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Equivariant determinant bundles and supersymmetric closed strings
Volume 203, number 4
PHYSICS LETTERS B
7 April 1988
EQUIVARIANT DETERMINANT BUNDLES AND SUPERSYMMETRIC CLOSED STRINGS Miao LI 1 ICRA - International Centerfor Relativistic Astrophysics, Rome, Italy and Dipartimento di Fisica, Universitit "La Sapienza'" 1-00185 Rome, Italy Received 23 November 1987
This work is to point out the relationship between the geometric approach to the string theories of Bowick and Rajeev, and Pilch and Warner and the recent research on the index of the Dirac operator in loop space. We consider character-valued Dirac index bundles instead of the vacuum bundle over some parameter space such as DiffS~/S ~.Thus, some necessary conditions for the absence of the Virasoro anomaly can be derived from some formulas presented here.
Recently intense interest has arisen in geometric approaches to string theories. One of them was discovered by Bowick and Rajeev [ 1 ]. They showed that the anomaly o f the Virasoro algebra corresponds to the curvature o f a certain vector bundle over the coset space DiffS ~/S ~. The fibre o f this bundle is just the Hilbert space of strings, and the bundle is twisted by transformations in DiffS1/S 1. To obtain a "gauge" invariant theory, one must add the contribution to the curvature from ghost vacua to cancel the Virasoro anomaly, then the correct critical dimensions are obtained [ 1-3 ]. Pilch and Warner [4 ] consider the homogeneous v a c u u m vector bundle instead o f the infinite-dimensional vector bundle over Diff S~/S I, in this new bundle the fibre now consists of vacua. They also derived the correct result in that the curvature o f the bundle is just the Virasoro anomaly. In ref. [4] Li extends their work to the supersymmetric case and closed strings, namely when considering vacuum bundles on Super DiffS~/S 1, then the critical dimensions and levels of the ghost vacua are derived. All works mentioned above deal only with flat space-time cases, except for the string field theory o f Bowick and Rajeev. We believe that the formalism o f Pilch and Warner should also be applicable when space-time is curved, i.e., if we define a string theory On leave from Center for Astrophysics, University of Science and Technology of China, Hefei, Anhui 23029, P.R. China. 360
in which strings propagate in curved space-time by a nonlinear a-model, then in each sector the v a c u u m bundle over some space such as DiffS1/S l has a curvature which should be cancelled by a ghost v a c u u m bundle. In this spirit, if we can derive some proper formulae o f curvature, then o f course one can derive from these the result obtained in ref. [ 2 ] for group manifolds as a specific case. Let us start with the N = ½ supersymmetric nonlinear a-model, with ghosts neglected. The action is I = ; d2a[gijO+XiO_X j + i~vi (goO_ + O_ x kO~ko ) ~ ] ,
0+=0/0a ±, a+-=(l/x/~)(z+a).
(1)
It is well known that in order to be conformally anomaly free, some extra left-moving degrees should be included in the theory. Let V be a vector bundle over the space-time manifold M, endowed with a metric gaB; we introduce the fermions 2 "~, the section o f the pull-back bundle x * ( V ) on the world sheet. A term is added to eq. ( 1 ): A / = f dEtT[i,~A(gAS0+ +Ai.aBO+Xi)~, B - ½FoABVg~V:2A2BI,
(2)
where A~ and F~j are the connection and the curvature of the vector bundle V. N o w we have the nonlinear a-model of heterotic strings, provided we assume that
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the world sheet takes the form S IxR. To quantize this theory, introduce the commutators [pi(a), xJ( a ' ) ] = -igOf( a - a ' ) ,
gti( a), ~d ( a' )} =gijc~( a - a ' ) , •J.A(a), ..~B(a' ) } = g A e • ( a - - a '
) .
(3)
We see that the configuration space of the field x ( a ) is the loop space LM and its corresponding phase space is the tangent bundle T ( L M ) . The commutators of v / s h o w that the v / m a y be treated as y-matrices of T ( L M ) . As for the fields .~A, given a point x in LM, all sections of the pull-back bundle x * ( V ) over S ~ form on this point a vector space, when x varies then a vector bundle v is obtained. The commutators in eq. (3) tell us that the 2 'a are y-matrices of the bundle v so its corresponding Hilbert space consists of sections of the spinor bundle A(v) of v. let A(LM) denote the spinor bundle of LM, then the Hilbert space of the system is composed of sections of
A(LM)®A(v). It is obvious that the inner product of the Hilbert space is just the metric in the A(LM) lift from the metric of T ( L M ) : (fix, 6y) = f da g,j 6xi6y j ,
(4)
Define the metric in v by (21,22)=
A8, f dagA82122
(5)
then the inner product in A(v) is the lift from the metric defined above. Finally, from the metrics in A(LM) and A(v) we obtain the inner product of the Hilbert space, the metric of A (LM) ®A ( v ). So far, we completed the discussion about the geometric description of the Hilbert space of the N = ½ non-linear a-model. The next step is to introduce the super-Virasoro algebra, L,,, /5,, G~, which statisfies some Poisson brackets [ 4 ]. The hamiltonian H and m o m e n t u m P are H = L o +/5o,
P=Lo-/5o •
(6)
We assume that H and P are properly regularized. H is not invariant under transformations generated by L,,,/5, and Gt, so the vacua defined by HI g2) = 0 also change by these transformations except for those generated by Lo, /50 and Go (in the case of the Ra-
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mond sector). Thus, we obtain a vacuum bundle over Super DiffS l/S' ®DiffS ~/S', with a metric induced from the metric of A ( L M ) ® A ( v ) . Its curvature should be cancelled by that coming from the contribution of ghosts. However, in the case of curved space-time it is impossible to calculate the explicit form of the curvature. Happily, even though we can not do so we can still turn to topological considerations, then some powerful tools make it possible to obtain some information from the restriction that the curvature must vanish. To show this, first let us introduce the supercharge:
Q = x f~ f da giJ~'iO+xj
,
D
0x J)
D 6 D x i ( a ) - 8xi(a) +Ogok~J~ k.
(7)
We have the relations Q--Go, L o = Q 2. It was first pointed out by Witten [5 ] that Q is just the Dirac operator in loop space. Its character-valued index was recently discussed in refs. [ 6-8 ]. Now the vacua defined by H [ g 2 ) = 0 satisfy L0[g2)=/5o[I2)=0, so the following condition is equivalent to HI s'2) = 0: Q[£2) =0,
P[g2)--0.
