal transformations for small perturbations in supergravity.
We can decompose the fields into constant background fields (Gmn , Bmn ) and small perturbations
(hmn , b̌mn ) as follows,
gmn = Gmn + hmn ,
bmn = Bmn + b̌mn .
(4.38)
And, taking into account that derivatives of the constant fields, though not the lie derivatives,
are zero one finds,
δhmn = Lξ Gmn + Lξ hmn = ∂m n + ∂n m + Lξ hmn ,
∂ b̌mn = Lξ Bmn + Lξ b̌mn = ∂m ˜n + ∂n ˜m + Lξ b̌mn .
(4.39)
Where we have defined the following parameters,
m ≡ Gmn ξ n ,
˜m ≡ Λm − Bmn ξ n .
40
(4.40)
4.5
Cubic action and undoubled limit
If one now defines the background fluctuations field,
ěmn = hmn + b̌mn ,
(4.41)
it is easy to see that it transforms as follows,
δěmn = (∂m n + ∂n m ) + (∂m ˜n + ∂n ˜m ) + (∂m ξ p ěpn + ∂n ξ p ěmp + ξ p ∂p ěmn ).
(4.42)
This is the transformation we will compare with the transformations of DFT coming from
string field theory. In order to perform this comparison we need to restrict the fields to have
√
no X̃ dependence. This is realised by setting ∂˜ = 0. Then, if one absorbs the α0 in Eq.
(4.16) by setting Y → (α0 )−1/2 Y , one finds that the left- and right-handed derivatives become
Dm = D̃m = ∂m and so the transformations of DFT in Eq. (4.33) can be rewritten as:
1
δemn =∂n λm + ∂m λ̄n + ∂m λp epn + ∂n λ̄p emp + (λ̄p ∂ p + λp ∂ p )emn −
2
1
1
− (∂m λp + ∂ p λm )epn − (∂n λ̄p + ∂ p λ̄n )emp ,
2
2
1
1
δd = − (∂m λm + ∂m λ̄m ) + (λm ∂m + λ̄m ∂m )d.
4
2
(4.43)
We now need to find the transformations with parameters , ˜. To find the transformations with
parameter , one needs to set λm = λ̄m = m . Then, it can be shown that [5],
1
∂ (emn + epm epn ) = ∂n m + ∂m n + (∂m p )epn + (∂n p )emp + p ∂ p emn .
2
(4.44)
Hence, if we define the following field,
1 p
e+
mn ≡ emn + em epn .
2
(4.45)
One sees that it transforms as,
+
+ 2
δ (e+
mn ) = ∂n m + ∂m n + L emn = ∂m n + ∂n m + L emn + O(e ) =
= ∂m n + ∂n m + L e+
mn .
(4.46)
One can also calculate transformations with respect to the parameter ˜ by choosing λm = −λ̃m =
˜m . Then it can be shown [5] that the transformation becomes,
δ˜e+
˜m − ∂m ˜n .
mn = ∂n (4.47)
Then, observing Eqs. (4.46) and (4.47), one sees that we have indeed recovered the Supergravity
transformations in Eq. (4.42). This can be done by taking ěmn = e+
mn and identifying the third
parentheses in Eq. (4.42) as the Lie derivative. Hence, we have shown that the transformations
from the cubic action of Double Field Theory indeed reduce to those of supergravity when the
dependence in the dual coordinates is removed.
41
Chapter 4
4.6
Double Field Theory
Full action and generalized Lie derivatives
In this section we will review the construction of the full O(D, D) covariant and gauge invariant
action, as well as its symmetries. We will follow the original papers where these constructions
where developed: [6] and [7]. We will also use the reviews [18] and [35]. We will start presenting
the full background independent action for DFT derived in [6]. Then we will rewrite it in a fully
O(D, D) covariant fashion following [7] and this path will lead us the construction of O(D, D)
covariant generalized Lie derivatives in doubled space.
4.6.1
Background independent full action
In section 4.5 we presented the cubic action for double field theory derived from string field
theory. The next step would be to find the complete action, that is, to all orders. Quantum
gravity theories are commonly required to be background independent. Hence, string theory
should satisfy this property and we would like the action of double field theory to inherit it.
However, in the action (4.29) there is explicit background dependence entering through the
covariant derivatives in Eq. (4.16), as they depend on the background field Emn . The fact that
the indices are raised and lowered using Gmn is also background dependent. In what follows, we
construct a background independent full action for DFT with the strong constraint following
[6].
Background independence implies that a constant part of the fluctuation field emn can be absorbed as a change in the background field Emn . Mathematically, for an infinitesimal change
χmn , background independence is realised in an action S if it verifies:
S[Emn , emn + χmn ] = S[Emn + χmn , emn + fmn (χ, e)].
(4.48)
Where fmn is a function linear in χ and, to leading order, in e. It can be explicitly shown that the
action (4.29) is background independent to quadratic order [6]. However, as we have discussed
this independence is hidden. We would like to construct and explicitly background independent
action. Before doing so, let us study if we can write the gauge transformations in a background
independent form. Firstly, one would like to have a background independent field to start with.
