World-sheet anomalies and loop geometry

Nuclear Physics B288 (1987) 578-588 North-Holland, Amsterdam

W O R L D - S H E E T A N O M A L I E S AND L O O P G E O M E T R Y T.P. KILLINGBACK* Joseph Henry Laboratories, Princeton University, Princeton NJ 08544, USA

Received 2 January 1987

Perturbative and global word-sheet anomalies are discussed in terms of an infinite dimensional geometry which generalizes the finite dimensional geometry of spin structures. This geometrical structure is related to the Dirac-Ramond operator and provides an insight into the symmetries of string theory.

One of the most interesting aspects of string theory is the structure of the anomalies present in the theory. The first type of anomaly to be considered in string theories was the conformal anomaly which arises in Polyakov's formulation [1]. The requirement that this anomaly cancels fixes the space-time dimension of the theory to be 26 or 10. A second type of anomaly which can arise is a perturbative anomaly in the field theory limit of the chiral superstring theories. These anomalies were studied for the type II closed string [2] and for the type I open string [3]. It is the requirement that perturbative anomalies in the field theory limit of the type I string cancel which forces the gauge group to be SO(32). Perturbative field theory anomalies also cancel for the gauge group E 8 × E 8 although this cannot arise from an open string theory. The subsequent discovery of the heterotic string [4] allowed both these two anomaly free field theories to be realized as the low-energy limits of superstring theories with gauge groups SO(32) and E 8 × E s. In order for the space-time anomalies to cancel it is necessary to introduce a 2-form B, which transforms as 6B = tr(A OA) - tr(to dO)

(1)

under an infinitesimal gauge transformation in A parametrized by A and an infinitesimal local Lorentz transformation in to parametrized by 0. It follows from (1) that the gauge invariant field strength H of B satisfies d H = T r F 2 - t r R 2,

(2)

* Supported by a Science and Engineering Research Council Postdoctoral Fellowship. Address after January 1, 1987: DAMTP, Silver Street, Cambridge, CB3 9EW, UK. 0550-3213/87/$03.50©E1sevier Science Publishers B.V. (North-HoUand Physics Pubfishing Division)

T.P. Killingback /

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World-sheet anomalies

where F is the Yang-Mills field strength 2-form and R the Riemann curvature 2-form. An interesting feature of (2) is that it implies a topological constraint on the gauge theory and the space-time M [5]. If V ~ M is the SO(32) or E 8 × E 8 bundle defining the gauge theory and T is the tangent bundle of M, then integrating (2) over an arbitrary dosed 4-dimensional submanifold of M results in pl(V) - pl(T) = 0.

(3)

In this expression px(V) and pl(T) denote the first Pontryagin classes of V and T, respectively, in the de Rham cohomology group H41~(M) of M. Thus for the theory to be consistent (3) must hold for the bundles V and T. So far we have only mentioned perturbative anomalies in the space-time; however, there are also anomalies on the world-sheet of the string and it is these anomalies which are the subject of the present paper. Requiring that a theory be free of perturbative anomalies means that it is invariant under transformations of the theory which are homotopic to the identity. However, most interesting theories also contain transformations which are not homotopic to the identity. In this case one must also require that the theory be invariant under the group of components of these transformations. The failure of a theory to be invariant under a transformation which is not homotopic to the identity is referred to as a global anomaly. It is known that an SU(2) gauge theory with an odd number of fermion doublets in four dimensions is inconsistent due to a global anomaly [6]. A global anomaly also occurs in general relativity with an odd number of Majorana fermions in 8k or 8k + 1 dimensions [2]. Global anomalies have been studied in detail in [7] and [8]. The prototypical global anomaly occurs in the supersymmetric quantum mechahics of a point particle moving in a space-time M. This global anomaly was discussed in the supersymmetric derivation of the index theorem [9] and it has been studied in detail in ref. [8]. The action for a point particle with world-line supersymmetry is