(8)
The first condition in eq. (14) implies that all vacua fall into Ker Q, but usually the dimension of Ker Q is infinite and we have to put the second condition to specify the vacua. By transformations in Super D i f f S ' / S 1®DiffS 1/S' we may obtain a fibre bundle Ker Q over this space. But we encounter here some difficulties. First, this bundle is infinite-dimensional, hence is not easy to he controlled. Secondly, by requesting that P i g 2 ) = 0 , the dimension of the fibre may j u m p (on a more general base space). To solve this problem, we introduce the fermionic operator ( - 1 )F satisfying ( -- 1 ) r Q + Q ( _ 1 )F=0, to define the index bundle Ind Q
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Ind Q = K e r Ql - C o k e r Ql , 0 =(Q,
(9)
Then, the seco :d difficulty is overcome. We shall stress that the operator ( - 1 )F is just ?5 in loop space, so eq. (9) defines the usual index bundle of the Dirac operator. To reduce this bundle in order to be finitedimensional, we need to put the restriction PI £27 = 0. Note that P is the generator of the rotation group S ~, which is the isometry group of LM and P commutes with both Q and ( - 1 )v. Thus we are able to introduce the character-valued index bundle IndqQ= ~ ) q~(Ker Qi - Coker Ql )~, 2
(10)
where (Ker QI - C o k e r QI )4 = { I q/):[ gt) ~Ind Q, P[ gt) =21 ~ 7 } .
(11)
Suppose the ghost vacuum bundle is G, it should be pointed out that this bundle does not depend on curved space-time, fixed once the sector is fixed. For heterotic strings there are two sectors of ghosts, but here we are only interested in the Dirac-Ramond operator, i.e., consider the Ramond sector for rightmoving q/only. Therefore, we only need to consider the ghost vacuum bundle of the Ramond sector. Now our requirement is that the curvature of the bundle (Ker Q~ + Coker Q~ )a=o®G (if well-defined, this is the case when the base space is the coset space of the super-Virasoro group) should be zero. Since (Ker Q~)x=o®G and (Coker Q~)a=o®G are orthogonal their curvature must vanish, so must that of (Ker Q~ - Coker Q~ )x=o®G. We deduce then that the first Chern class of the bundle ( K e r Q ~ Coker Q~)~=o®G must be zero, for the first Chern class is the trace of the curvature. Therefore, the problem is simplified to solving the following topological problem: to calculate the qO term of the first Chern class of the character-valued index bundle in eq. (10). The character-valued (or equivalent) first Chern class is the two-form in the character-valued Chern character: ChqQ= ~ q~Ch(Ker Q~ - C o k e r Q~ )a.
(12)
2
It is a dosed form on base space, the first term (pure
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number) is the character-valued index of Q, already calculated in refs. [6-8 ]. The second term, the twoform is what we want to calculate. In refs. [9,10], Bismut and Freed proved that in finite dimensions the curvature of the Quillen connection of the determinant bundle DET D of the Dirac operator D is equal to the first Chern class of the corresponding index bundle Ind D, following the work of Quillen [ 11 ]. This result can be extended straightforwardly to the character-valued case [ 12 ]. Then, the qO term in c~ (q) is just the curvature of the equivariant determinant bundle DEToQ, so called first by Witten [5], where DEToQ is the bundle so defined that the equivariant determinant det0Q, the determinant of Q restricted in the subspace P = 0, is a section of it. As shown in ref. [ 13 ], also in refs. [ 9,10], the anomaly of the determinant is related to its family index, so the qo term in the first Chern class of the character-valued index bundle represents the anomaly of the equivariant determinant. In ref. [ 5 ] Witten questioned the anomaly of the equivariant determinant being a natural concept in string field theory. By our arguments presented so far, it seems that it should be. Before proceeding further, we establish a framework in which one can derive a formula of the character-valued family index (indeed, in ref. [ 6 ] Pilch et al. have already touched this). Now each of the operators Q, P and the generators of the superVirasoro algebra may be viewed as a family operator parametrized by the space B=Super DiffS~/S~® DiffS1/S I. An element in the coset space B acts on LM, therefore on all operators defined on LM. Given a point x in LM and an infinitesimal generator L in B, x transforms as (13)
x'=x+{x,L},
where {, } is the Poisson bracket. We have then a fibred space X - , B whose fibre space is LM. Still, we use the symbol T ( L M ) to denote the tangent bundle on X along the fibre LM. The inner product (4) can be used to define the metric of T (LM). Note that if locally X is written as U × LM (U is an open subset of B), on the point around the origin specified by L, the metric is (Sx, By) = f d a g i : S x i ' S y j' . d
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The metric then also depends on B. Along the same way, the vector bundles v and A(v) defined on LM extend to be on X. The same symbols are used to denote their extensions. Thus, the family operator Q is now the Dirac operator acting on the bundle A(LM)®A(v) over X. The character-valued index theorem can be formulated as an integration over the fixed-point set, this will be true for family ones. Now the operator P is lifted to act on fibres of X, in each fibre its fixed-point set is M. Then the fixed-point set of X is a fibred space Y--,B with fibre space M. Bundless T ( L M ) , v are reduced to bundles "i'M and ~r over B, with corresponding reduced metrices. We denote their curvature twoforms as R and F. Bismut and Freed [ 9 ] proved that for the Dirac operator D coupled to A (TM) ® A(V) ( it is assumed that d = d i m M is even and M is compact without boundary), the Chern character of the index bundle is Ch(Xnd D) = [ ,,i('i~M)Ch (Xr),
(14)
M
n/2
ChqQ=q'o-d'/i2f,4(R) n 2 c o s h - ~ M
B=t
× f i d e t ( l + q a~ e x p ( i F / 2 g ) ) n=l det (I-q 2" exp(iR/2~) )
~ ! fl=l ~ ( 02 (yfl/27~ ~--~ [~') t~]
a/e(O(x~/2~lO)-'
×I-I ,~=~
x,:l(r)
'
t / ( z ) = q 1/'2 f i ( l - q 2 " ) .