This field would have to depend on both the background Emn and the fluctuations emn . It can
be shown [6, 34] that the appropriate background independent combination is:
Emn ≡ gmn + bmn = Emn + Fmp (e)epn .
Where,
1
F ≡ 1− e
2
42
−1
.
(4.49)
(4.50)
4.6
Full action and generalized Lie derivatives
Let us now introduce the following derivatives,
1
D̂m ≡ Dm − emn D̄n ,
2
ˆ ≡ D̄ − 1 e Dm .
D̄
m
m
nm
2
(4.51)
One can use these derivatives to rewrite the gauge transformations of the fluctuation field emn
in (4.33) as,
1
1
1
δemn =(D̄n − epn Dp )λm + (Dm − emp D̄p )λ̄n + (Dm λp )epn +
2
2
2
1
1
+ (D̄n λ̄p )emp + (λ̄p D̄p + λp Dp )emn =
2
2
(4.52)
ˆ λ + D̂ λ¯ + 1 (D λp )e + 1 (D̄ λ̄p )e + 1 (λ̄ D̄p + λ Dp )e .
= D̄
n m
m n
m
pn
n
mp
p
p
mn
2
2
2
(4.53)
Furthermore, it can be shown that the gauge variations of the background independent field are
related to these variations by [6]:
δE = FδeF.
(4.54)
Now that we have a background independent field and we know how it transforms, we would
like to find background independent derivatives that we can use to construct the action. The
following derivatives are indeed explicitly background independent,
Dm ≡ Fmn D̂n = ∂m − Emn ∂˜n ,
ˆ = ∂ + E ∂˜m .
D̄m ≡ Fmn D̄
mn
m
mn
(4.55)
Then, the gauge transformations of the background independent field can be written in a manifestly background independent manner [6]:
δξ Emn = ∂m ξ˜n − ∂n ξ˜n + Lξ Emn + Emp (∂˜k ξ p − ∂˜p ξ k )Ekn + Lξ̃ Emn .
(4.56)
Where we have introduced a dual Lie derivative defined as follows,
Lξ̃ Emn ≡ ξ˜P ∂˜P Emn − ∂˜P ξ˜m Epn − ∂˜P ξ˜n Emp ,
(4.57)
and we have also used new gauge parameters defined as:
λm ≡ Emn ξ n − ξ˜m ,
λ̄m ≡ Emn ξ n + ξ˜m .
(4.58)
Now, if the fields are independent of the doubled coordinates X̃, the gauge transformations
become,
δξ Emn = Lξ Emn + ∂m ξ˜n − ∂n ξ˜m .
(4.59)
And this transformation is precisely that of a diffeomorphism with infinitesimal parameter ξm
and a two-form gauge principle with parameter ξ˜m , which is what we expected from supergravity,
43
Chapter 4
Double Field Theory
confirming once again that DFT correctly reproduces the supergravity limit. Finally, the gauge
transformations for the fields depending both on the X, X̃ coordinates can be rewritten as:
δEmn = Dm ξ˜n − D̃n ξ˜m + Dm ξ p Epn + D̃n ξ p Emp + ξ p Emp + ξ i ∂i Emn ,
(4.60)
1
δd = − ∂i ξ i + ξ I ∂i d.
2
Where, again, the indices H, I, J, K are O(D, D) indices, while L, M, N, O, P are usual Ddimensional space-time indices. The action can now be constructed. This is done by looking
for terms with two background independent derivatives of the background independent fields
E, d. Then the action is required to reproduce (4.29) to cubic order. Doing this exercise the full
action can be found to be [6]:
SB.I. =
Z
1
1
dXdX̃e−2d − g mp g nl Dk Epl Dk Emn + g pl (Dn Emp Dm Enl +
4
4
(4.61)
i
+ D̄ Epm D̄ Eln + D dD̄ Emn + D̄ dD Emn + 4D dDm d .
n
m
m
m
m
n
m
Where the indices in the D derivatives are raised and lowered using the full metric gmn and
the indices in the background field E with the background metric Gmn . This action is explicitly
background invariant and can be shown to be T-duality invariant, as is done in [6]. Furthermore,
this action can be shown to reduce to standard supergravity when fields do not depend on the
doubled coordinates X̃, as one would expect.
As we have seen, the invariance of this action only holds if the strong constraint is used. This
means that this action is not the action of full double field theory, but it is a restriction of DFT
to a null subspace. The full DFT should be a generalization is the theory presented here and its
full action is yet unknown.
4.6.2
Generalized Lie derivatives, C-brackets and D-brackets
So far, we have presented the full explicitly background independent action for double field
theory. Even though this action is already O(D, D) invariant, we would like to formulate the
theory in such a way that the O(D, D) symmetry is explicit. In this action we will study the
properties of O(D, D) tensors and, in particular, of the generalized metric introduced in (3.64).
Studying the transformations of this object will lead us to the notion of generalized Lie derivative.
Finally, the study of this Lie derivative will lead to the introduction of C-brackets and D-brackets.
We will follow the original work by Hohm, Hull and Zwiebach [7].
Since we want to develop the theory in an O(D, D) covariant fashion, we will need to use objects
transforming in O(D, D) representations, that is, with H, I, J, K indices. We have already seen
44
4.6
Full action and generalized Lie derivatives
that, the generalized metric in Eq. (3.64) is an O(D, D) tensor. Let us redefine it for the full
fields as,


−g mp bpn
g mn




(4.62)
Hij = 
.