I

=

fat

[

dxidx j ( d --d~COkij(x(~))~bJ(r')]d,xk ~ ' - d½gij(x('r)) d'r -d1" + ½i~bi(r)"\ g i j - o'r

~

(4)

where Xi are coordinates on M, ~k~j is the spin connection on M and ~i are real anticommuting variables which take values in the pull-back of the tangent bundle of M to the world-line parametrized by z. This action has the world-line supersymmetry 8x i = i~iE, dx i 8qJi= - dr e - t~%'jg,k~Je,

(5)

580

T.P. Killingback / WorM-sheetanomalies

where e is an anticommuting constant. The conversed quantity corresponding to these transformations is Q = q~i dxi/dz- When quantized, the #i's obey (qJi(~-), qJj(~) ) = gij (x (I-)) and may therefore be viewed as gamma matrices. Consequently, the wave functions must be spinor fields on M and a simple argument shows that the conserved charge Q is the Dirac operator on M. However, we know that spinors and the Dirac operator can only be defined on a manifold if it admits a spin structure. Thus, the above construction must contain an inconsistency unless M is a spin manifold. This inconsistency can be seen in the path integral quantization of (4), in which we must define the square root of the fermion determinant (det y)1/2, where Y is the world-line Dirac operator

d

dx~

i1

Y = i --63d'r+ --~T ~kj )" Consider the case in which the world-line of the particle is a circle S1. Then for a given world-line 7, we can define (det y)1/2 to be positive, say, and the sign of (det y)1/2 for some other world-line 7' is determined by requiring that (det y)1/2 should vary smoothly under a smooth interpolation from 7 to y'. Such an interpolation is given by a mapping q0: S1 × [0,1] ~ M, where S1 represents the world-line parametrized by T, u ~ [0,1], and q0(~,0)= 7(~') and q0(~-,1 ) = ~/'(~-). It is shown in [8] that there exists an interpolation qo(r, u) from 7 to itself such that [det Y(u = 0)] 1/2 = - [ d e t Y(u = 1) 1/2 if and only if M does not admit a spin structure. Thus, if M is not a spin manifold there is a global anomaly which prevents the fermion effective action from being consistently defined. Therefore, the quantization of a single particle may be inconsistent due to the presence of a global anomaly on the particle world-line. The analogue of this in string theory is the occurance of a global anomaly on the string world-sheet. This has been studied in some detail for the heterotic string in [8]. Let 2~ be a Riemann surface which represents the string world-sheet and let q0: X ~ M be a map into the space-time M. Given a diffeomorphism f ~ Diff X, then the metric g on X will be transformed into g f = f * g , and there is a 1-parameter family of metrics gU = (1 - u)g + ug / on X. The effective world-sheet measure for the heterotic superstring is [4]

S=[det' '0( 1

tdet'/01-T-I I-)]16 1 -

~

16

[c,

-

-

i

,:6)

where a, r , 7 denote spin structures on 2~ (for the E 8 × E 8 gauge group/3 and Y are independent; for the SO(32) gauge group fl = y), ia9 is the Dirac operator, (1 + ~ ) / 2 is the chirality projection operator and ~ is the Rarita-Schwinger operator. S is free

T.P. Killingbaek / World-sheetanomalies

581

of perturbative anomalies and requiring at one loop that S is invariant under diffeomorphisms f: ~ ---,~ not homotopic to the identity forces the gauge group to be SO(32) or E 8 × E 8 [4]. Moreover, it was shown in [8] that S is invariant under any diffeomorphism f ~ Diff ~, for Z a Riemann surface of arbitrary genus. These global anomalies involve only a 1-parameter family of metrics on 2~, the generalization of this, which is the string analogue of the propagation of a point particle, is to consider also a non-trivial map ¢p: Z - * M. In this case the effective measure becomes (in the notation of [8]) S=