n=l
(16)
In principle, the above equation enables us to evaluate the curvature of the equivariant determinant bundle. Introducing the ghost vacuum bundle G, construct the bundle IndqQ~)G = ( ~ qa(Ker Q~ - Coker QI )a (~) G ,
2
where the integral is performed on fibre space, it is the direct extension of the original index theorem. While in refs. [ 6-8 ], the character-valued index of Q was verified to be
IndqQ=q("-a)/12
its Chern character is Ch (IndqQ(~) G)
/=1
m=l
(15) where TM, V are the original bundles over M,
SoT=(~q"S"T , ^ q V = ( ~ q t ^ ' T , 1>/0
n is the dimension of V, supposed even. The generalization of eqs. (14) and ( 15 ) to the character-valued family index is straightforward, it should take a form as in eq. (15) (in finite dimensions this can be proved by use of the SUSY quantum mechanics [ 12 ] ), TM and V are placed by "i'M and ~r. Let x~, y# be skew eigen-values of iR/2n and iF/2rt, respectively, and q=exp(inr), then the character-valued Chern character can be cast into the form
=ChqQ C h ( G )
= [F(q) +c, (q) +... ] [2 +c~ (G) +... l = 2 F ( q ) + [2c,
X (.4(TM)Ch(~Sq2, T M ( ~ A ( V ) ( ~ A q2,,V,M)
n>~0
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(q)+F(q)c, ( G ) ] +...
(17)
where F(q) is the character-valued index and 2 the dimension of G, ct (G) is already known [4]. Vanishing of the total curvature yields [2c~ (q)+F(q)c, (G)]~=o = 0 ,
(18)
the subscript 2 = 0 indicates the qOterm in the bracket. Settle it in another way, if it is possible to define a Dirac operator Q' on loop space in the transverse directions, the ghost effect is included. We need only to calculate ChqQ' and put the restriction el (q,Q') =0. Notice that in the calculation of the character-valued index of Q', one just multiplies the character-valued index of Q by a f a c t o r f ( q ) , the determinant of the ghost sector in the path integral representation [ 8 ]. In the same spirit, we get the Chern character of IndqQ' by multiplying ChqQwith a f a c t o r f ( q )
ChqQ' = ChqQf( q) . (19) c~(q)f(q) is the two-form in ChqQf(q). Even when 363
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we can not define Q' in the transverse directions, eq. (18) tells that we can use ChqQf(q) instead of Ch (IndqQ® G). We obtain an equation equivalent to eq. (18):
(c, (q)f(q) )4=0 = 0 .
(20)
The ghost contribution f ( q ) in the heterotic string is
[8] f(q)=gl '/6 f i (1--q2n)2=rl2(T) .
(21)
n=l
The quantity in eq. (19) is a modular form of weight - d / 2 + 1 under the level-two subgroup F(2 ) ofPSL(2,2~), i f T r R 2 - T r F 2 = 0 (under F ( 2 ) , R and F transform as R - , R / ( c r + d ) and F--,F/(c~+d). This condition was also obtained in ref. [ 14 ] as the condition being gravitational anomaly-free. There one constructs the anomaly generating function on the one-loop level. For a given sector, the generating function is a modular form of weight - d / 2 + 1 too, provided T r R 2 - T r F 2 = 0 . Here R and F are ordinary curvatures. The reason that ChqQf(q) is a modular form under the level-two subgroup only is that this subgroup only keeps spin structures or sectors intact. The author ofref. [ 14] showed that if the theory is modular invariant, the condition Tr R 2- - Tr F 2 insures the anomaly generating function of a certain combination of sectors to be a modular form of weight - d/2 + 1 of the group PSL (2,Z). Only the ( d + 2 ) - f o r m in the generating function contributes to the anomaly, its coefficient is a modular form of weight 2, the qO term vanishes. In our case, we are interested in the two-form in ChqQf(q), or in the ( d + 2)-form in the integrant in (19). But now this is a form which has d degrees in the direction of fibre M and two degrees in the direction of base space. We can not use the argument in ref. [ 14 ] to derive the same condition when we deal with one sector only. Properly combining different sectors, we also get a modular form under the whole group, then the argument in ref. [ 14 ] applies. Our anomaly-free (or curvature-vanishing) condition is ( T r R e - T r F 2 ) ¢ ° , l . 2 ) = 0 , here (0,1,2) means that only terms of degree 0,1 and 2 are included in base space. However, the vanishing of the curvature of the independent sector will give more restrictions than this one, the derivation of all restrictions is the task of our next work. We describe shortly how to draw the above restriction. 364
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In the derivation of eqs. (16) and (17) we have assumed that the fields 2A are in the Ramond sector, and the fermion number F which enables us to make a decomposition of Q as in eq. (9) does not take that of 2A into account. In a path integral calculation, the 2A are antiperiodic in the time direction, they correspond to spin structure ( +, - ) and give rise to functions 02. When the 2A are still in the Ramond sector, while the fermion number contains a contribution from the fields 2A, the spin structure ( + , + ) for 2" is then introduced in the path integral calculation. In a similar manner the spin structures ( - , - ) and ( - , + ) can also be introduced. Use index i to denote four spin structures of 2, and Q~ denotes the Dirac operator in each case. Let 2 i be either + 1 o r - 1, we define a new fibre bundle E = ~i2iIndqQ i, its Chern character and first Chern class are C h ( E ) = Zi2~ChqQ j, el(E) = Z~2'cl (q,Q~). The latter corresponds to the curvature of the equivariant determinant bundle ®~(DEToQ~) z' (because el (®i(DEToQi) ~' ) = Y~)t~c~(DEToQ ~) ). To take the contribution of the ghost into account, we simply mutiply Ch(E) b y f ( q ) as before. In case of the Spin(32)/Z2 string, let d = 10. The theory is modular invariant provided • 2 = 2 3 = • 4 = 1 and 2~ i s + l o r - 1 . Denote the integrant in C h ( E ) f ( q ) as F(q,R,F), it is expressible in terms of functions 0~. Define
F~ (q,R,F) = exp [ ( 1 / 64~z4 ) G2 (q 2 ) X ( T r R 2 - T r F 2) ]F(q,R,F), G2(q2)=~ -
1 - 2 4 ~ a~(n)q 2n .
