The Dirac-Ramond operator in string theory and loop space index theorems

Nuclear Physics B (Proc. Suppl.) 1A (1987) 189-216 189 North-Holland, Amsterdam

T H E D I R A C - R A M O N D O P E R A T O R I N S T R I N G T H E O R Y A N D L O O P S P A C E I N D E X T H E O R E M S * , t

O r l a n d o A L V A R E Z

Department of Physics and Lawrence Berkeley Laboratory, University of California, Berkeley, CA 9~7~0, USA

T.P. K I L L I N G B A C K

DAMTP, University of Cambridge, England CB3 9EW

Miche lange lo M A N G A N O

Fermi National Accelerator Laboratory, Batavia, IL 60510, USA

Pau l W l N D E Y

Department of Physics and Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA

The index of the Dirac-Ramond opera tor is computed and analyzed. It is shown to be the extension of the Atiyah-Singer index theorem for loop space. It can also be seen as a generating function for the Atiyah-Singer index for the s tates of the string. Its existence depends on the Green-Schwarz anomaly cancellation condition, pl(M) = 0, which defines an analog of a spin structure for the loop space. One also finds topological invariants for the loop space which correspond to different twistings of the Di rac-Ramond operator . All of them can be expressed in terms of 3acobi elliptic functions.

1 I n t r o d u c t i o n

One of the lessons tha t can be drawn from recent work in field theory and in string theory is tha t one can gain a lot of insight about their structure

*Invited talk presented by P.W. at the Irvine conference on non-perturbative methods in physics. tThis work was supported in part by the Director, Office of Energy Research, Office of High Energy and

Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE- AC03-76SF00098 and in part by the National Science Foundation under grant PHY85-15857. Fermilab is operated by the Universities Research Association, Inc., under contract with the United States Department of Energy.

**Address after October 87: LPTHE, Universitd Pierre et Marie Curie, Tour 16, let dtage, Paris VI, pl. Jussieu, 17-753200 Paris CEDE)( 05, FRANCE

0920-5632/87/$03 .50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

190 O. Alvarez et al. / The Dirac-Ramond operator in string theory

through the s tudy of the anomalies which plague them. Indeed the re- moval of these anomalies imposes strong constraints tha t any meaningful theory should satisfy. Another important fact which was noticed is tha t all anomalies are governed by one version or another of the Atiyah-Singer index theorem. It seemed then natural to t ry to derive a version of tha t theorem which would be of interest for string theories. By this I mean an index theorem which could be derived directly from a string theory and would then contain informations about the spectrum of operators appearing nat- urally in string theory rather than in the limiting field theories associated with it. In field theories index theorems govern the spectrum of massless point particles and link it with the topology of the configuration space M (space-time) where these evolve. Similar theorems in string theory should then probe the structure of the loop space of M (the configuration space of the string) and relate its topological properties with the string spectrum. More precisely such index theorems should give at least two broad type of constraints on string theories. Firstly the constraint must be related to the tradit ional field theory anomalies. Indeed one might expect to recover at once all the known anomalies of the field theories which are implicitly con- tained in a given string theory. The naive reason is as follows. Potential field theory anomalies result from the coupling of elementary particles with gauge or gravitational fields supported by M. Since string theory dynami- cally generates these fields - - i t is just the massless part of its spec t rum-- one expects tha t the information about their anomalies should be encoded in the structure of the loop space of M. The second type of constraint con- cerns the definition of the index itself. Let us recall what happens in the context the Atiyah-Singer index theorem seen from a physicist's view point. It is known tha t the suitably modified part i t ion function of a spinning point particle gives the index of the Dirac operator. This part i t ion function is equivalent to a one dimensional pa th integral. One can show [1] tha t this pa th integral has a potential global anomaly. The vanishing of tha t anomaly is a sine qua n o n condition for the existence of the pa th integral and consequently for the definition of the index of the Dirac operator. It is of course shown to be the same condition which guarantees the existence of a spin structure for M. Of course it is a pr ior i unnatura l to t ry to put a spinning particle or define the index of a Dirac operator on a manifold which is known not to have a spin structure. But for our purpose this is the point of view we will adopt. Indeed such an obstruction to the definition of an index for an operator in string theory would simply imply tha t the manifold M cannot support a string just like it could not support a spinor before. This of course is of particular interest since if one believes tha t strings exist

O. Alvarez et al. / The Dirac-Ramond operator in string theory 191

just as spinors do it would provide us with some constraint on the structure of space-time permit ted by string theory. I will show tha t there exists an simple analogue of this constraint in string theory which arises when one tries to define a meaningful topological invariant on loop space. The same constraint was studied in a more general context in [2]. There the concept of a string structure was introduced in analogy with the idea of spin structure for ordinary manifolds. The existence of a string structure on loop space is guaranteed by the vanishing of the first Pontrjagin class, p l ( M ) in integral cohomology. This is then interpreted as the condition for the existence of fermions in string theory. Finally one can hope tha t an index theorem for string theory would reveal some new and unexpected structures.