[detaTi~(~-~)]l°[detl3vti~( ~ )]16 r " 1 - ~ ~'1161[ 1 "q- p -1 ×[detrv/~J(---~) ] [det,8~---~) ]

exp(ifff*B),

(7)

where det~z is the determinant for the ten fight-moving fermions (with spin structure a) interacting with the pull-back to 2~ of the spin connection on the tangent bundle T of M. Similarly, V = Va • V2 is the E 8 × E 8 bundle in space-time and detav ' (respectively, detBv2) denotes the determinant for fermions interacting with the pull-back of the gauge connection on V1 (respectively, V2). This action may possess both perturbative and global anomalies; however, the perturbative o-model anomalies cancel if B transforms as in (1). It follows from (1) that the bundles V and T must satisfy p l ( V ) - p l ( T ) = 0, where pl(V) and pl(T) denote the first Pontryagin classes of V and T, respectively, in the de Rham cohomology of M. The de Rham cohomology groups HPR(M) are isomorphic to the singular cohomology groups HP(M; R) of M with real coefficients [10]; thus we can take p~(V), px(T) H4(M;R). It is possible to generalize the cohomology groups HP(M;R) to the cohomology groups HP(M; Z) of M with integer coefficients; the difference between these two groups is, essentially, that HP(M; Z) may contain torsion elements. Thus, we may contemplate the natural generalization of (3), i.e.,

pa(V) - pl(T) = 0,

(8)

where pl(V), p l ( T ) ~ H4(M; Z), as the actual consistency condition of the string theory. In refs. [11] and [8] some evidence was produced to support this view - the idea is to show that (8) is required for the cancelation of perturbative and global anomalies in the effective world-sheet measure (7). As we know that the cancelation of perturbative world-sheet anomalies requires (8) to hold for the rational first Pontryagin classes of V and T, it only remains to show that (8) holds also for torsion classes. In the case of the point-particle discussed above the global anomaly in the measure was detected by the mapping ¢p: S1 × Sa ~ M, which represents an interpolation from the world-line "t to itself. In the string case we must consider a

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T.P. Killingback / World-sheetanomalies

1-parameter family of maps from a 1-parameter family of Riemann surfaces into space-time. If f ~ Diff 2~ then the mapping torus Zf = (27 × S1)f [(27 × S1)f is obtained from 27 × [0,1] by the identification (x, 0) - (f(x), 1), x E 2~] represents a 1-parameter family of Riemann surfaces and we consider a map qo: 2 f - o M. Thus, the image qo(~f) is a 3-dimensional submanifold of M and so represents an element of the 3-dimensional homology group H3(M; Z) of M. It follows from the universal coefficient theorem [10] that the free and torsion parts of the homology and cohomology of M are related by

HV(M; Z)free~

He(M;

Z)free,

HP(M; Z)tor~ He_x(M; Z)to .

(9)

Thus, if q0(2~y) represents a torsion class in H3(M; Z) then it is related to a torsion class in H4(M; Z), and in certain cases this can be used to show that a global anomaly cancels if (8) is satisfied for torsion classes. A more general argument has also been given in ref. [12] which shows that the effective measure is free of global anomalies if (8) holds for pl(V) and pl(T) torsion classes. So, to recapitulate, we have seen that in the propagation of the single particle there occurs a global world-line anomaly and for the single string there occurs both a perturbative and a global world-sheet anomaly. For the single particle this global anomaly cancels if the space-time M admits a spin structure. This inconsistency in the quantization of the single particle is perhaps the simplest way to see physically why it is important for the space time to be a spin manifold. However, we know, that the existence of a spin structure on a manifold M is an important geometrical and topological property of the manifold, and it is only through understanding this geometry and topology that we fully understand the global properties of spinors and the Dirac operator on M. Our goal in this paper is to achieve a similar geometrical understanding of world-sheet anomalies in string theory. First recall the definition of a spin structure on a manifold M. If M is an orientable n-dimensional manifold then let F---, M be the principal SO( n )-bundle of orthonormal frames on M. Recall that the group SO(n) has a double covering group Spin(n) which fits into the short exact sequence 0 -o Z 2 ~ Spin(n) --, SO(n) ~ 0.