(22)
Under the modular transformation ~ ( a ~ + b ) /
(cT+d)~PSL(2,7/), R-~R/c~+d, F--,F/cT+d, Fj ( q,R,F) --, ( c~ + d) - 4F~ ( q,R,F) .
( 23)
Now a certain ( d + 2 ) - f o r m in F is the sum of a ( d + 2 ) - f o r m in F~ and a ( d - 2 ) - f o r m in F~ times T r R 2 - T r F 2, etc. If (TrR2-TrF2)(°,~.2)=O, the ( d + 2)-form in F we are interested in is just that in F~, this is zero because the coefficient g(~) of this term is a modular form of weight two with vanishing qO term (see ref. [ 141 ). Therefore, when the following condition holds: (Tr R 2 - T r F2) (°.~'2)= 0 ,
(24)
the curvature of equivariant bundle ®~(DEToQ~) a'
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cancels that from the ghost contribution. At present, we do not k n o w i f this also guarantees that the curvature o f each factor DEToQ i cancels the ghost contribution. This argument applies to all theories which are m o d u l a r invariant. All known such theories are classified in ref. [14]. N o t e that here the c o n d i t i o n ( T r R 2 _ Tr F 2) ~o~= 0 is the anomaly-free c o n d i t i o n in the o r d i n a r y sense. This tells us that starting from the e q u i v a r i a n t d e t e r m i n a n t b u n d l e one can d r a w some correct result which coincides with what was o b t a i n e d by other arguments used before. We turn to type-II strings. T h e r e are two supersymmetries Q~,Q2 Lo = Q ~ ,
/So=Q~ •
(25)
The H i l b e r t space is no longer the space o f sections o f the s p i n o r b u n d l e A ( L M ) , but the space o f sections o f the tangent b u n d l e T ( L M ) (or other twisting versions o f this bundle, in the case o f other sectors). N e i t h e r Q~ nor Q2 is the D i r a c o p e r a t o r on LM. Using ( - 1 )FR we split Q~ into two parts, here FR is the f e r m i o n n u m b e r o f the right-moving sector. Again the vacua are defined by the requirements Q~ I-Q ) =P112) = 0. Define the following charactervalued index bundle: ( ~ q ~ ( K e r Q1 - C o k e r Q~ )~ .
Taking that both the right sector a n d the left sector are periodic, the index f o r m u l a
F(q)=q-a/S(L(TM)Ch(~Sq2nTM(~ ^ q2,TM,M) 1=1
(27) generalizes to the family index ChqQt =
f ~ x,~O2(x,J2nl'r) ~=l
01(x,/2~rlr)
(28)
M
To include the ghost contribution, simple multiply eq. ( 3 4 ) by [8]
f(q)= n=l l~I (l-q2n) 2 (1 +q2n) 2"
m o d u l a r invariant. So a nontrivial restriction should be d e r i v e d from a single sector. O f course, a nontrivial restriction exists in some cases. Let us consider a special case, M = T a and be it endowed with a flat metric. N o w the ( d + 2 ) - f o r m in the integrant of ChqQlf(q) is zero, because o n l y R ~1) and R ~2~ m a y be non-zero, b y these terms we can not construct the ( d + 2 ) - f o r m we need i f d ) 2 . This should be so because now F(q)=0, this means that d i m K e r Q~ = d i m Coker Q~, the contributions from the two parts cancel. Therefore we get no information from ChqQ~f(q). But if we start from v a c u u m bundle, we can gain a nontrivial restriction, d = 10. Summarizing, our work o f reducing the geometric restriction to the topological one gives a powerful tool to investigate anomalies such as the Virasoro one. But this m e t h o d also loses some information. F o r instance, in the case we presented above. I a m grateful to Dr. H.B. G a o for helpful discussions; I would like to thank Professor R. Ruffini for his k i n d hospitality at I C R A and the D e p a r t m e n t o f Physics, R o m e University.
References
(26)
2
n=l
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(29)
ChqQ~f(q) is a m o d u l a r form o f weight -d/2+ 1 o f the level-two subgroup F ( 2 ) without any condition. I f we c o m b i n e all sectors p r o p e r l y to form an e q u i v a r i a n t d e t e r m i n a n t bundle, then its curvature a u t o m a t i c a l l y vanishes ( n o w we have to m u l t i p l y the different ghost v a c u u m bundles G i ) , i f the theory is
[1 ] B.J. Bowick and S.G. Rajeev, Phys. Rev. Lett. 58 (1987) 535; Nucl. Phys. B 293 (1987) 348. [2] J. Mickelson, Commun. Math. Phys. 112 (1987) 653. [3] D. Harari, D.K. Hong, P. Ramond and V. Rodgers, Nucl. Phys. B 294 (1987) 556; Z.Y. Zhao, K. Wu and T. Saito, SISSA preprint (1987). [4] K. Pilch and N.P. Warner, MIT preprint CTP-1457 ( 1987); M. Li, ICTP preprint ( 1987 ); M. Lauer, MIT preprint (1987). [ 5 ] E. Witten, in: Syrup. on Anomalies, geometry, topology, eds. W.A. Bardeen and A.R. White (World Scientific, Singapore, 1985 ). [6] K. Pilch, A.N. Schellekens and N.P. Warner, Nucl. Phys. B 287 (1987) 362; K.-K. Li, MIT preprint CTP-1460 (1987). [7] E. Witten, Commun. Math. Phys. 109 (1987) 525; Princeton preprint PUPT-1050 (1987). [8] O. Alvarez, T.P. Killingback, M. Mangano and P. Windey, Commun. Math. Phys. 111 (1987) 1. [9] J.M. Bismut and D.S. Freed, Commun. Math. Phys. 106 (1986) 159; 107 (1986) 103. [ 10] D.S. Freed, Commun. Math. Phys. 107 (1987) 483. [ 111 D. Quillen, Funct. Anal. Appl. 19 ( 1985 ) 31, [ 12 ] M. Li, in prepration. 365
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[13]M.F. Atiyah and I.M. Singer, Proc. Natl. Acad. Sci. 81 (1984) 2597; O. Alvarez, I.M. Singer and B. Zumino, Commun. Math.
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Phys. 96 (1984) 409. [ 14 ] A.N. Schellekens and N.P. Warner, Phys. Lett. B 171 (1986) 317; Nucl. Phys. B 287 ( 1987 ) 317.