pn
pl
bmp g
gmn − bmp g bln
We have also seen that O(D, D) indices are raised and lowered using the invariant O(D, D)
metric J in Eq. (4.12). Using this matrix it is easy to see that the generalized metric verifies:
Hij Hjk = δki .
(4.63)
So it can indeed be seen as a metric in the doubled space. The generalized metric encodes the
information of the space-time metric and the two form in an O(D, D) invariant tensor, making
it a natural candidate to formulate the O(D, D) covariant version of DFT. Let us study how it
transforms. Reminding the definition of the E field in Eq. (4.49), it can be seen from Eq. (4.60)
that the transformations for the fields gmn and bmn are given by [7],
δξ gmn =Lξ gmn + Lξ̄ gmn + ∂˜P ξ l − ∂˜l ξ p (gpm bnl + gpn bml )
δξ g mn =Lξ g mn + Lξ̄ g mn − ∂˜m ξ p − ∂˜p ξ m g nl blp + ∂˜n ξ p − ∂˜p ξ n g ml blp
δξ Bmn
=Lξ bmn + Lξ̄ bmn + ∂m ξ˜ − ∂n ξ˜m + gmp ∂˜l ξ p − ∂˜p ξ l gln +
(4.64)
+ bmp ∂˜l ξ p − ∂˜P ξ l bln .
And, from these transformations, it can be shown that the transformations of Hij might be
written in the following O(D, D) invariant form [7],
δξ Hij = ξ k ∂k Hij − ∂k ξ i Hjk + ∂ i ξk Hkj − ∂k ξ j Hik + ∂ j ξk Hik .
(4.65)
Which may be rewritten in the more suggestive fashion,
δξ Hij = ξ k ∂k Hij + (∂ i ξk − ∂k ξ i )Hkj + (∂ j ξk − ∂k ξ j )Hik .
(4.66)
And this looks like a Lie derivative with some extra terms, see for instance the lie derivative of
the metric in Eq. (2.100). Moreover, we have said that Hij can be seen as a generalized metric
on doubled space. The gauge transformations of a metric are just the diffeomorphisms described
by the Lie derivative. This motivates the following identification,
δξ Hij = L̂ξ Hij .
(4.67)
Where we have implicitly defined the generalized Lie derivative as,
L̂ξ Aji ≡ ξ k ∂k Aji + (∂i ξ k − ∂ k ξi )Ajk + (∂ i ξk − ∂k ξ j )Aki .
45
(4.68)
Chapter 4
Double Field Theory
For an O(D, D) tensor Aji . It can be seen that this Lie derivative indeed satisfies the Leibniz
rule. An important result regarding the generalized Lie derivative is seen by applying it to both
sides of the O(D, D) relation, HJH = J −1 ,
L̂ξ (HJH) = 0
⇒
(L̂ξ H)JH + HJ(L̂ξ H) = 0
| {z }
⇒
δξ (HJH) = 0.
(4.69)
| {z }
δξ H
δξ H
So the O(D, D) relation is conserved. This implies that the O(D, D) transformations are compatible with the gauge symmetry of H. The action of the generalized Lie derivative on an O(D, D)
vector can be found using the Leibniz rule:
L̂ξ Vi = ξ k ∂k Ai + (∂i ξ k − ∂ k ξi )Ak .
(4.70)
We know that the commutator algebra of usual Lie derivatives is governed by the Lie bracket,
that is [17],
[LX , LY ] = L[X,Y ] .
(4.71)
[X, Y ] = LX Y.
(4.72)
Where the Lie bracket id defined as,
One may now wonder whether there is a similar structure governing the commutator algebra of
the generalized Lie derivative. The calculation can be carried out and gives the following result
[7],
h
i
L̂ξ1 , L̂ξ2 Vi = −L̂[ξ1 ,ξ2 ]C Vi + ∂
q
k
ξ[1i ∂q ξ2]
Vk
1
j
∂q Vi .
− ξ[1j ∂ q ξ2]
2
(4.73)
Where [ξ1 , ξ2 ]C stands for the C-bracket, defined as [3],
1 j i
j
i
[ξ1 , ξ2 ]iC ≡ ξ[1
∂j ξ2]
− ξ[1
∂ ξ2]i .
2
(4.74)
If the strong constraint is assumed, the commutator algebra reduces to,
[L̂ξ1 , L̂ξ2 ]Vi = −L̂[ξ1 ,ξ2 ]C .
(4.75)
Now, putting together Eqs. (4.67) and (4.75), the gauge gauge algebra of the generalized metric
is found to be,
[δξ1 , δξ2 ]Hij = δ−[ξ1 ,ξ2 ]C Hij .
(4.76)
That is, the gauge transformations of the generalized metric close according to the C-bracket.
The usual Lie derivative naturally defines the Lie bracket as in Eq. (4.72). One may wonder
whether a similar structure can be defined using the generalized Lie derivative. This is indeed
the case, and one finds an analogous bracket called D-bracket and defined as follows[7],
[A, B]D ≡ L̂A B.