(10)

The manifold M has a spin structure if F rifts to a principal Spin(n) bundle ~" -o M via this sequence. It is well known that such a lifting exists if and only if the second Stiefel-Whitney class w2(M)~ H2(M;Z2) vanishes [13]. If w2(M ) = 0, then the inequivalent liftings of F to F are classified by the group HI(M;Z2). Thus, if w z ( M ) = 0 a spin structure on M is determined by a cohomology class o ~ Hi(M; Z2). More generally, if P is any principal SO(m) bundle on an oriented

T.P. Killingback / World-sheetanomalies

583

n-dimensional manifold M, then P admits a spin structure if P lifts to a principal Spin(m) bundle via the short exact sequence (10). The necessary and sufficient condition for this to occur is that w2(P ) ~ H2(M; Z2) should vanish, and in this case the different spin structures are classified by Hi(M; Z2) as before. If a manifold M admits a spin structure then it is possible to define spinors and the Dirac operator on M, and thus to discuss the physics of fermions on M. To obtain a geometrical understanding of world-sheet anomalies in string theory we must replace the finite dimensional geometry associated with spin structures by a suitable infinite dimensional geometry involving loop spaces. If M is a compact oriented n-dimensional manifold, representing space-time, then let LM = Map(St; M) be the free loop space of M (Map(St; M) denotes the space of all smooth maps f: S1 ~ M). LM is a smooth infinite dimensional manifold (in fact, if f is in the Sobolev space having square integrable first derivatives then LM is a Hilbert manifold) with two important properties: (i) LM is an S1 manifold where the S 1 action is given by rotating loops, and (ii) LM has a natural riemannian structure. These properties will be discussed more fully below. It should be noted that it is also possible to define the based loop space ElM of M which consists of all maps f : S1 ~ M with f(1) = Xo, for 1 ~ S1 and x 0 ~ M. The free and based loop spaces of M are related by the fibration ElM --* LM --* M,

(11)

where the projection is given by the evaluation map. In the present paper we will work with the free loop space LM. For a compact Lie group G the free loop space L G forms a group under pointwise multiplication; thus, L G is an infinite dimensional Lie group. Similarly, the based loop group 9 G of G is an infinite dimensional Lie group. The constant loops in L G give G c L G and the quotient space can be identified with 9 G . 9G = LG/G.

(12)

N o w consider a principal G-bundle P over the manifold M G ~ P ~ M.

(13)

Such a bundle would occur, for instance, in defining a gauge theory on M with gauge group G. It is easy to see that by taking free loops on the above bundle we obtain a princible LG-bundle LP over LM L G ~ LP ~ L M .

(14)

In the case of spin structures we had a principal SO(m) bundle over M and we

584

T.P. Killingback / WorM-sheetanomalies

wished to lift this to a principal Spin(m) bundle on M, where SO(m) and Spin(m) are related by the exact sequence (10). In the present case the analogue of SO(m) is L G and we need to find the analogue of Spin(m). Recall that the group Spin(m) is the central extension of SO(m) by the two-element group Z 2. The loop group L G has a canonical extension L G by the circle group S1 (see [14], for example). The central extension is a non-trivial S1 bundle SI~L-GLLG,

(15)

which is determined by the extension of Lie algebras R ~ Lie(L-'G) ~ Lie(LG)

(16)

A

in the following way: if the group extension S 1 ~ L G ~ L G corresponds to the Lie algebra cohomology class to ~ H2(Lie(LG); R), then the image of ~/2~r under the map H2(Lie(LG); R) ~ H2(LG; R) (in which an antisymmetric multilinear form on Lie(LG) is interpreted as a left-invariant differential form on the manifold LG) represents the first Chern class of the S 1 bundle L G ~ LG. Thus, in the case at hand, the central extension L G plays the role of the group Spin(m) and we should consider lifting the L G bundle LP ~ LM to a L G bundle L O ~ LP ~ LM

(17)

via the sequence (15). The exact sequence (15) induces an exact sequence of sheaves of local cross sections

S1 -~ ~

~ LG.