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EQUIVARIANT DETERMINANT BUNDLES AND SUPERSYMMETRIC CLOSED STRINGS Miao LI 1 ICRA - International Centerfor Relativistic Astrophysics, Rome, Italy and Dipartimento di Fisica, Universitit "La Sapienza'" 1-00185 Rome, Italy Received 23 November 1987
This work is to point out the relationship between the geometric approach to the string theories of Bowick and Rajeev, and Pilch and Warner and the recent research on the index of the Dirac operator in loop space. We consider character-valued Dirac index bundles instead of the vacuum bundle over some parameter space such as DiffS~/S ~.Thus, some necessary conditions for the absence of the Virasoro anomaly can be derived from some formulas presented here.
Recently intense interest has arisen in geometric approaches to string theories. One of them was discovered by Bowick and Rajeev [ 1 ]. They showed that the anomaly o f the Virasoro algebra corresponds to the curvature o f a certain vector bundle over the coset space DiffS ~/S ~. The fibre o f this bundle is just the Hilbert space of strings, and the bundle is twisted by transformations in DiffS1/S 1. To obtain a "gauge" invariant theory, one must add the contribution to the curvature from ghost vacua to cancel the Virasoro anomaly, then the correct critical dimensions are obtained [ 1-3 ]. Pilch and Warner [4 ] consider the homogeneous v a c u u m vector bundle instead o f the infinite-dimensional vector bundle over Diff S~/S I, in this new bundle the fibre now consists of vacua. They also derived the correct result in that the curvature o f the bundle is just the Virasoro anomaly. In ref. [4] Li extends their work to the supersymmetric case and closed strings, namely when considering vacuum bundles on Super DiffS~/S 1, then the critical dimensions and levels of the ghost vacua are derived. All works mentioned above deal only with flat space-time cases, except for the string field theory o f Bowick and Rajeev. We believe that the formalism o f Pilch and Warner should also be applicable when space-time is curved, i.e., if we define a string theory On leave from Center for Astrophysics, University of Science and Technology of China, Hefei, Anhui 23029, P.R. China. 360
in which strings propagate in curved space-time by a nonlinear a-model, then in each sector the v a c u u m bundle over some space such as DiffS1/S l has a curvature which should be cancelled by a ghost v a c u u m bundle. In this spirit, if we can derive some proper formulae o f curvature, then o f course one can derive from these the result obtained in ref. [ 2 ] for group manifolds as a specific case. Let us start with the N = ½ supersymmetric nonlinear a-model, with ghosts neglected. The action is I = ; d2a[gijO+XiO_X j + i~vi (goO_ + O_ x kO~ko ) ~ ] ,
0+=0/0a ±, a+-=(l/x/~)(z+a).
(1)
It is well known that in order to be conformally anomaly free, some extra left-moving degrees should be included in the theory. Let V be a vector bundle over the space-time manifold M, endowed with a metric gaB; we introduce the fermions 2 "~, the section o f the pull-back bundle x * ( V ) on the world sheet. A term is added to eq. ( 1 ): A / = f dEtT[i,~A(gAS0+ +Ai.aBO+Xi)~, B - ½FoABVg~V:2A2BI,
(2)
where A~ and F~j are the connection and the curvature of the vector bundle V. N o w we have the nonlinear a-model of heterotic strings, provided we assume that
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the world sheet takes the form S IxR. To quantize this theory, introduce the commutators [pi(a), xJ( a ' ) ] = -igOf( a - a ' ) ,
gti( a), ~d ( a' )} =gijc~( a - a ' ) , •J.A(a), ..~B(a' ) } = g A e • ( a - - a '
) .
(3)
We see that the configuration space of the field x ( a ) is the loop space LM and its corresponding phase space is the tangent bundle T ( L M ) . The commutators of v / s h o w that the v / m a y be treated as y-matrices of T ( L M ) . As for the fields .~A, given a point x in LM, all sections of the pull-back bundle x * ( V ) over S ~ form on this point a vector space, when x varies then a vector bundle v is obtained. The commutators in eq. (3) tell us that the 2 'a are y-matrices of the bundle v so its corresponding Hilbert space consists of sections of the spinor bundle A(v) of v. let A(LM) denote the spinor bundle of LM, then the Hilbert space of the system is composed of sections of
A(LM)®A(v). It is obvious that the inner product of the Hilbert space is just the metric in the A(LM) lift from the metric of T ( L M ) : (fix, 6y) = f da g,j 6xi6y j ,
(4)
Define the metric in v by (21,22)=
A8, f dagA82122
(5)
then the inner product in A(v) is the lift from the metric defined above. Finally, from the metrics in A(LM) and A(v) we obtain the inner product of the Hilbert space, the metric of A (LM) ®A ( v ). So far, we completed the discussion about the geometric description of the Hilbert space of the N = ½ non-linear a-model. The next step is to introduce the super-Virasoro algebra, L,,, /5,, G~, which statisfies some Poisson brackets [ 4 ]. The hamiltonian H and m o m e n t u m P are H = L o +/5o,
P=Lo-/5o •
(6)
We assume that H and P are properly regularized. H is not invariant under transformations generated by L,,,/5, and Gt, so the vacua defined by HI g2) = 0 also change by these transformations except for those generated by Lo, /50 and Go (in the case of the Ra-
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mond sector). Thus, we obtain a vacuum bundle over Super DiffS l/S' ®DiffS ~/S', with a metric induced from the metric of A ( L M ) ® A ( v ) . Its curvature should be cancelled by that coming from the contribution of ghosts. However, in the case of curved space-time it is impossible to calculate the explicit form of the curvature. Happily, even though we can not do so we can still turn to topological considerations, then some powerful tools make it possible to obtain some information from the restriction that the curvature must vanish. To show this, first let us introduce the supercharge:
Q = x f~ f da giJ~'iO+xj
,
D
0x J)
D 6 D x i ( a ) - 8xi(a) +Ogok~J~ k.
(7)
We have the relations Q--Go, L o = Q 2. It was first pointed out by Witten [5 ] that Q is just the Dirac operator in loop space. Its character-valued index was recently discussed in refs. [ 6-8 ]. Now the vacua defined by H [ g 2 ) = 0 satisfy L0[g2)=/5o[I2)=0, so the following condition is equivalent to HI s'2) = 0: Q[£2) =0,
P[g2)--0.