Wha t I choose to review today is some rather old unpublished work done in collaboration with Orlando Alvarez, T.P. Killingback and Michelangelo Mangano [3] which precisely realizes the program outlined above. The idea was to find an extension of the Atiyah-Singer index theorem on the loop space of M through the use of the same type of field theory techniques (appropriately adapted to string theory) which had been used in the supersymmetric derivation of the ordinary index theorem [1,4]. We simply looked for the analogue of the Dirac operator in string theories, i.e. the Dirac-Ramond operator, and computed its index. At first sight the result doesn't look like any known index formula, but a careful analysis reveals tha t it is simply a character valued index, properly extended to the infinite dimensional case. However the expression for the index is valid only if the Green-Schwarz anomaly cancellation occurs [5]. Otherwise, the index is not well defined. This anomaly comes as an obstruction to the index being a well behaved modular function. When tha t condition is satisfied the general index formula - -der ived in section 3 - - takes the form of a generating func- t ion for the index of each Dirac operator associated with the different states of the spectrum of the string. Most known results are derived in a unifying

setting.

Recently A. ScheUekens and N. Warner have published a series of papers [6,7,8] int imately related to the work presented here, al though stemming from different motivations. They introduce the one-loop string part i t ion in arbi trary gauge and gravitational background fields and with the help of the Atiyah-Singer index theorem find the generating functional for the field theory anomalies. They also gave with K. Pilch a pa th integral derivation of these results [9]. In a rather different context some of the formulae that we derive have appeared in the mathematics l i terature [10,11,12]. These lat ter works, motivated by a conjecture of E. Wi t ten [13], are based on an extension of the Hirzebruch definition of a genus and called elliptic genus.

192 O. Alvarez et al. / The Dirae-Ramond operator in string theory

The link between elliptic genera and the index of the Ramond operator has been made most recently in a paper by E. Wit ten [14]. We hope tha t our derivation will be of interest equally to string theorists and mathematicians, and it will shed some light on the connection between string theories, elliptic genera and loop space index theorems. A simple geometric interpretat ion of our work and its relation with character index theorems can be found in [15].

Since the Atiyah-Singer theorem is the paradigm of all other index theorems, a brief review of its physicist 's derivation is in order. This will be done in section 2. It will give us the opportuni ty to introduce most of the methods to be used subsequently in this paper and establish some of the notations. The exact parallelism between the t reatment of the first and second section will show tha t mutatis mutandis the Ramond operator is to the loop space of a closed compact manifold M without boundaries what the Dirac operator is to M. Every concept pertaining to the lat ter case will find its natura l generalization in the former. We also hope tha t this review will convince the reader tha t an in depth s tudy of the Ramond operator is likely to unravel some of the intricacies of the structure of string theories as was the case for the Dirac operator with field theory, from instantons to anomalies.

In section 3 we derive the index of the Ramond operator in the simplest sett ing of a N ---- 1//2 supersymmetric sigma model,i .e.in the Ramond sector of a gauged fixed heterotic string, with a supersymmetric right moving sector, embedded in an arbi trary external background gravitational field. Finally in section 4 we generalize our t reatment to include the coupling of the original N = 1//2 system to a left moving sector with the four boundary conditions corresponding to the spin structures of the torus. The four resulting topological invariants are simply expressed as a ratio of two the ta functions or equivalently in terms of the Jacobi elliptic functions. A simple modification of the left-moving sector would lead to equivalent index theorems for the Dirac-Ramond operator twisted by a coupling to external gauge fields. The above mentioned consistency condition Tr R 2 = 0 would simply become Tr R 2 - T r F 2 = 0 .

