Volume 250, number 1,2
PHYSICS LETTERS B
1 November 1990
Anomalies in non-polynomial closed string field theory
Michio K a k u 1,2
Institute for Advanced Study, Princeton, NJ 08540, USA
Received 17 June 1990
The complete classicalaction for the non-polynomialclosed string field theory was written down last year by the author and the
Kyoto group. It successfullyreproduces all closed string tree diagrams, but fails to reproduce modular invariant loop amplitudes.
In this paper we show that the classical action is also riddled with gauge anomalies. Thus, the classical action is not really gauge
invariant and fails as a quantum theory. The presence of gauge anomalies and the violation of modular invariance appear to be a
disaster for the theory. Actually,this is a blessingin disguise.We show that by adding new non-polynomialterms to the action, we
can simultaneously eliminate both the gauge anomalies and the modular-violating loop diagrams. We show this explicitly at the
one loop level and also for an infinite class of p-puncture, genus-gamplitudes, making use of a series of non-trivial identities. The
theory is thus an acceptable quantum theory. We comment on the origin of this strange link between local gauge anomalies and
global modular invariance.
1. Introduction
Recently, the author a n d the Kyoto group proposed the complete classical action [ 1,2 ] for the n o n - p o l y n o m ial closed string field theory, thus solving a long-standing problem in string field theory [ 3,4 ]. We presented two
proofs. The first proof was by brute force, calculating the lagrangian to the N = eighth level [ 5,6 ], furthering the
calculation of the N = 4 case [ 7 ], and then generalizing this to all levels. The second proof was more elegant,
using purely group theoretic arguments from geometric string field theory [8,9] to calculate the action to all
orders in N.
Although the theory successfully reproduced all tree amplitudes, it failed to provide a m o d u l a r i n v a r i a n t quant u m theory at the loop level, either at the one-puncture level [ 10 ] or at the two-puncture level [ t 1 ]. The conformal m a p for the p-puncture, genus-g amplitude was written down in ref. [ 11 ], a n d apparently this violation
of m o d u l a r invariance persists to higher loop levels.
However, it has been pointed out to us [ 12 ] that a string field theory which is truly gauge i n v a r i a n t should
also, in some sense, be automatically m o d u l a r invariant. Thus we are led to suspect that the theory is not really
gauge invariant.
In this paper, we verify this conjecture. We show that the classical theory is riddled with gauge anomalies. By
carefully analyzing the j a c o b i a n of the functional measure using Fujikawa's method [ 13 ], we find an infinite set
of gauge-violating anomalies.
However, in this paper we also propose the simultaneous solution to both problems. By adding a new infinite
set of new n o n - p o l y n o m i a l terms to the action, we find that we can cure both problems, i.e. we can eliminate the
a n o m a l o u s diagrams a n d automatically remove the loop diagrams which violate m o d u l a r invariance. We show
this explicitly at the one loop level and also for an infinite class of multi-loop diagrams. It is straightforward to
Permanent address: City College of the City University of New York, New York, NY 10031, USA.
2 Worksupported by NSF-PHY-8615338,NSF-INT-8715626,CUNY-FRAP-6-669347,and DE-FG02-90ER40542.
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0370-2693/90/$ 03.50 © 1990 - Elsevier SciencePublishers B.V. ( North-Holland )
Volume 250, n u m b e r 1,2
PHYSICS LETTERS B
1 November 1990
generalize this analysis to all possible multi-loop diagrams using a powerful recursion relation, although we have
not yet done this.
Using Fujikawa's approach, we also show that certain string field theories that have been proposed are actually
anomalous and hence unacceptable. This gives us a powerful tool by which to eliminate many of the string field
theories that have been proposed in the past.
Witten's original open string field theory can be shown to be anomaly-free. If • is an open string field, then
8~=QA+A,dp-~,A .
(1.1)
Then the jacobian appearing in the measure, to lowest order, is
lndet ~
=Tr(A,-,A)=0,
(1.2)
which vanishes because of anti-symmetry of the • operation.
However, for the covariantized light cone theory [14,15 ], the theory is anomalous. If q~(~P) represents an
open (closed) string field, then
8~=QA+A,~-dP,A+(~oCboA), 8~=QA+2~P,A,
(1.3)
then the jacobian does not cancel to lowest order:
In dett~)=Tr[A,-,A
+2(~oA)] ,0, In dett~)=Tr(2A,)@0,
(1.4)
and the actions are anomalous.