(18)

This in turn gives rise to the exact sequence of sheaf cohomology groups on LM

... ~ n l ( t M ; S 1 ) --i, H x( LM; L ~ ) p* a, H2(LM;S1 ) ~ "'" ~ H I ( L M ; L G ) --

(19)

Recall that for any space Y and group ~ the cohomology group Hi(Y; ~ ) classifies q-bundles over Y. Now let 4 ~ HI(LM; LG) be the principal L G bundle LP ~ LM, then this bundle lifts to a L G bundle L P ~ LM if and only if there exists a ~"~ HI(LM; LG) such that = 4.

(20)

From the exact sequence (19) this occurs if and only if 4 ~ Ker 8.. Thus, the bundle L G ~ LP ~ LM classified by 4 lifts if and only if 8.4 ~ H2(LM; S~) vanishes. By considering the exact sequence of sheaf cohomology groups on LM induced by the

T.P. Killingback / WorM-sheetanomalies

585

short exact sequence

i

exp

Z ~ C ~ S1

(21)

H2(LM; s_ - W(LM; z)

(22)

one finds that

and, therefore, that the obstruction to lifting L G ---, LP ~ LM to L G ~ LP ~ LM is = ~,~ ~ H3(LM; Z). For any compact M we can define elements a ~ H3(LM; Z) by the transgression of elements fl ~ H4(M; Z). The transgression map T: H4(M; Z) ---, H3(LM; Z) is induced by the evaluation map e: S 1 × LM ~ M, (o,x) ~ x ( o ) , by taking e*(fl) ~ H4(S 1 x LM; Z) and "integrating over S 1'' to get a ~ H3(LM; Z). Using essentially standard topological methods (explained in greater detail in [15]) one finds that the first Pontryagin class pl(P) E Ha(M; Z) of the G bundle P ~ M transgresses under ~- to the obstruction class 2,, i.e. ~-(pl(P))= ~. Thus, the L G bundle LP ~ L M lifts to an LG-bundle LP ~ LM if

el(P) =0.

(23)

If the bundle G ~ P ~ M is such that pl(P) = 0 then it follows from the exact sequence (19) that the inequivalent liftings of L G ~ LP ~ LM to L G ~ L P ~ LM are classified by the elements of the cohomology group H I ( L M ; S~). By the same argument involving the exact sequence (21) that led to the isomorphism (22) one finds that H I ( L M ; S~)---H2(LM; Z). As such a lifting is the analogue in string theory of a spin structure in the theory of a point particle we may refer to it as a string structure. A manifold M admits a string structure if and only if its frame bundle F ~ M does. Alternatively, taking E ~ M to be a vector bundle associated to P ~ M, then E admits a string structure if pl(E) = 0. If px(T) = 0, for T the tangent bundle of M, then M admits a string structure. More generally, for V an arbitrary vector bundle over M, we may take E = V - T to be the difference bundle in the K theory group K(M); E admits a string structure if pl(E) = 0, i.e. if P a(V) - p l(T) = 0,

(24)

which we know to be the condition for the cancelation of perturbative and global world-sheet anomalies in string theory. Before discussing the significance of this result there is a technical point to make. The argument that the bundle L G ~ LP ~ L M lifts to L G ~ LP---, LM if pl(P) = 0 only works in a straightforward way if the group G is simply connected*. In general the bundle G ~ P ~ M need not have a simply connected G; however, if w2(P ) --* In fact, it is possible to modify the argument to allow for a non-simply connected group G by including the effort of a non-zero first Chern class q(P).