2 T h e p o i n t p a r t i c l e c a s e

The present section follows the t reatment of the Atiyah-Singer index theorem [16] introduced in [1] (see also [4]). The principle of the derivation rests on some simple but profound remarks made by Wi t ten [17] about the gen-

O. Alvarez et al. / The Dirac-Rarnond operator in string theory 193

era] structure of supersymmetric theories. Let (--1) F be the fermion parity operator (or any Z2 grading) such tha t the Hilbert space 7-/of a theory is split into ~ = S+ @ S_, such tha t ( -1 ) F has +1 ( - 1 ) eigenvalue on S+ (respectively S_). Let Q : S+ --* S_ be an operator such that:

H = Q t Q

and

(2.1)

Q* = - @ , (2.2)

@ ( - 1 ) F + ( - 1 ) F @ = 0 . (2.3)

It is easy to see tha t any eigenstate of H ]¢+) E S+ with non zero eigenvalue is paired with an eigenstate Q I¢+) = I¢-) E S_ with the same eigenvalue. Obviously the pairing does not hold for the states annihilated by Q (respectively Qt) i.e. for states belonging to the kernel of Q (Qt). These zero energy states satisfy H -- 0. It is then clear, as was pointed out by Witten, tha t :

I ---- Ind Q = dim kerQ - dim coker Q = T r ( - - 1 ) f l g = o (2.4)

is a topological invariant. States can only reach or leave zero energy in pairs. This results from the alluded pairing which also implies tha t one can equally well compute the index through:

I = T r ( - 1 ) F e - . H (2.5)

since only the zero energy states contribute to the trace. The situation just described is realized in any supersymmetric theory where Q is the symmetry generator and H the Hamiltonian. So to any such theory one can associate an index. The index of any operator Q satisfying the relations (2.1) can be computed through pa th integral techniques [18] provided tha t the second quantized expression of the supersymmetric charge coincide with Q and the Hamiltonian with H.

The case of the Dirac operator is of primary interest to us. Let consider a supersymmetric quantum mechanical system with the following commu- ta t ion relations among the dynamical variables:

[ipu, x ~] = 6~ (2.8)

= , (2.7)

where x ~ and p" are respectively the position and momentum operator in a flat space M (# = 1 . . . n), while ¢ " correspond to Dirac matrices. If we

__ _= i -n l2 ~ e ¢ul . .¢u" , then choose Q ipUCu, H pe a n d ( - 1 ) f - - - -9 ' s : n. " , ' "~-~ "

the relations (2.1) are satisfied. The above operators appears after second quantization of the Lagrangian:


1 t, loh~, 0 ~/.t, L = + (2.8)

This Lagrangian is invariant under the supersymmetry transformations

&~ = ~¢~ (2.9) 6d2 t' = - e a t x ~' . (2.10)

We now generalize the above to include coupling to an arbi t rary background gravitational field gz~ describing an arbi trary spin manifold M. Following [1] we will use superfield notations and sometimes suppress the indices when no confusion is possible. Let X ~ = x z + 0¢ ~ where 0 is the Grassmann variable associated with the time t. Defining D = oat - 0o we have:

S = f d t d O lg , ,~(X)OtX~'DX'~ with (2.11)

(2.12)

In this form the action is explicitly supersymmetric. The transformation of a field is given by its commutator with eQ = cOOt + Oo. In components we have

L ,.~ g~,~(x)Otx~Otx ~' + g~,,(x)¢t '(Ot¢ ~" + Otx '~F~¢~) , (2.13)

which leads through canonical quantization to Q = 40 where 40 is the usual Dirac operator on spinors. To compute the index of the Dirac operator we just have to follow the steps explained at the beginning of this section and compute I = Tr( - -1 )Fe -~H. With the usual correspondence between the pa th integral and the canonical formalism this is very easily done. We refer the reader to [1] for the details and just mention the main logical steps of the computat ion as well as some formulae which will turn out to be useful in the sequel.

.

.

First notice tha t the ( -1 ) F f a c t o r implies computing the part i t ion function with periodic boundary conditions in time for both the x and the

¢ fields.

Since the index is known to be a topological invariant independent of T - - it is just an in teger- - we can perform an expansion of the action

for small r . We will expand the fields about the constant solutions of the equations of motion and keep only the terms of lowest order in the background metric. Let x ~ = x~+~ t' and ¢" -- ¢~-t-~ ". Using Riemann normal coordinate expansion about x0 we have

gu~,(Xo) = 6,~, (2.14)


O~gut,(Xo) = 0 (2.15)

+ (2.16) apO~g~,~(xo) - 3

The component lagrangian becomes:

L ,,., - c9 ,~uc~¢ u + nu~ ,a t¢~ ' { ~' + c9~¢'¢ z" (2.17)

w h e r e 7P~# 1 B .~.U.~.u = ~ ft#ua/3"V/O'~U 0 .

3. We compute the properly regularized determinants (-Dr - 7E) paying attention to the t reatment of the zero modes which will essentially provide the volume element.