The same situation applies for the classical non-polynomial action. The action and its gauge invariance are
given by [ 1,5]
to=½(~lOTX>+~
n=3
½an(~"),
167Xo)=lQA)+ ~ fl, 17U"A).
n=l
(1.5)
However, the jacobian now includes terms (see fig. 1 ) which do not cancel:
In det~g-~- )= 2 , n/7, Tri ( 7s"- ~A) @0.
(1.6)
Thus, the classical non-polynomial action is unacceptable at the quantum level, and new terms must be added
to cancel this anomaly.
~F
8~
Tr ~:
8~'
A
~F
Fig. 1. The functional measure is not gauge invariant, and its
variation produces these non-invariant graphs.
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2. One-loop anomaly calculation
To cancel this anomalous term, we add the following infinite terms to the action and the gauge invariance
(see figs. 2 and 3; l equals the loop level ):
L=Lo+~
Lt,
187'>=15~Vo)+~ 15~).
l=1
(2.1)
l=1
In this paper, we will be concerned with the following sub-class of infinite corrections to the action and gauge
invariance:
Ll=
~
a~Trl(Dl~"),
15~t)=
n= 1;1= 1
fl~TrtlD/T"A),
~
n=0;l=
(2.2)
1
where DI is an operator insertion (to be determined shortly), and the trace operation is defined by
Trt(Dt V") - (2l+, ~.i sy~mp__lJ,
)
(kpl2p_,(kplzpD p < ~1211 < ~12/+ ~..-< ~121+n I V21+n)
,
(2.3)
where I V, ) satisfies the G o t o - N a k a conditions for the n-faced polyhedron (see refs. [ 1,5 ] for notations).
We now demand that the gauge variation coming from the jacobian cancels against the term coming from the
modified action:
In det ~
+ ~
1,5'/-'>= 0 .
(2.4)
For the lth loop level, our cancellation is based on the key recursion relation
Oo °
L~:
II o
qJ
Ln•
q.,
Fig. 2. To cancel the non-invariant terms arising from the measure, the lagrangian must be modified to include multi-loopgraphs. Some
of them are shown here.
~F
\
ud
° o
Fig. 3. The gauge variation of the field must also be supplemented to include the multi-loopgraphs shown here.
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1 November1990
Now let us first analyse the cancellation at the one loop level. The first term, coming from the jacobian of the
measure (fig. l ), is
67'o
oo
Tr,(-g-~;)= ~= n[3°Tr,( ~n-'A> .
(2.6)
One term coming from the variation of the action is given by (see fig. 4)
=
na~fl~n Trl na,fl,, <D~n-ll I ~A> .
n=0
(2.7)
n=3;m=0
The key to reducing this expression is to realize that the vertex function is not BRST invariant, but instead
obeys the following formula [ 3 ] (see fig. 5 ):
( - 1 ) n < ~wl [QA>+n<Q~II~n-IA>+
n--2
~ C~,<7'~-Vl I 7'va> = 0 .
(2.8)
p=l
We will multiply this identity with a D insertion and trace, arriving at (see fig. 6)
½ n ( n - 1 ) ( - l ) ~ W r z <QAII
+½n(n- 1 ) ( n - 2 )
~J'-2D>+ln(n-1)
Tr~ <all ~ - 2 { a , D} )
Tr~ (Q~[ [D ~"-3A )
n--2
= - ~ ½(n-p)(n-p-1)C~ Trt (
~w-v-2Ol I 7'PA>-½p(p-1)C~Trl
( ~-211D~v-EA>
p=l
-p(n-2)C~, Trl < ~n-P-ll
ID[ I UP-'A>.
(2.9)
Lastly, we have the contribution (see fig. 7)
<~
16~,> =
~Tr, flt~<Q~IID~A>+
n=0
~
0 I
na.fl,,,
n=3;m=O
Tr~ < ~r,/n-- 1 I ID ~"A>
.
(2.10)
We now add (2.6), (2.7), and (2.10), using (2.9). The system of equations is now highly over-constrained,
giving us numerous counterchecks of the correctness of our calculation. Solving the recursion relation (2.5), we
can now reassemble all terms and cancel the anomaly if we have
QA
Fig. 4. The variation of L1, contractedon IG~o>,gives rise to
these graphs.
Fig. 5. T h e p o l y h e d r a l v e r t e x f u n c t i o n is n o t B R S T i n v a r i a n t , b u t
o b e y s t h i s vertex identity.
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Volume 250, n u m b e r 1,2
PHYSICS LETTERS B
1 November 1990
Q~FN~
...