(8)
The first condition in eq. (14) implies that all vacua fall into Ker Q, but usually the dimension of Ker Q is infinite and we have to put the second condition to specify the vacua. By transformations in Super D i f f S ' / S 1®DiffS 1/S' we may obtain a fibre bundle Ker Q over this space. But we encounter here some difficulties. First, this bundle is infinite-dimensional, hence is not easy to he controlled. Secondly, by requesting that P i g 2 ) = 0 , the dimension of the fibre may j u m p (on a more general base space). To solve this problem, we introduce the fermionic operator ( - 1 )F satisfying ( -- 1 ) r Q + Q ( _ 1 )F=0, to define the index bundle Ind Q
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Ind Q = K e r Ql - C o k e r Ql , 0 =(Q,
(9)
Then, the seco :d difficulty is overcome. We shall stress that the operator ( - 1 )F is just ?5 in loop space, so eq. (9) defines the usual index bundle of the Dirac operator. To reduce this bundle in order to be finitedimensional, we need to put the restriction PI £27 = 0. Note that P is the generator of the rotation group S ~, which is the isometry group of LM and P commutes with both Q and ( - 1 )v. Thus we are able to introduce the character-valued index bundle IndqQ= ~ ) q~(Ker Qi - Coker Ql )~, 2
(10)
where (Ker QI - C o k e r QI )4 = { I q/):[ gt) ~Ind Q, P[ gt) =21 ~ 7 } .
(11)
Suppose the ghost vacuum bundle is G, it should be pointed out that this bundle does not depend on curved space-time, fixed once the sector is fixed. For heterotic strings there are two sectors of ghosts, but here we are only interested in the Dirac-Ramond operator, i.e., consider the Ramond sector for rightmoving q/only. Therefore, we only need to consider the ghost vacuum bundle of the Ramond sector. Now our requirement is that the curvature of the bundle (Ker Q~ + Coker Q~ )a=o®G (if well-defined, this is the case when the base space is the coset space of the super-Virasoro group) should be zero. Since (Ker Q~)x=o®G and (Coker Q~)a=o®G are orthogonal their curvature must vanish, so must that of (Ker Q~ - Coker Q~ )x=o®G. We deduce then that the first Chern class of the bundle ( K e r Q ~ Coker Q~)~=o®G must be zero, for the first Chern class is the trace of the curvature. Therefore, the problem is simplified to solving the following topological problem: to calculate the qO term of the first Chern class of the character-valued index bundle in eq. (10). The character-valued (or equivalent) first Chern class is the two-form in the character-valued Chern character: ChqQ= ~ q~Ch(Ker Q~ - C o k e r Q~ )a.
(12)
2
It is a dosed form on base space, the first term (pure
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number) is the character-valued index of Q, already calculated in refs. [6-8 ]. The second term, the twoform is what we want to calculate. In refs. [9,10], Bismut and Freed proved that in finite dimensions the curvature of the Quillen connection of the determinant bundle DET D of the Dirac operator D is equal to the first Chern class of the corresponding index bundle Ind D, following the work of Quillen [ 11 ]. This result can be extended straightforwardly to the character-valued case [ 12 ]. Then, the qO term in c~ (q) is just the curvature of the equivariant determinant bundle DEToQ, so called first by Witten [5], where DEToQ is the bundle so defined that the equivariant determinant det0Q, the determinant of Q restricted in the subspace P = 0, is a section of it. As shown in ref. [ 13 ], also in refs. [ 9,10], the anomaly of the determinant is related to its family index, so the qo term in the first Chern class of the character-valued index bundle represents the anomaly of the equivariant determinant. In ref. [ 5 ] Witten questioned the anomaly of the equivariant determinant being a natural concept in string field theory. By our arguments presented so far, it seems that it should be. Before proceeding further, we establish a framework in which one can derive a formula of the character-valued family index (indeed, in ref. [ 6 ] Pilch et al. have already touched this). Now each of the operators Q, P and the generators of the superVirasoro algebra may be viewed as a family operator parametrized by the space B=Super DiffS~/S~® DiffS1/S I. An element in the coset space B acts on LM, therefore on all operators defined on LM. Given a point x in LM and an infinitesimal generator L in B, x transforms as (13)
x'=x+{x,L},
where {, } is the Poisson bracket. We have then a fibred space X - , B whose fibre space is LM. Still, we use the symbol T ( L M ) to denote the tangent bundle on X along the fibre LM. The inner product (4) can be used to define the metric of T (LM). Note that if locally X is written as U × LM (U is an open subset of B), on the point around the origin specified by L, the metric is (Sx, By) = f d a g i : S x i ' S y j' . d
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The metric then also depends on B. Along the same way, the vector bundles v and A(v) defined on LM extend to be on X. The same symbols are used to denote their extensions. Thus, the family operator Q is now the Dirac operator acting on the bundle A(LM)®A(v) over X. The character-valued index theorem can be formulated as an integration over the fixed-point set, this will be true for family ones. Now the operator P is lifted to act on fibres of X, in each fibre its fixed-point set is M. Then the fixed-point set of X is a fibred space Y--,B with fibre space M. Bundless T ( L M ) , v are reduced to bundles "i'M and ~r over B, with corresponding reduced metrices. We denote their curvature twoforms as R and F. Bismut and Freed [ 9 ] proved that for the Dirac operator D coupled to A (TM) ® A(V) ( it is assumed that d = d i m M is even and M is compact without boundary), the Chern character of the index bundle is Ch(Xnd D) = [ ,,i('i~M)Ch (Xr),
(14)
M
n/2
ChqQ=q'o-d'/i2f,4(R) n 2 c o s h - ~ M
B=t
× f i d e t ( l + q a~ e x p ( i F / 2 g ) ) n=l det (I-q 2" exp(iR/2~) )
~ ! fl=l ~ ( 02 (yfl/27~ ~--~ [~') t~]
a/e(O(x~/2~lO)-'
×I-I ,~=~
x,:l(r)
'
t / ( z ) = q 1/'2 f i ( l - q 2 " ) .