(4.77)
It can be shown that, in the case of fields not depending on the doubled coordinates X̃, the
D-bracket reduces to the Dorfman bracket, see [27]. It is important to notice that the gauge
transformations in (4.66) are not infinitesimal diffeomorphisms in the doubled space, as they do
not close with the Lie bracket but with the C-bracket [36].
46
4.7
Doubled Geometry
The explicitly O(D, D) invariant action.
4.6.3
We are now equipped to find the explicitly O(D, D) invariant action in therms of the generalized
metric, encoding the space-time metric, the two-form and the dilaton field. In this section we will
present the full O(D, D) explicitly invariant action following [7] using the machinery developed
so far. The terms in the action should be O(D, D) scalars, that is, they must have all their
O(D, D) indices contracted. Furthermore, the terms should be invariant under the action of the
Z2 symmetry E −→ E T that we described in section 3.3.2. The terms should have two derivatives
and the resulting action should be invariant under gauge transformations. It can be shown the
only terms fulfilling these criteria are the four following ones [7],
∂i d∂j Hij ,
Hij ∂i d∂j d,
Hij ∂i Hkh ∂j Hkh ,
(4.78)
Hij ∂j Hkh ∂h Hik .
Then, the action should be a linear combination of these terms multiplies by e−2d . It can be
seen that the appropriate combination satisfying gauge invariance is,
S=
Z
dXdX̃e−2d
1
1 ij
H ∂i Hkh ∂j Hkh − Hij ∂j Hkh ∂h Hik − 2∂i d∂j Hij + 4Hij ∂i d∂j d .
8
2
(4.79)
It can be seen that this action is equivalent to (4.61), as it should [7]. Hence, this is the full
O(D, D) action for DFT with the strong constraint applied. This action may be rewritten, upon
parts integration as [7],
S=
4.7
Z
dXdX̃e−2d
1 ij
1
H ∂i Hkh ∂j Hkh − Hij ∂j Hkh ∂h Hik + 2Hij ∂i ∂j d .
8
2
(4.80)
Doubled Geometry
So far we have described the theory is an O(D, D) covariant manner. In doing so we introduced
a generalized metric and a generalized Lie derivative to describe its transformations. As we saw
in section 2.2 a metric and a Lie derivative are the basic elements of Riemannian geometry. From
these elements the remaining structure is constructed and connections, curvature and torsion
tensors follow. One may now ask whether DFT can be given such a treatment. In other words,
is there an underlying geometry in DFT? Can DFT be formulated in a more geometrical way?
We will see that there is such a formulation, but its geometry differs significantly from the
Riemannian one. In this section we will mainly follow [18], [6], [7] and [8].
4.7.1
Generalized curvature and torsion tensors
In Eq. (4.61) we presented the full action for DFT with the strong constraint. We also know that,
in Riemannian geometry, the action of general relativity can be written in a purely geometrical
47
Chapter 4
Double Field Theory
way (2.74). This leads us to ask whether a similar formulation exists for DFT. This formulation
would also allow us to start investigating the geometry of DFT as we would have a "generalized
scalar of curvature". The appropriate action is of the following form,
0
SB.I.
=
Z
dXdX̃e−2d R(E, d).
(4.81)
Where R is the generalized Ricci scalar. It turns out that such a generalized curvature scalar
exists and was found in [6]. However, since we have also found a manifestly covariant version of
the action one may wonder whether a similar formulation exists for the action 4.80. From Eq.
(4.81) one expects to find the generalized curvature scalar from the dilaton equation of motion,
which should be,
R = 0.
(4.82)
Applying this assumption to (4.80) one finds [7],
1
1
R ≡ 4Hij ∂i ∂j d − ∂i ∂j Hij − 4Hij ∂i d∂j d + 4∂i Hij ∂j d + Hij ∂i Hkl ∂j Hkl − Hij ∂i Hkl ∂k Hjl . (4.83)
8
2
The action in Eq. (4.81) is manifestly O(D, D) invariant, but for it to be a gauge invariant, R
should be a gauge scalar. This is indeed the case, as is shown in [7]. Furthermore, it can be
proven that, with this scalar curvature, the action is equivalent to (4.80), up to total derivatives.
Let us now study the derivatives. We already introduced a generalized Lie derivative in Eq.
(4.70). Let us now introduce a covariant derivative with connection Γ in analogy with the
connection of Riemannian geometry, as done in 2.2,
∇i V j ≡ ∂i V j + Γjik V k ,
(4.84)
∇i Vj ≡ ∂i Vj − Γkij Vk .
(4.85)
For an O(D, D) vector V i . The connection transforms as required for the covariant derivatives
to be generalized tensors [8], that is,
δξ Γkij = ∂i ∂j ξ k − ∂i ∂ k ξj .
(4.86)
The last term does not appear in Riemannian geometry and it implies that the connection can
not be chosen to be symmetric in its lower indices. We can now define a generalized Riemann
and torsion tensors in analogy with (2.67), using the generalized covariant derivative,
[∇i , ∇j ]Vk ≡ −Rlijk Vl − Tij l ∇l Vk .