The result is the well known Atiyah-Singer index for the Dirac operator:

I = fM k2cr] det-1/2 T~ sinh T~ , (2.18)

where now T~% = ½R%,~dx'~dx~. A simple modification of the above leads to the index of a Dirac operator coupled to external gauge fields Au(x ). One simply introduces new fermionic operators ~a and ffa with the commutation relations [ff~, Vs] = Us. The Hilbert space is now the tensor product of spinors with antisymmetric tensor of arbi t rary rank in internal space. The new covariant Dirac operator corresponds to the generator of supersymmetry modified by the addition of the following Lagrangian to (2.11):

L, 7 = f dOIVDAN + i a N N , (2.19)

with the fermionic superfield N = ~7 + 0¢ and DA = D + DXUAu(X) . Here cr is a Lagrange multiplier. Elimination of the auxiliary field ¢ leads to

1 L,7 = fl(Ot + OtxUAu)~ - ~/Fu~Ou¢ .7/+ ic~¢/r/. (2.20)

Notice tha t since the generator of supersymmetry is quadratic in the 77 fields, they may be chosen to be either periodic or antiperiodic in time. One can then evaluate the character valued index I(c~) = Tr( - -1) fe -rH+~N, = ~k Ike -~k, where Nn is the number operator for the r/field. The last equality results from [Nn,/-~ = 0. It is obvious tha t Ik is the index of the Dirac operator coupled to an antisymmetric tensor of rank k in internal space. The computat ion is performed (with antiperiodic boundary conditions for r/) by expanding around A,(xo), and computing a simple fermion determinant d e t ( O t - 1 u ~F~,~¢o ¢o)" The result is:

196 O. Alvarez et al. / The Dirae-Ramond operator in string theory

( i ~ n/2 eF_io~ ) _1/5 [(1 ) -1 1 ] I(a) = fM \-{-~] det(1 + det T~ sinh T4 , (2.21)

where F = 1/2Fu, dxUdx% Notice tha t the above implies the following Tr e -r~n = det(1 + e ~) for any antisymmetric matr ix w. We list similar useful formulae, which follow from equivalent fermion determinant computations when the appropriate (--1) f is inserted:

Tr e -~"m = det(1 + e ~) (2.22)

T r ( - 1 ) Re - ~ v = det(1 - e ~) (2.23)

Tre¼77¢~ = (2) n/2 det (coshT~/2)} (2.24)

T r ( - 1 ) Re ¼7Te7 = (2i) "/2 det (sinh 74/2) ½ (2.25)

These equations can of course be checked directly. Finally we give the result 1 corresponding to the choice r /system corresponding to Fu, ---- - ~/qYfh,,~,~-y e.

Computing the pa th integral with periodic or anti-periodic boundary conditions or using (2.22) gives:

X = fM /2-~) det-1/2 sinh ~ Tr3,sei ~R~ (2.26)

f 1 = . . . . e m ' " ~ m m - . . 7 ~ 1~,, (2.27) (47r)n/2(n/2)! - ,

sign(M) = fM \2--~r] det-X/2 T4 sinh T4 Tre ¼~R7 (2.28)

= f(i)~/~det-1/~ [ (17~) -XtanhlT4] . (2.29)

These correspond respectively to the Euler number and the Hirzebruch signature. This will be explained in more detailed in section 4 where their infinite dimensional case counterpart will be analyzed.

3 T h e i n d e x of t he R a m o n d o p e r a t o r

We now generalize the preceding analysis of the index theorem with functional integration methods from the point particle to the string case. The goal is to uncover new topological invariants relevant to string theory rather than to field theory. It should also reveal the properties of the loop space of


the manifold M in which the string is embedded. The evolution in a finite proper time of a - - supe r symmet r i c - - point particle along a closed loop in space-time (a manifold M) will be replaced by the displacement of a closed string sweeping a two dimensional torus embedded in M. A fixed value of the moduli parameter ~" = T1 + iT2 will uniquely characterize the torus. In fact we consider a two dimensional field theory in some fixed background - - a - m o d e l - - rather than a string theory per se. The start ing point of the analysis will be the simplest generahzation to two dimensions of the N - 1