+
A
A
A
A
o ,
I'lOln•m=
,8~= ( - 1 )n+l
(m+2)(m+l)
(m+n+l)
,
A
Fig. 7. The variation of Lo, contracted onto 1 5 ~ ), produces these
graphs.
Fig. 6. When we contract the vertex identity in fig. 5 and form a
single loop, we find this new vertex identity.
flo=(__l ) ,,+t 2a,,,t
+
(n+2)(n+l)a~+2,
r,+m+~
Olm+n'~m+2
1 o
'
nOlnt~m=
(n+m)(n+l)(n+2)
(n+m+l)
1
c,,,+m+l
Oln+m~m
(2.11)
'
and {Q, D ] = 2 when sandwiched between string fields. Fortunately, a consistent solution of these overconstrained equations is given by (we set o~° = ~o~,)
f l ~ - ( n°+24n) l!
a,,+4,
ot,=~ - ½ ( n + l ) ( n + 2 ) a ° + 2 .
(2.12)
3. Multi-loop anomaly cancellation
The solution of the recursion relation (2.5) for the multi-loop case is now straightforward. Let us examine
the contribution to the anomaly cancellation from the following terms:
Trl (
~ ~'~--
2 ~
l,l'
~
8tPr = ~
nfllTu(D,~n-lA)
na~Trt(D,~pn-~l I Q / ) +
n= I;1= 1
2
2
Tr,(D,g~"-lll BmDt'
'" g-'mA),
re=O;/' = 1 n= l;l= 1
(3.1)
n=0;l=
l
m=O;l=
l n=3
The sum of these three terms must be zero by the recursion relation (2.5). Let us now contract the vertex
identity (2.8) with Dt and trace over the relevant legs. We arrive at the generalized vertex identity, shown in fig.
8. Using this identity, we can now set all terms equal to zero in the recursion relation (2.5), which yields the
following constraints:
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Volume250,number1,2
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PHYSICS LETTERS B
A
+A
+ /1",
+ ~
+...=0
Fig. 8. When the vertex identity in fig. 5 is contracted on multiloops, we find this generalized vertex identity.
fit = ( _ 1 ) . + , 2 ( / + 1)or.t+l , f f . = ( - l ) " ( n + l ) ( n + 2 ) a t . + 2 ,
ncGfl~= (n+m)p~n+2/(p~+m-I+Zl) -Irn+m-l+21
1
*-'m+21
OLn+m
l l'
(~..L~Dm+21"Dn+2l--l(Dm+n+21+21'--l~--l~l+l'
-z
~-t+t'
~ O l n f l m ~- . . . . . . J - t '
)
,
g'~m+n+21'+2l--1
t.t n + m X...,m + 2l,
.
(3.2)
where
n!
PT- (n_2i)!2ti!.
(3.3)
The fact that a solution exists o f this overconstrained set o f equations is highly non-trivial. The solution is
given by
fl~-
(n+21+2)!
2ln!l!
o
an+21+2
,
1
(n+2l)! o
otn = ( - 1)1 Un!l! OLn+21"
(3.4)
The calculation also yields the following non-trivial constraint:
p,.+.+2i+2j+l
"'-'m+2j
--
i'Jv. fl~fl~(m+n+2i+ 2j+l)(m+n n)
(j+j)! ]~i++Jmk
m+Zj
--1
(3.5,
which we can show is exactly satisfied, which is another check on our results.
4. Modular invariance
N o w that we have e l i m i n a t e d a certain infinite class o f anomalies, let us now check that our m o d i f i e d action
is also m o d u l a r invariant.
In particular, let us calculate the contribution o f all diagrams to t h e / - l o o p . N-puncture amplitude. The relevant terms in the action are given by
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Volume 250, number 1,2
o~ff-2u Trl (Dpl ~ - 2 p )
PHYSICS LETTERSB
l November 1990
.
(4.1)
p=0
Let us now trace over the external legs and calculate the amplitude for N string scattering with l loops, which
violates modular invariance. After calculating the appropriate combinatorial factors, we find that the amplitude
is proportional to
(± OllN_2pLIN----P2p) Trl (Dr[ ~_JN--21>
\p=O
(4.2)
,
/
where
L~=
M
(N_2l)!2tl ! .
(4.3)
Plugging in the value of o~~ found in ( 3.3 ) into (4.2), we find that the amplitude is proportional to
p=o
j
,v
,v-2p-
oo
,v ( N - 2 1 ) 2 q !
L oC)
]
( - I)P = 0 .