(4.87)
And one finds the following candidates to generalized curvature and torsion tensors,
Rlijk ≡ ∂i Γljk − ∂j Γlik + Γlih Γhjk − Γljh Γhik ,
T kij ≡ 2Γk[ij] .
(4.88)
(4.89)
48
4.7
Doubled Geometry
Where R satisfies Rijkl = −Rjikl . However, while their addition does, neither of the above
quantities transform as generalized tensors. It can be shown that the appropriate curvature
tensor, transforming as a generalized tensor, is [8],
Rijkl ≡ Rijkl + Rklij + Γhij Γhkl .
(4.90)
This can be interpreted as a generalized Riemann tensor and it naturally verifies,
Rijkl = Rklij .
(4.91)
Analogously, it can be seen that the appropriate definition of a generalized torsion is defined by
[37] [38],
j k
L̂∇
ξ − L̂ξ Vi = Tijk ξ V .
(4.92)
Where L̂∇ is the generalized Lie derivative with the partial derivative substituted by a covariant
one. For the case at hand, the coordinate form of the generalized torsion is [8],
Tijk ≡ Γijk − Γjik + Γkij = Tijk + Γkij .
4.7.2
(4.93)
Constraints in the connection
We saw that, in Riemannian geometry, a torsionless connection is completely determined by
the metric1 . This followed from metric compatibility and vanishing torsion. In DFT four such
constraints exist; compatibility with the metric J and the generalized metric H, vanishing of
the generalized torsion and covariant partial integration. Let us study them,
1. Compatibility with Jij . For the raising and lowering of indices to be compatible with the
connection one requires,
: 0 − Γ i J − Γl J = 0.
∇i Jjk = 0 ⇒ ∂i Jjk
ij lk
ik jl
(4.94)
And so one finds that the connection is antisymmetric in the last two indices,
Γijk = −Γikj .
(4.95)
One may use this result to rewrite the R tensor in Eq. (4.88) as follows,
R ijkl ≡ ∂i Γjkl − ∂j Γikl − Γilh Γjkh + Γikh Γjlh .
(4.96)
So Rijkl is antisymmetric in its two last indices. Together with its antisymmetry in the first
two indices, this implies that the generalized Riemann tensor has the familiar properties
of the Riemann tensor given in Eq. (2.69),
Rijkl = −Rijlk ,
1
Rijkl = −Rjikl ,
See sec. 2.2
49
Rijkl = Rklij .
(4.97)
Chapter 4
Double Field Theory
2. Vanishing generalized torsion. The vanishing of T as defined in Eq. (4.93) implies,
Tijk = 0 ⇒ Γijk − Γjik + Γkij = 0.
(4.98)
Which together with (4.95) implies the vanishing of the totally antisymmetric part of the
connection,
Γ[ijk] = 0.
(4.99)
An important consequence of this constraint, together with the first constraint, is that
the generalized Riemann tensor enjoys a Bianchi identity (BI), analogous to the general
relativity BI in (2.70),
R[ijkl] = 0.
(4.100)
∇i Hjk = ∂i Hjk − Γijl Hlk − Γikl Hjl = 0.
(4.101)
3. Generalized metric compatibility:
4. Covariant partial integration. This last condition requires the that partial integration is
verified for the covariant derivative in presence of the factor e−2d :
Z
e−2d Aj ∇i V i = −
Z
e−2d V i ∇i Aj .
(4.102)
This condition implies that [8],
∇i V i = e2d ∂I e−2d V i = ∂i V i + Γiji V j .
(4.103)
And, this in turn implies,
Γijj = −2∂i d.
(4.104)
To find which additional components of the connection are specified by conditions 3 and 4, we
introduce the following projectors,
1
P̄i j ≡ (δi j + Hij ).
2
1
Pi j ≡ (δi j − Hij ),
2
(4.105)
It is easy to see that they are orthogonal projectors,
P + P̄ = 1,
P 2 = P,
P̄ 2 = P̄ ,
P P̄ = 0,
(4.106)
and that they are covariantly constant,
∇(δi j −Hij ) Pj k = ∇i P̄j k = 0.
(4.107)
Let us introduce under-barred and over-barred indices, meaning that they have been projected
as follows,
Wi ≡ Pi j Wj ,
Wi ≡ P̄i j Wj ,
50
Wi + Wi = Wi ,
W i Vi = 0.
(4.108)
4.7
Doubled Geometry
After some calculations it can be shown that the connection may be written as [8],
Γijk = Γ̂ijk + Σijk ,
(4.109)
where Γ̂ijk is the part of the connection determined by the four constraints detailed above,
Γ̂ijk = − 2(P ∂i P )[jk] − 2 P̄[j l P̄k]h − P[j l Pk]h ∂l Phi +
4 +
Pi[j Pk]h + P̄i[j P̄k]h · (∂h d + (P ∂ l P ))[hl] ,
D−1
(4.110)
and Σijk remains undetermined,
Σijk = Γijk +
|
2
2
Pi[J Pk]l φl + Γijk +
P̄i[J P̄k]l φl .
D −{z1
D
−
1
} |
{z
}
Γ̃ijk
(4.111)
Γ̃ijk
Where,
φi = −2 ∂i d + (P ∂j P )[ji] .