supersymmetric Lagrangian which gave us the .4-genus •

s = f - + (3.1/

where we have used the usual complex variable notation. The integration is over the torus (taken to be a parallelogram with sides (1, T) in the complex plane) and xU(z ,2 ) and CU(z) are respectively two dimensional scalars and right-moving Majorana-Weyl spinors. The rest of the notat ion follows section 2. We have to unders tand what operator plays the role of the Dirac operator. We expect it to be the generator of supersymmetry of (3.1) expressed in terms of second quantized fields. Given this operator we have to compute its analytic index and express it in terms of the characteristic classes of M or bet ter some functions of these tha t should be like characteristic classes of the loop manifold of M. 1 The lat ter step will require the evaluation of a functional integral with appropriate boundary conditions while the former, to which we now turn , necessitates the understanding of the extension of Tr exp(- /3H) to two dimensions. We focus only on the features directly relevant to our problem. A more detailed account can be found in a remarkable paper by J. Cardy [19] on conformally invariant sta- tistical mechanics. A naive guess is simply to replace the one dimensional Hamiltonian by its two dimensional equivalent. This will correctly describe the proper-time evolution of the system but ignores the fact tha t the string can slide on itself during its time evolution. This motion, which corresponds to the existence of a Killing vector in loop space, is generated by an operator P such tha t [H, P] = 0. The loop will then be rotated by a finite twist after it has completed its sweep of the torus. The resulting part i t ion function is:

Tr exp(--21rT2H + 2zriTiP) (3.2)

where T = ~'1 + iT2 is the modular parameter of the torus. Modular transformations are generated by T --* T + 1 and ~" ---* - -1/r . Their role will be stressed later on. Notice tha t since [H, P] -- 0 the states of the system

1In fac t we will get a S l - i n d e x , i.e. we will get a n expans ion in t e r m s of t he fixed po in t set M of an S 1 ac t ion on t he loop space o f M . A more de ta i led accoun t o f th i s a r g u m e n t can be found in [15].


can be labelled by both the eigenvalues of H and P. Let us go back to the 1 supersymmetric a -model introduced in (3.1). It is analysis of the N =

more convenient to use superfield notation. Let X ~ = x u + 0¢ ~. Defining D = cOo + 0cOz we have:

S : f d2zdO gu~,(X)cO~XUDX ~'

g.~.(x) = g~(x) + o¢~o~g~v(x)

with (3.3)

(3.4) In the superfield formulation the Lagrangian is explicitly supersymmetric. The holomorphic supersymmetry transformations of the right-moving sector axe 6x u = e'¢ u and 6¢ u = eO~x u with e = e(z). They can be derived as usual by computing [eQ, X] = e 6 X with Q = 00 - OOz. For the time being we will. only consider the Ramond sector of the theory where all the fields obey periodic boundary conditions along any closed loop on the torus. These boundary conditions respect supersymmetry. Canonical quantization gives:

~., ~°] = [p.~o~: -, ~°] = ~

[n., ¢-] = [a.v¢ ~, ¢-] = ~ The generator of supersymmetry is

Q = f dzgm'¢uOz x~

(3.5)

(3.6)

(3.7) as can easily be checked using-the canonical commutation relations. There is another way to look at the operator Q. The energy-momentum superfield is given by

T ( z , O) = TF + OTB

with Tp = gu~,¢UOzx ~'

and Ts = gu~,OzxUO~x ~' + guv'CUDz¢ ~'

(3.8) (3.9)

(3.10) Considering for now the theory as defined on the complex plane rather than

on a torus we have:

TF = Ez -" -~C.

TB = ~_, z-n-2L,~ ;

(3.11)

(3.12) n

where the Gn and Ln axe respectively the odd and even generators of the Neveu-Schwaxz-Ramond algebra of superconformal holomorphic transformations. They satisfy well known graded commutation relations with a central charge c whose value here is -~. For the left-moving sector which is not supersymmetric, we jus t have ~' = gu~,O~xUO~x v and 7 ~ = ~2~ ~ ' -~-2L, and c = 1. The radial quantization Hamiltonian of the theory is given by H = L0 + L0 while the rotation operator mentioned before is P = L0 - L0. We have


[ H , P ] = 0 [H, G 0 ] = 0 [P, G 0 ] - 0 and (3.13)

[L0, H ] - - 0 [L0, P ] = 0 [L0, G 0 ] - - 0 (3.14)

This par t of the Virasoro algebra is all what we need. It is of course satisfied whether or not conformal invaxiance is maintained at the quantum level. It is easy to verify tha t Go - Q. This operator is the well known Ramond operator. It is the generalization to string theory of the Dirac operator. In fact one has Go = 0 + " " as one may see by using the expansions cOzx =