(4.4)
Notice that the term on the right is proportional to ( 1 - 1 )l, and hence vanishes. Thus, the terms (2.2) which
restore gauge invariance are also precisely the terms which restore modular invariance. What is remarkable
about this calculation is that the elimination of anomalies is so intimately related to the question of modular
invariance. At first, this seems puzzling, because the anomaly cancellation arises when restoring local gauge
invariance, while modular invariance strictly a global phenomenon on the world sheet.
The answer to this puzzle lies in the regularization scheme. Anomalies arise when we consider the ultra-violet
behavior of certain graphs in quantum field theory. However, the origin of this ultra-violet behavior is precisely
the infinite overcounting of moduli space (for z, the modular parameter, near the origin ). Thus, when we restore
modular invariance, we automatically eliminate the ultra-violet behavior of the loops graphs, which in turn
eliminates the anomalies. Thus, the origin of this curious puzzle, linking the local and global properties of string
theory, lies in a crucial feature common to both: the ultra-violet behavior of the loop graphs. By eliminating the
diseases of one, we automatically eliminate the diseases of the other.
5. Conclusion
Anomaly cancellation has proven to be one of the most productive and illuminating ways in which to construct (and eliminate) certain quantum field theories. By applying this method to the field theory of strings, we
find that several proposed string field theory actions are actually anomalous, including the classical action proposed in refs. [ 1,2] and also in refs. [ 14,15].
In this paper we added a new, non-polynomial term to the action which cancelled the anomalies coming from
the jacobian in the measure. We showed this explicitly at the one-loop level, and also for a certain infinite class
of multi-loop diagrams.
What is remarkable is that the addition of this new term also restores modular invariance to the theory. We
showed that one class of infinite diagrams which violate modular invariance cancels exactly.
There are still some loose ends, however.
Our recursion relation (2.5) is so general that we can easily construct more general infinite series of nonpolynomial loop terms and check for gauge invariance. However, we have only considered one such infinite set,
and are still checking others. Thus, the question of modular invariance to all orders is still being investigated,
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1 November 1990
although we feel confident that the solution o f (2.5) is so stringent that it will also fix the a m p l i t u d e s to be
m o d u l a r invariant to all orders.
Second, we are also investigating the regularization o f these graphs. We have implicitly assumed one type o f
regularization, in which the o p e r a t o r D is equal to a closed string propagator, multiplied by factors o f the zero
m o d e s o f the b ghost field, multiplied by the o p e r a t o r exp[~ (L0 + L0 - 2 ) ]. However, there are other choices,
such as using a gauge-invariant regularization.
Third, is also the p r o b l e m o f rewriting the a n o m a l y calculation in a way which reveals an underlying topological structure, as in o r d i n a r y gauge theory.
Fourth, it m a y be possible to rewrite some o f our field theory results in the language o f refs. [ 16,17 ], which
are based on conformal, rather than field theoretical, arguments, although it is not clear how to do this (since
our m o d u l a r p a r a m e t e r s are different for the loops).
A n d lastly, we note that geometric string field theory [ 8 - l 1 ] can be shown to be anomaly-free. F o r the open
string case, the theory is anomaly-free because the structure constant is anti-symmetric. F o r the closed loop case,
the calculation resembles closely the work o f refs. [ 18,19 ] on the cancellation o f Lorentz anomalies in light cone
field theory.
Acknowledgement
We would like to thank E. Witten, K. Kikkawa, S. Sawada, a n d L. H u a for crucial discussions on gauge invariance, m o d u l a r invariance, a n d Lorentz anomalies. We would like to thank the hospitality o f the Institute for
A d v a n c e d Study and Osaka University, where this work was conducted.
References
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Haba and J. Sobczyk (Birkhauser, Basel ).
[2] T. Kugo, H. Kunitomo and K. Suehiro, Phys. Len. B 226 (1989) 48;
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[3] M. Kaku and K. Kikkawa, Phys. Rev. D 10 (1974) 1110, 1823.
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[ 5 ] M. Kaku, Non-polynomial string field theory, preprint CCNY-HEP-89-6, Phys. Rev. D, to appear;
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[6] T. Kugo and K. Suehiro, preprint KUNS 988 HE(TH) 89/08.
[7] M. Kaku and J. Lykken, Phys. Rev. D 38 (1988) 3067.
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[ 13] K. Fujikawa, Phys. Rev. Lett. 42 (1979) 1195.
[ 14 ] H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D 34 (1986) 1360; Phys. Rev. D 35 (1987) 1318, 1356;
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