(4.112)
The connection in DFT is thus not completely determined by the consistency constraints, as
opposed to Riemannian geometry.
4.7.3
Generalized Ricci tensor and scalar
As we have seen, the connection is not fully determined in DFT. This implies that the generalized
Riemann tensor may itself be not fully determined. The generalized Riemann tensor can be
decomposed in under-barred and over-barred components to study which components remain
undetermined. Using the symmetries in Eq. (4.97) one can see that only the following projections
need to be studied,
Rijkl ,
Rijkl ,
Rijkl ,
Rijkl ,
Rijkl ,
Rijkl .
(4.113)
It can be shown that some of these projections vanish, while the rest contain undetermined
connections [8],
Rijkl = Rijkl = 0.
(4.114)
The rest have undetermined connections. Hence, one can not find a generalized Riemann tensor
in terms of the physical fields. We have, however, found a Ricci scalar in (4.83). Demanding
these two observations to be consistent implies that the undetermined connections should drop
out when tracing the generalized Riemann tensor to find the Ricci scalar. In fact, one can show
that the naive Ricci scalar constructed from tracing the Riemann tensor vanishes [8],
Rijij = Γijk (Γkij − Γjik + Γijk ) = 0.
51
(4.115)
Chapter 4
Double Field Theory
Where we have used Eq. (4.99). Actually, it can be shown that the appropriate contraction
producing the Ricci scalar in (4.83) is,
ij
ij
R≡R
= −Rijij .
(4.116)
Further analysis of the generalized Riemann tensor shows that the only well defined contractions
producing a second rank generalized tensor are,
k
(4.117)
k
Rij ≡ Rkij ,
Rji ≡ Rkji .
And the Bianchi identity implies that they are in fact equal,
Rij = Rji .
(4.118)
Which is exactly what one would expect for a normal Ricci tensor, as in Eq. (2.73). Then, we
can treat Rji as a generalized Ricci tensor. Furthermore, it can be shown that the undetermined
connections drop and that the generalized Ricci tensor in uniquely defined [8].
We have thus found the generalized torsion and curvature tensors. The underlying geometry of
DFT shares similarities with Riemannian geometry, like the symmetries of these objects or the
Bianchi identity. The most striking difference is that, in DFT, the connection is not uniquely
specified and the Riemann tensor cannot be written in terms of the fields. The Ricci tensor and
scalar are uniquely determined and can be written in terms of the fields.
4.7.4
Coordinate independent formulation of DFT
We have thus far formulated DFT in a coordinate-dependent fashion. However, we saw in section
2.2 that Riemannian geometry can be formulated in a more general coordinate independent way.
The foundations of a similar formulation of DFT were set in [9]. In this section we will present
the results of this paper, analogous to the coordinate independent Riemannian geometry. Let
us start by generalizing the definition of Riemann tensor in Eq. (2.93). We want it to define a
generalized tensor, so we must substitute the Lie bracket by either a C-bracket or a D-bracket.
Furthermore, we want the map to scale correctly under (X, Y, Z) → (f X, gY, hZ). In [9], the Dbracket is preferred for its simplicity as it rescales properly under Y → gY , while the C-bracket
does not. Then, one suggests,
R(X, Y, Z, W ) ≡
D
E
∇X ∇Y − ∇Y ∇X − ∇[X,Y ]D Z, W + ...
(4.119)
Where extra terms are needed to ensure that R(X, Y, Z, W ) scales correctly under X → f X.
The final expression found in [9] with correct rescaling for all variables is the following,
R(X, Y, Z, W ) ≡
D
+
E
∇X ∇Y − ∇Y ∇X − ∇[X,Y ]D Z, W +
D
∇Z ∇W − ∇W ∇Z − ∇[Z,W ]D X, Y
52
E
+ hY, ∇ZM Xi hW, ∇Z M Zi .
(4.120)
4.7
Doubled Geometry
Where we have used the fact that we may expand any vector field X as X = X M ZM . It can
be shown that the component expression of this tensor is indeed (4.90). An expression for the
generalized torsion can be found analogously. One finds [9],
T (X, Y, Z) ≡ h∇X Y − ∇Y X − [X, Y ]D , Zi + hY, ∇Z Xi .
(4.121)
And one can again shown that the coordinate expression of this tensor reduces to (4.93).
53
Chapter 5: Conclusions
In this work we have introduced T-duality and double field theory. Starting from string theory
we have encountered T-duality, a symmetry characteristic of string theory and directly related to
the ability of strings to wind around compact dimensions. This symmetry is lost in supergravity
and trying to preserve it in a field theory naturally led us to the introductions of dual coordinates
and the development of double field theory. We have formulated DFT in a number of ways, and
in particular, in a fully O(D, D) covariant fashion. We have further studied its geometry. By
now we have hopefully understood the usefulness of double field theory, as well as its machinery
and formulation.
Double field theory is a very active area of research and many aspects of the theory are yet to
be fully understood. In the following, we will try to account for some of the latest developments
in DFT and provide references for further reading. We do not intend to provide an exhaustive
reference guide and, for such a guide, the reader may consult [18] and [35].