_ n _ 1 ~E nz - '~ - l xn and ¢ = ~ z 2¢n and the canonical commutation relations for the x u and Cu fields. So Go is to the loop space of M what 40 is to M. The Ramond operator is precisely the operator we were looking for and we have now to compute its index. The Ramond operator shares many properties with the Dirac operator. It commutes with the Hamiltonian and anticommutes with the world-sheet fermion parity (--1)F:

(--1)FG0 -4- G0( -1 ) F ----- 0 . (3.15)

However the relation between Go 2 and the Hamiltonian is slightly complicated by the presence of the central charge:

G~ - Lo - c / 2 4 . (3.16)

All the supersymmetric invaxiant states I¢0) satisfy

Go [¢0) = 0 and (3.17)

L0 [¢0) = - c / 2 4 I¢0) (3.18)

while the non-supersymmetric states ICh) with non-zero L0 eigenvalues h and -4-1 fermion parity axe paired by Go:

L0 ICh)+ ---- h ICh) (3.19)

( -1) ICh)+ = ICh>+ (3.20)

Go ICh)+ ---- ICh)- (3.21)

L0 ICh)- = h ]¢h)_ (3.22)

(--1) F ICh)- = --ICh)- (3.23)

All these states will be the direct product of the Ramond vacuum defined above with symmetric tensors of any rank in the tangent bundle of M. They are created by repeated application of the creation operators x-n . We axe now in position to assert tha t the index of the Ramond operator is given by:

Ind Go = Ind Q = Tr ( - 1 ) vqL°+'(q)L°+g (3.24)

= Tr exp[-2~-'r2(U + e +~) + 21ri'rx(P + e - g)~3.25)


where we have used q = e 2'~i~. Several comments are in order about the precise form of this formula before we can understand why it gives the index of the Ramond operator. Besides the insertion of the (--1) F factor it differs from (3.2) by a shift in the Hamiltonian. This shift is the universal shift in energy caused by the inhomogeneous transformation of the energy- momentum tensor under a change of coordinates zi ---, zj:

(Ozj~ 2 c Ti = \-~zi] Tj + -~ S,j , (3.26)

where Sij is the usual Schwarzian derivative given by:

Sis__ S ( z , , z j ) _ ~O~zi 3 ( ~ / 2

Since the L0 and L0 modes were defined for the complex plane they will be shifted by going to the torus coordinates (z ~ e2~'i"). This is reflected by the appearance of the e's in (3.24). Let us examine (3.24) and use (3.16). One gets:

Ind ( G o ) = Tr(--1)FqGg(q) L°+~ (3.28)

= q~+h y] I k ~ . (3.29) kEN

The first line is just a rewriting of (3.24). The second equation follows from:

• the pairing and the ( - 1 ) F factor in the trace imply tha t only the supersymmetric states contribute. From (3.16) their L0 eigenvalue is - e = c/24;

• since Go and L0 commute, each of these states contributes with a weight qLo+~;

• since the energy is bounded from below, there is a state of lowest energy H = L0 + J~0 = - e + h. For the N = ½ system considered above it will of course have the quantum numbers of a spinor on M as usual for the Ramond sector. On this ground state the index of Go will simply be the ordinary A-genus . All the higher states will be obtained by repeated application of the creation operators and form the traditional spectrum of the corresponding string theory. This gives the summation over the positive integer in (3.28). For each level we will get the index of the Dirac operator coupled to the corresponding vector bundle.


Since P is the generator of an isometry, we can view (3.28) as a sort of G-index formula and each Ik in the index formula is an integer given by the index of Go on the corresponding representation of P. In other words it is a character valued U(1) index formula (very much like the one obtained in section 2.)

3.1 A n a l y t i c a l i n d e x

We now turn to the determination of the analytical index, tha t is the evaluation of the functional integral which will give us an expression of (3.28) as a function of q and characteristic classes of M. This section is mainly devoted to the computat ion of some infinite dimensional determinants on a torus. The reader who is not interested in the details can go directly to the main result of this paper which is given in equation (3.57).

As in the point particle case we will expand the fields about the constant solutions of the equations of motion and keep only the terms of lowest order in the background metric. Let x" = z~ + ~u and Cu = ¢~ + Cu. Riemann normal coordinate expansion about x0 we have

Using

gu,,(Xo) = 6u~, (3.30)

cO),g#,,(xo) = 0 (3.31)

1 R a,,a;,g,,,(xo) = + ( 3 . 3 2 )

The component lagrangian becomes:

(3.33)

where 1 # v 2 TEa~ = ~Ruv~/~ ¢ ¢ . We have

Ind G0 = (27r) -d /2 fM dazaz¢ [det '(-0~0z + TED,)] -½ [det '(0e)] ½ (3.34)

We have used an obvious matr ix notat ion and dropped the subscript 0 for the zero modes. The prime over the determinant means tha t the zero modes have to be excluded.