T.P. Killingback / World-sheet anomalies

586

cl(P) mod 2 ~ HZ(M; Z2) is zero, then we can always work with the covering bundle ~ [~ ~ M, where G is simply connected. If, for example, P = F the SO(n) bundle of frames on M, then F is the Spin(n) bundle discussed above. In this case requiring wz(F ) = 0 simply means that M is a spin manifold. In general, demanding that w z ( P ) = 0 means that P admits a "spin structure" in the sense that there exists a suitable covering bundle P of P. The requirement that the bundles involved admit a spin structure is physically quite r e a s o n a b l e - in the case of the frame bundle F ~ M this is necessary just to have a consistent quantum theory of a supersymmetric point particle in M. I would now like to make some remarks concerning the interpretation of the above result. We have seen that a supersymmetric point particle can be consistently quantized only if the space-time M admits a spin structure. If M is not a spin manifold then the quantum theory is inconsistent due to a global anomaly. The requirement that M should be a spin manifold is, of course, perfectly reasonable as otherwise space-time fermions cannot be defined and neither can the Dirac operator. So in this case information obtained from a world-line anomaly can be understood in terms of physically important objects (spinors and the Dirac operator) on the space-time M. Similarly, we would expect the information obtained from world-sheet anomalies to be comprehensible in terms of physical objects on the loop space LM. The natural object to consider is the Dirac-Ramond operator. This is the generalization of the Dirac operator in string theory and it may be interpreted as a Dirac-like operator on the loop space LM [8]. As was mentioned above a riemannian metric gij on M induces a riemannian metric on LM: let 8x ~ be a tangent vector at a point 3' ~ LM (i.e., 3' corresponds to a loop x~(o), o ~ S1) then the metric on LM is defined by (Sx, 8x) = fZ~do gij(x(o))Sxi(o)SxJ(o). The loop space LM also has a natural S1 action given by mapping a loop x~(o) to xi(o + O) for 0 ~ S1. The infinitesimal form of this action is 8xi(o)= Oxi/Oo. Given a differential operator ~ on a manifold Y together with an S 1 action on Y generated by a vector field K, we can define the equivariant generalization ~K of ~ to be ~ K - ~ + iK, where i K denotes inner product with the vector field K (see ref. [16], for example). As LM is a riemannian manifold we can formally define the Dirac operator iD on LM as (see ref. [8])

iff) = - i f 2x do ~lli(O ) Jo

D

Dxi( a

),

(25)

where the ~ki(o) are essentially gamma matrices and D

8

Dxi(o~- 8xi(o~ + o~ijk(x(o))~J(o)~k(a).

(26)

We also have an S1 action on LM and thus we can define the equivariant extension

T.P. Killingback / World-sheetanomalies

587

of the Dirac operator iff)r = iO + i r which is explicitly (see ref. [8])

[2~r d

D

il~r=Jo o~i(o)- i ~

Ox j + giJw

(27)