We have discussed the geometry of DFT, but in doing so we have only considered "geometric"
backgrounds, those with diffeomorphisms as transition functions. However, T-duality allows us
to consider backgrounds where the diffeomorphism group is extended with T-duality transformations. This is an example of "non-geometry". Non-geometric backgrounds are a feature of
string theory incorporating the various symmetries from string theory to the diffeomorphism
group. This points to the development of "stringy geometry", in contrast to the usual "particle"
(Riemannian) geometry. As we mentioned, strings see a very different geometry from the one
particles do, but it has yet to be formulated. These stringy geometry is dubbed "non-geometry".
For a review in non-geometry, see [39]. Thus, DFT should allow non-geometric backgrounds and
their study would shed light in the nature of stringy geometry. This approach is taken in [40] and
[41], for instance. The notion of T-duality transformations being transitions functions leads, in
particular, to T-folds. T-fold backgrounds in string theory are studied in [42] and [43]. Finding
this geometry for non-geometry is a very active and promising field of research.
Another field with recent developments is that of studying large gauge transformations in DFT.
Throughout this work we have only considered infinitesimal transformations, but an understanding of the large (non infinitesimal) version of them is very important. These transformations were
first studied in [36]. A formula for the large transformations in DFT is proposed and further
shown that it can be obtained by exponentiation of the generalized Lie derivative. However, how
this can be extended to non-geometric background is yet to be understood.
54
Chapter 5
Conclusions
Another limitation of the theory so far described is that it does not take into account any α0
corrections. This issue was first studied in [8] and [9]. A better understanding of the modifications
that these corrections might imply to the theory is needed. For instance, this might elucidate
the issue of the undetermined components of the generalized curvature tensor.
A further generalization of the theory is the inclusion of supersymmetry and fermions. This has
been pursued in [44] and [45]. Further generalization also leads to extending the framework to
other string theory or M-theory dualities and, in particular, to U-duality. This effort has led to
"exceptional field theory" in analogy with DFT, see [46] and [47]. Also see [48].
Another obvious generalization to the theory we have presented so far is trying to relax the
strong constraint to obtain a truly doubled DFT. Some solutions restricting the form of the
fields, rather than their coordinate dependence, have already been explored, see for instance
[49].
We thus see that DFT is still a rather young framework that is being actively researched upon.
Many different research fields are finding connections with DFT. In this sense, it is clear how
powerful DFT is and there is reason to be hopeful that DFT will help us to better understand
string theory and its geometry.
55
Bibliography
[1]
M.J. Duff. Duality rotations in string theory. Nuclear Physics B, 335(3):610 – 620, 1990.
[2]
A. A. Tseytlin. Duality-symmetric string theory and the cosmological-constant problem.
Phys. Rev. Lett., 66:545–548, Feb 1991.
[3]
W. Siegel. Superspace duality in low-energy superstrings. Phys. Rev. D, 48:2826–2837, Sep
1993.
[4]
W. Siegel.
Two-vierbein formalism for string-inspired axionic gravity.
Phys. Rev. D,
47:5453–5459, Jun 1993.
[5]
Chris Hull and Barton Zwiebach. Double field theory. Journal of High Energy Physics,
2009(09):099, 2009.
[6]
Olaf Hohm, Chris Hull, and Barton Zwiebach. Background independent action for double
field theory. Journal of High Energy Physics, 2010(7), 2010.
[7]
Olaf Hohm, Chris Hull, and Barton Zwiebach. Generalized metric formulation of double
field theory. Journal of High Energy Physics, 2010(8), 2010.
[8]
Olaf Hohm and Barton Zwiebach. On the riemann tensor in double field theory. Journal
of High Energy Physics, 2012(5), 2012.
[9]
Olaf Hohm and Barton Zwiebach. Towards an invariant geometry of double field theory.
Journal of Mathematical Physics, 54(3):–, 2013.
[10] M. B. Green, J. H. Schwarz, and E. Witten. Superstring Theory. Cambridge University
Press, 1987.
[11] L.E. Ibáñez and A.M. Uranga. String Theory and Particle Physics: An Introduction to
String Phenomenology. Cambridge University Press, 2012.
[12] K. Becker, M. Becker, and J.H. Schwarz. String Theory and M-Theory: A Modern Introduction. Cambridge University Press, 2006.
[13] J. Polchinski. String Theory: Volume 1, An Introduction to the Bosonic String. Cambridge
Monographs on Mathematical Physics. Cambridge University Press, 1998.
[14] Steven Weinberg. Ultraviolet divergences in quantum theories of gravitation. pages 790–831,
1980.
56
Bibliography
[15] R. Blumenhagen and E. Plauschinn. Introduction to Conformal Field Theory: With Applications to String Theory. Lecture Notes in Physics. Springer, 2009.
[16] Peter West. Introduction to strings and branes. Cambridge Univ. Press, Cambridge, 2012.
[17] Mikio Nakahara. Geometry, topology and physics. Graduate student series in physics.
Hilger, Bristol, 1990.
[18] Gerardo Aldazabal, Diego Marqués, and Carmen Núñez. Double field theory: a pedagogical
review. Classical and Quantum Gravity, 30(16):163001, 2013.
[19] G. Nordström. Über die möglichkeit, das elektromagnetische feld und das gravitationsfeld
zu vereinigen. Phys. Z., 15:504–506, 1914.