For later convenience the following notations are introduced. A complete set of normal modes with the appropriate periodicity in z --+ z + 1 and

2We will not consider here the possible global anomalies related to global sign ambiguities in the definition of the square roots of the determinants. An analysis of this problem and its relation to Stieffel-Whitney classes in the case of the point particle was done in [1].

202 O. Alvarez et al. / The Dirac-Rarnond operator in string theory

z ~ z + r is given by exp (az + fl~'):

Also let:

2~ri - - - ( m + h e )

21ri - - - ( m + ~,)

(3.35)

(3 .36)

r = { ~ = m + n ~ Ira, n e z } . (3.87)

Equivalently, P and ~ denote the replacement of r by ~ in (3.37), while the subscript in F0 indicates the omission of the point (m, n) = (0, 0) from the lattice P. We will often denote the product or the summation over all the sites of I'0 by a simple II' or E'.

It is clear at this point tha t if we keep only the n = 0 set of normal modes in the computation of the Raxnond index one should recover the particle case result i.e. the index of the Dirac operator. This fact can be used to easily check some of the determinant normalizations. As in the point particle case we could by paying attention to the normalization of the zero modes rescale the fields and the coordinates to get rid of the explicit 72 dependence in the subsequent calculations. We prefer not to do so.

The determination of the index in (3.34) involves the evaluation of the product of determinants:

"det ' ( -0~0z + T~0~)] -½ r d e t ' ( - 0 ~ 0 z + 7~0~)] -½ [ det'(0~) ]½ det '(0~) = [ d-~ ;~-- 0~-~z) [d~;(--S-0~Oz) ]

= de t ' ( -Oz + 7Z)-½. (3.38)

To go from the first to the second equation we have used the known values of the free boson and fermion determinants. They can be determined by writing the product of eigenvalues c~ and fl of (-OeOz) as:

I I 7rw 2 = expG'(0) ; (3.39) ~EPo q'2

with

0 ~ 7r ~ -2s G I (o) - at s=O (3.4o)

a s ~ o It2 ,

= - 2 In [ 2 T 2 r l ( q ) r l ( q ) ] . (3 .41)

Here rl(q) = q~ I]~=x(1 - q~) is the Dedekind function We then obtain for the parti t ion function of a free periodic right moving fermion whose ground

O. Alvarez et al. / The Dirac-Rarnond operator in string theory 203

state energy (Fermi level) in the Ramond sector is known to be 1/16:

o o _.L1 _1_ Tr ( - 1 ) F q Lo+~ - -q 48q,8 I I ( 1 - q n) = r l ( q ) - (2~-2)-½det'(0~) ½ (3.42)

Here L0 is the Hamiltonian of a c = 1/2 free right moving fermion field. Similarly the Bose-Einstein distribution gives:

- . _ ! 1 '15: qLo-~qLo-~ _ [r/(q)r/(q)]-I 12~-21 det (-0~0~)-2 (3.43)

In d dimensions the properly normalized result reads:

. det ' (a , ) ~ 1 d det'(-a~a.)] = [ 2~rl(q)] (3.44)

We have to evaluate a determinant of the type:

d e t ' ( 0 ~ - F ) = I I I ~ ( m + n ' r ) - F 1 (3.45) Fo LI2 J

Now let us recall the definition of the Weierstrass a- funct ion [20]:

a(z) = zII ' (1 - ~ ) e x p + ~ . (3.47)

Notice the similarity between the a function and the sine function - - i n Net the subsequent results will merely follow from a substitution of the sine by a in the formulae of section 2-- . This is most easily seen by writing down another representation for the a function3:

sin(~'z). ~ (1 - q'~e 2"~iz) (1 - qne-2"~iz) ~ ( z ) = e , , z ' • ~ , (1 - ¢ ) = ( 3 . 4 s )

O(z,"r) (3.49) - O'(O,r) e~ 'z ' '

where rh = ¢(1/2) is the value of the logarithmic derivative ( (z) of a(z) at the half per iod [20]. Compar ing (3.47) and (3.45) we have:

1 d e t ' ( - 0 ~ - F ) = -2Ar/2(q) • ~ - ~ . a , (3.50)

where log A ---- -52 ' - ~ - ~ . The lack of absolute convergence of this

aMost definitions and conventions can be found in the appendix.