This is just the Dirac-Ramond operator in string theory. For a supersymmetric point particle it is only possible to define the Dirac operator if M is a spin manifold. The natural generalization of this to superstrings is that it is only possible to define the Dirac-Ramond operator satisfactorily if the space-time admits a string structure, i.e. if p l ( T ) = 0. The geometrical structure behind this involves the lifting from L G ~ LP ~ LM to L G ~ LP ~ LM discussed above and the geometry of strings fields and the Dirac-Ramond operator which is studied in greater detail in ref. [15]. We will not here discuss the geometry of the Dirac-Ramond operator any further, but rather mention one situation in which the existence of a string structure arise s naturally. This is the calculation of the index of the Dirac-Ramond operator. The index of this operator was studied in ref. [17] and has already been discussed in some detail [18]. It is found that in computing the index of the Dirac-Ramond operator (by path integral methods, for instance) one requires that the tangent bundle T of M satisfies pl(T) = 0. In the calculation of the index of the Dirac operator on M given in [9] the corresponding condition is that w2(T ) = 0, i.e. that M should be a spin manifold. In both cases these conditions arise from cancelling anomalies in the respective path integrals, as we have discussed above; however, they may also be viewed as the conditions necessary to define the Dirac operator and the Dirac-Ramond operator, respectively. Finally, in conclusion, I would like to consider symmetries in string theory. First let us start with a supersymmetric point particle moving in a space-time M. We know that M must be a spin manifold and the Dirac operator on M is determined by the choice of spin structure s ~ HI(M;Z2). If ~ r ~ D i f f M is an orientation preserving diffeomorphism of M then for ~r to be a symmetry of our theory we require that ~r should preserve the spin structure s, i.e. rr *s = s. This result can be interpreted as a generalized world-line anomaly as was discussed in [8]. Instead of considering a map cp: S1 × S1 ~ M we consider a map q~: S1 x Sx ~ M . , where M~ is the mapping torus of M under the diffeomorphism ~r. If we have a global anomaly in this situation than it follows that M,, is not a spin manifold. However, if rr*s = s then the spin structure s on M extends to one on M~,. Thus, if there is a global anomaly ~r must be such that 7r *s 4: s and ~r is not a symmetry of our theory. This anomaly argument can also be applied to string theory. We now consider a map ~: ~ f ~ M,,, where f ~ Diff 2, ~r ~ DiffM. Imposing the anomaly cancelation condition in this situation (ignoring the vector bundle V ---, M) would then require that pl(T,,) = 0, where T,, is the tangent bundle of M~. This equation expresses the condition for the transformation 7r to be a symmetry of string theory. It is the

588

T.P. Killingback / World-sheet anomalies

analogue of the condition w2(Y~r ) = 0 for 7r to be a symmetry of a single particle. It was shown above that a string structure is determined by a cohomology class o ~ H 2 ( L M ; Z ) . It is natural to conjecture, in analogy with the case of spin structures, that a transformation H ~ Diff LM is a symmetry of string theory only if 17"o = o, i.e. i f / 7 preserves the string structure. In the case when H is induced by a diffeomorphism ~r ~ D i f f M it then follows that p l ( T . ) = 0. Thus the idea of string structure m a y prove to be important in understanding both the anomalies and the symmetries of string theory. I would like to thank G.B. Segal for discussions.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

A.M. Polyakov, Phys. Lett. 103B (1981) 207 L. Alvarez-Gaum6 and E. Witten, Nucl. Phys. B234 (1983) 269 M.B. Green and J.H. Schwarz, Phys. lett. 149B (1984) 117 D.J. Gross, J. Harvey, E. Martinec and R. Rohm, Nucl. Phys. B256 (1985) 253, B267 (1986) 75 E. Witten, Phys. Lett. 149B (1984) 351 E. Witten, Phys. Lett. l17B (1982) 324 E. Witten, Commun. Math. Phys. 100 (1985) 197 E. Witten, Symp. on Anomalies, Geometry and Topology (1985) L. Alvarez-Gaum~, Commun. Math. Phys. 90 (1983) 161; D. Friedan and P. Windey, Nucl. Phys. B235 (1984) 395 R. Bott and L.W. Tu, Differential forms in algebraic topology (Springer, 1982) X.-G. Wen and E. Witten, Nucl. Phys. B261 (1985) 651 D. Freed, MIT preprint (1986) J. Milnor, L'Ens. Math. 9 (1963) 198 G. B. Segal, Commun. Math. Phys. 80 (1981) 301 T.P. Killingback, in preparation E. Witten, J. Diff. Geom. 17 (1982) 661 O. Alvarez, T.P. Killingback, M. Mangano and P. Windey, unpublished (August 1985) E. Witten, Princeton preprint PUPT-1024 (1986); O. Alvarez, T. P. Killingback, M. Mangano and P. Windey, Berkeley preprint (February 1987)