[20] T. Kaluza. Zum unitätsproblem der physik. Sitzungsber. Preuss. Akad. Wiss., Phys.-Math.
Kl., 1921:966–972, 1921.
[21] Oskar Klein.
Quantentheorie und fünfdimensionale relativitätstheorie.
Zeitschrift für
Physik, 37(12):895–906, 1926.
[22] E. Cremmer and J. Scherk. Dual models in four dimensions with internal symmetries.
Nuclear Physics B, 103(3):399 – 425, 1976.
[23] Amit Giveon, Massimo Porrati, and Eliezer Rabinovici. Target space duality in string
theory. Phys.Rept., 244:77–202, 1994.
[24] Norisuke Sakai and Ikuo Senda. Vacuum energies of string compactified on torus. Progress
of Theoretical Physics, 75(3):692–705, 1986.
[25] Michael B. Green, John H. Schwarz, and Lars Brink. N = 4 yang-mills and n = 8 supergravity as limits of string theories. Nuclear Physics B, 198(3):474 – 492, 1982.
[26] David J. Gross, Jeffrey A. Harvey, Emil Martinec, and Ryan Rohm. Heterotic string theory
(i). the free heterotic string. Nuclear Physics B, 256:253 – 284, 1985.
[27] M. Gualtieri. Generalized complex geometry. PhD thesis, January 2004.
[28] A. Giveon, N. Malkin, and E. Rabinovici. On discrete symmetries and fundamental domains
of target space. Physics Letters B, 238(1):57 – 64, 1990.
[29] T.H. Buscher. A symmetry of the string background field equations. Physics Letters B,
194(1):59 – 62, 1987.
[30] T.H. Buscher. Path-integral derivation of quantum duality in nonlinear sigma-models.
Physics Letters B, 201(4):466 – 472, 1988.
57
Bibliography
[31] E. Alvarez, L. Alvarez-Gaume, and Y. Lozano. An introduction to t-duality in string theory.
Nuclear Physics B - Proceedings Supplements, 41(1 – 3):1 – 20, 1995.
[32] N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roček. Hyper-kåhler metrics and supersymmetry. Comm. Math. Phys., 108(4):535–589, 1987.
[33] C. Montonen and David I. Olive. Magnetic monopoles as gauge particles? Phys. Lett.,
B72:117, 1977.
[34] Chris Hull and Barton Zwiebach. The gauge algebra of double field theory and courant
brackets. Journal of High Energy Physics, 2009(09):090, 2009.
[35] O. Hohm, D. Lust, and B. Zwiebach. The spacetime of double field theory: Review, remarks,
and outlook. Fortschritte der Physik, 61(10):926–966, 2013.
[36] Olaf Hohm and Barton Zwiebach. Large gauge transformations in double field theory.
Journal of High Energy Physics, 2013(2), 2013.
[37] A. Coimbra, Charles Strickland-Constable, and Daniel Waldram. Supergravity as generalised geometry i: type ii theories. Journal of High Energy Physics, 2011(11), 2011.
[38] A. Coimbra, C. Strickland-Constable, and D. Waldram. Generalised geometry and type ii
supergravity. Fortschritte der Physik, 60(9-10):982–986, 2012.
[39] David Andriot, Magdalena Larfors, Dieter Lüst, and Peter Patalong. A ten-dimensional
action for non-geometric fluxes. Journal of High Energy Physics, 2011(9), 2011.
[40] C.M. Hull and R.A. Reid-Edwards. Non-geometric backgrounds, doubled geometry and
generalised t-duality. Journal of High Energy Physics, 2009(09):014, 2009.
[41] D. Andriot, O. Hohm, M. Larfors, D. Lüst, and P. Patalong. Non-geometric fluxes in
supergravity and double field theory. Fortschritte der Physik, 60(11-12):1150–1186, 2012.
[42] Christofer M. Hull. A geometry for non-geometric string backgrounds. Journal of High
Energy Physics, 2005(10):065, 2005.
[43] Christopher M. Hull. Doubled geometry and t-folds. Journal of High Energy Physics,
2007(07):080, 2007.
[44] Imtak Jeon, Kanghoon Lee, and Jeong-Hyuck Park. Supersymmetric double field theory:
A stringy reformulation of supergravity. Phys. Rev. D, 85:081501, Apr 2012.
[45] Imtak Jeon, Kanghoon Lee, and Jeong-Hyuck Park. Incorporation of fermions into double
field theory. Journal of High Energy Physics, 2011(11), 2011.
58
Bibliography
[46] Olaf Hohm and Henning Samtleben. U-duality covariant gravity. Journal of High Energy
Physics, 2013(9), 2013.
[47] Olaf Hohm and Henning Samtleben. Exceptional form of d = 11 supergravity. Phys. Rev.
Lett., 111:231601, Dec 2013.
[48] David S. Berman and Daniel C. Thompson.
Duality symmetric string and m-theory.
Phys.Rept., 566:1–60, 2014.
[49] Gerardo Aldazabal, Walter Baron, Diego Marqués, and Carmen Núñez. The effective action
of double field theory. Journal of High Energy Physics, 2011(11), 2011.
59