n=l
(16)
In principle, the above equation enables us to evaluate the curvature of the equivariant determinant bundle. Introducing the ghost vacuum bundle G, construct the bundle IndqQ~)G = ( ~ qa(Ker Q~ - Coker QI )a (~) G ,
2
where the integral is performed on fibre space, it is the direct extension of the original index theorem. While in refs. [ 6-8 ], the character-valued index of Q was verified to be
IndqQ=q("-a)/12
its Chern character is Ch (IndqQ(~) G)
/=1
m=l
(15) where TM, V are the original bundles over M,
SoT=(~q"S"T , ^ q V = ( ~ q t ^ ' T , 1>/0
n is the dimension of V, supposed even. The generalization of eqs. (14) and ( 15 ) to the character-valued family index is straightforward, it should take a form as in eq. (15) (in finite dimensions this can be proved by use of the SUSY quantum mechanics [ 12 ] ), TM and V are placed by "i'M and ~r. Let x~, y# be skew eigen-values of iR/2n and iF/2rt, respectively, and q=exp(inr), then the character-valued Chern character can be cast into the form
=ChqQ C h ( G )
= [F(q) +c, (q) +... ] [2 +c~ (G) +... l = 2 F ( q ) + [2c,
X (.4(TM)Ch(~Sq2, T M ( ~ A ( V ) ( ~ A q2,,V,M)
n>~0
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(q)+F(q)c, ( G ) ] +...
(17)
where F(q) is the character-valued index and 2 the dimension of G, ct (G) is already known [4]. Vanishing of the total curvature yields [2c~ (q)+F(q)c, (G)]~=o = 0 ,
(18)
the subscript 2 = 0 indicates the qOterm in the bracket. Settle it in another way, if it is possible to define a Dirac operator Q' on loop space in the transverse directions, the ghost effect is included. We need only to calculate ChqQ' and put the restriction el (q,Q') =0. Notice that in the calculation of the character-valued index of Q', one just multiplies the character-valued index of Q by a f a c t o r f ( q ) , the determinant of the ghost sector in the path integral representation [ 8 ]. In the same spirit, we get the Chern character of IndqQ' by multiplying ChqQwith a f a c t o r f ( q )
ChqQ' = ChqQf( q) . (19) c~(q)f(q) is the two-form in ChqQf(q). Even when 363
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we can not define Q' in the transverse directions, eq. (18) tells that we can use ChqQf(q) instead of Ch (IndqQ® G). We obtain an equation equivalent to eq. (18):
(c, (q)f(q) )4=0 = 0 .
(20)
The ghost contribution f ( q ) in the heterotic string is
[8] f(q)=gl '/6 f i (1--q2n)2=rl2(T) .
(21)
n=l
The quantity in eq. (19) is a modular form of weight - d / 2 + 1 under the level-two subgroup F(2 ) ofPSL(2,2~), i f T r R 2 - T r F 2 = 0 (under F ( 2 ) , R and F transform as R - , R / ( c r + d ) and F--,F/(c~+d). This condition was also obtained in ref. [ 14 ] as the condition being gravitational anomaly-free. There one constructs the anomaly generating function on the one-loop level. For a given sector, the generating function is a modular form of weight - d / 2 + 1 too, provided T r R 2 - T r F 2 = 0 . Here R and F are ordinary curvatures. The reason that ChqQf(q) is a modular form under the level-two subgroup only is that this subgroup only keeps spin structures or sectors intact. The author ofref. [ 14] showed that if the theory is modular invariant, the condition Tr R 2- - Tr F 2 insures the anomaly generating function of a certain combination of sectors to be a modular form of weight - d/2 + 1 of the group PSL (2,Z). Only the ( d + 2 ) - f o r m in the generating function contributes to the anomaly, its coefficient is a modular form of weight 2, the qO term vanishes. In our case, we are interested in the two-form in ChqQf(q), or in the ( d + 2)-form in the integrant in (19). But now this is a form which has d degrees in the direction of fibre M and two degrees in the direction of base space. We can not use the argument in ref. [ 14 ] to derive the same condition when we deal with one sector only. Properly combining different sectors, we also get a modular form under the whole group, then the argument in ref. [ 14 ] applies. Our anomaly-free (or curvature-vanishing) condition is ( T r R e - T r F 2 ) ¢ ° , l . 2 ) = 0 , here (0,1,2) means that only terms of degree 0,1 and 2 are included in base space. However, the vanishing of the curvature of the independent sector will give more restrictions than this one, the derivation of all restrictions is the task of our next work. We describe shortly how to draw the above restriction. 364
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In the derivation of eqs. (16) and (17) we have assumed that the fields 2A are in the Ramond sector, and the fermion number F which enables us to make a decomposition of Q as in eq. (9) does not take that of 2A into account. In a path integral calculation, the 2A are antiperiodic in the time direction, they correspond to spin structure ( +, - ) and give rise to functions 02. When the 2A are still in the Ramond sector, while the fermion number contains a contribution from the fields 2A, the spin structure ( + , + ) for 2" is then introduced in the path integral calculation. In a similar manner the spin structures ( - , - ) and ( - , + ) can also be introduced. Use index i to denote four spin structures of 2, and Q~ denotes the Dirac operator in each case. Let 2 i be either + 1 o r - 1, we define a new fibre bundle E = ~i2iIndqQ i, its Chern character and first Chern class are C h ( E ) = Zi2~ChqQ j, el(E) = Z~2'cl (q,Q~). The latter corresponds to the curvature of the equivariant determinant bundle ®~(DEToQ~) z' (because el (®i(DEToQi) ~' ) = Y~)t~c~(DEToQ ~) ). To take the contribution of the ghost into account, we simply mutiply Ch(E) b y f ( q ) as before. In case of the Spin(32)/Z2 string, let d = 10. The theory is modular invariant provided • 2 = 2 3 = • 4 = 1 and 2~ i s + l o r - 1 . Denote the integrant in C h ( E ) f ( q ) as F(q,R,F), it is expressible in terms of functions 0~. Define
F~ (q,R,F) = exp [ ( 1 / 64~z4 ) G2 (q 2 ) X ( T r R 2 - T r F 2) ]F(q,R,F), G2(q2)=~ -
1 - 2 4 ~ a~(n)q 2n .
(22)
Under the modular transformation ~ ( a ~ + b ) /
(cT+d)~PSL(2,7/), R-~R/c~+d, F--,F/cT+d, Fj ( q,R,F) --, ( c~ + d) - 4F~ ( q,R,F) .
( 23)
Now a certain ( d + 2 ) - f o r m in F is the sum of a ( d + 2 ) - f o r m in F~ and a ( d - 2 ) - f o r m in F~ times T r R 2 - T r F 2, etc. If (TrR2-TrF2)(°,~.2)=O, the ( d + 2)-form in F we are interested in is just that in F~, this is zero because the coefficient g(~) of this term is a modular form of weight two with vanishing qO term (see ref. [ 141 ). Therefore, when the following condition holds: (Tr R 2 - T r F2) (°.~'2)= 0 ,
(24)
the curvature of equivariant bundle ®~(DEToQ~) a'
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cancels that from the ghost contribution. At present, we do not k n o w i f this also guarantees that the curvature o f each factor DEToQ i cancels the ghost contribution. This argument applies to all theories which are m o d u l a r invariant. All known such theories are classified in ref. [14]. N o t e that here the c o n d i t i o n ( T r R 2 _ Tr F 2) ~o~= 0 is the anomaly-free c o n d i t i o n in the o r d i n a r y sense. This tells us that starting from the e q u i v a r i a n t d e t e r m i n a n t b u n d l e one can d r a w some correct result which coincides with what was o b t a i n e d by other arguments used before. We turn to type-II strings. T h e r e are two supersymmetries Q~,Q2 Lo = Q ~ ,
/So=Q~ •
(25)
The H i l b e r t space is no longer the space o f sections o f the s p i n o r b u n d l e A ( L M ) , but the space o f sections o f the tangent b u n d l e T ( L M ) (or other twisting versions o f this bundle, in the case o f other sectors). N e i t h e r Q~ nor Q2 is the D i r a c o p e r a t o r on LM. Using ( - 1 )FR we split Q~ into two parts, here FR is the f e r m i o n n u m b e r o f the right-moving sector. Again the vacua are defined by the requirements Q~ I-Q ) =P112) = 0. Define the following charactervalued index bundle: ( ~ q ~ ( K e r Q1 - C o k e r Q~ )~ .
Taking that both the right sector a n d the left sector are periodic, the index f o r m u l a
F(q)=q-a/S(L(TM)Ch(~Sq2nTM(~ ^ q2,TM,M) 1=1
(27) generalizes to the family index ChqQt =
f ~ x,~O2(x,J2nl'r) ~=l
01(x,/2~rlr)
(28)
M
To include the ghost contribution, simple multiply eq. ( 3 4 ) by [8]
f(q)= n=l l~I (l-q2n) 2 (1 +q2n) 2"
m o d u l a r invariant. So a nontrivial restriction should be d e r i v e d from a single sector. O f course, a nontrivial restriction exists in some cases. Let us consider a special case, M = T a and be it endowed with a flat metric. N o w the ( d + 2 ) - f o r m in the integrant of ChqQlf(q) is zero, because o n l y R ~1) and R ~2~ m a y be non-zero, b y these terms we can not construct the ( d + 2 ) - f o r m we need i f d ) 2 . This should be so because now F(q)=0, this means that d i m K e r Q~ = d i m Coker Q~, the contributions from the two parts cancel. Therefore we get no information from ChqQ~f(q). But if we start from v a c u u m bundle, we can gain a nontrivial restriction, d = 10. Summarizing, our work o f reducing the geometric restriction to the topological one gives a powerful tool to investigate anomalies such as the Virasoro one. But this m e t h o d also loses some information. F o r instance, in the case we presented above. I a m grateful to Dr. H.B. G a o for helpful discussions; I would like to thank Professor R. Ruffini for his k i n d hospitality at I C R A and the D e p a r t m e n t o f Physics, R o m e University.
References
(26)
2
n=l
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(29)
ChqQ~f(q) is a m o d u l a r form o f weight -d/2+ 1 o f the level-two subgroup F ( 2 ) without any condition. I f we c o m b i n e all sectors p r o p e r l y to form an e q u i v a r i a n t d e t e r m i n a n t bundle, then its curvature a u t o m a t i c a l l y vanishes ( n o w we have to m u l t i p l y the different ghost v a c u u m bundles G i ) , i f the theory is
[1 ] B.J. Bowick and S.G. Rajeev, Phys. Rev. Lett. 58 (1987) 535; Nucl. Phys. B 293 (1987) 348. [2] J. Mickelson, Commun. Math. Phys. 112 (1987) 653. [3] D. Harari, D.K. Hong, P. Ramond and V. Rodgers, Nucl. Phys. B 294 (1987) 556; Z.Y. Zhao, K. Wu and T. Saito, SISSA preprint (1987). [4] K. Pilch and N.P. Warner, MIT preprint CTP-1457 ( 1987); M. Li, ICTP preprint ( 1987 ); M. Lauer, MIT preprint (1987). [ 5 ] E. Witten, in: Syrup. on Anomalies, geometry, topology, eds. W.A. Bardeen and A.R. White (World Scientific, Singapore, 1985 ). [6] K. Pilch, A.N. Schellekens and N.P. Warner, Nucl. Phys. B 287 (1987) 362; K.-K. Li, MIT preprint CTP-1460 (1987). [7] E. Witten, Commun. Math. Phys. 109 (1987) 525; Princeton preprint PUPT-1050 (1987). [8] O. Alvarez, T.P. Killingback, M. Mangano and P. Windey, Commun. Math. Phys. 111 (1987) 1. [9] J.M. Bismut and D.S. Freed, Commun. Math. Phys. 106 (1986) 159; 107 (1986) 103. [ 10] D.S. Freed, Commun. Math. Phys. 107 (1987) 483. [ 111 D. Quillen, Funct. Anal. Appl. 19 ( 1985 ) 31, [ 12 ] M. Li, in prepration. 365
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[13]M.F. Atiyah and I.M. Singer, Proc. Natl. Acad. Sci. 81 (1984) 2597; O. Alvarez, I.M. Singer and B. Zumino, Commun. Math.
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Phys. 96 (1984) 409. [ 14 ] A.N. Schellekens and N.P. Warner, Phys. Lett. B 171 (1986) 317; Nucl. Phys. B 287 ( 1987 ) 317.