factor will play an important role in what follows. We can now repeat the same type of manipulation on eq. (3.38). Let the formal eigenvalues of 7~ be: TC2i+x,2i+2 -- -T¢2i+2,2i+1 --- 2Trxi (i -- 1 , . . . , d/2). Then the second bracket in (3.38) reads:

[de t ' ( -O~0z+~0~) l -* (1 2ir2xi (3.51)

ki=la(2ir2xi)j exp [1GI(~D ) ~ . z 2] (3.52)

In the last equation A has be rewritten in terms of the Eisenstein series Gx (see below). It is at this point that the role of A is important. Indeed since we are performing an index computation, no arbitrary process of regularization should be involved. There are many possible ways to deal with the problem of convergence of the series GI, but the only unambiguous one is simply to require that its multiplying coefficient in A vanishes. This introduces the

2 = Tr R 2 = 0. So the first conclu- most remarkable constraint: Pl (M) = E xi sion is that for the index to have a well defined meaning the first Pontrjagin class of the manifold must vanish. Also it should be rather obvious at this stage that had one introduced extra gauge fields this condition would have been replaced by Tr(R 2 - F 2) = 0, which is nothing but the Green-Schwarz anomaly cancellation condition in string theory [5]. Moreover we will see in a moment that once this condition is satisfied, the index will have very simple modular properties. This is of course the result which was expected in the introduction. The connection between the field theory anomaly and the modular properties was studied in full detail in [6,7,8]. From now on we will suppose that the above condition is satisfied. Let us now rearrange the above result. With the help of the Eisenstein series defined by:

1 a~(w)----- E w2~ (3.53)

the function a(z)/z has the expansion:

o(z) -- exp

Z

---- exp

---- exp

wEFo

_ _ . 3 ¢ _ - -

LwEFo W

[ oo l k 1 ] -EE zj k=3 wEPo

1 Z 2 + ~(w) ] (3.54)

(3.55)

(3.56)


Inserting (3.51) and (3.44) in (3.34) we have the final expression for the index of the Dirac-Ramond operator:

Ind ( G o ) - q~+h ~ Ikq k -" kEN

fM daxa/2 ixi/2rr 1 (3.57) II a(ixi/2~rl~) " ~(q)a i----1

dj2 ] 1 35s, = fM ddxi=tII exp (~ )2kGk(~) • 7?(q)d

,=1

Several remarks are in order. First the integration over the fermionic zero modes dd¢ has been performed. It results in replacing T~ by __R = Ru~,dxUd#' and interpreting the xi accordingly as 2-forms. Secondly the explicit T2 dependence has disappeared. This is because in the above formula only the terms which are forms of order d will contribute to the integral i.e. the terms homogeneous in xi of degree d/2. It is easy to check tha t such terms do not contain any residual T2 factor. It is important to notice the modular properties of the result. They are best seen in (3.58): if the dimension is a multiple of 24 we see tha t the index is modular invariant under the full modular group. Its behaviour at the cusp (q = 0) is easily seen to be q-d~24 which gives us g + h + d/24 = 0. Notice that the behaviour at the cusp corresponds to the ,4-genus of M. Finally we point out tha t (3.58) and (3.59) have been writ ten using the condition pl(M) = O. In the form (3.58) the result is readily seen to be a generating function for the topological index of the Dirac operator coupled to tensors of rank n where n is the level considered [6,7,14]. For example as we just saw, the zero mode correspond to the A genus ,i.e. the index of the Dirac operator for the spinors in the Ramond vacuum, while for higher levels the result is simply multiplied by the appropriate Chern character. In tha t sense we have a generating functional for the field theory anomalies. In the following we will loosely refer to this

generating functional as A('r).

We would like to point out that the result of interest is really A(r) itseff and not any of its truncations. The special form of the result comes from the product over all the modes of the lattice I'. It should really be seen as an extension of the Atiyah-Singer index theorem to the loop space and the link between the analytical index of the Dirac-Ramond operator and topological and cohomological properties of the loop space. Any new insight will come from considering this result in that light.


4 Euler number, signature, etc...

We are in position to analyze different topological invariants which arise from coupling the N ---- 1/2 "right" moving system with a leftmoving sector. This corresponds to introducing a twisting of the original Ramond-Dirac operator and in the point particle case to the introduction of the ~?, ~ system of section 2. The boundary conditions for the left sector can be chosen to be Ramond or Neveu-Schwarz in space and periodic or anti-periodic in time corresponding to the four spin structure on the torus. None of these choices will spoil the topological character of the result since as we saw in section 3 the topological invariance is entirely governed by the supersymmetry of the right sector. Of course if the new sector possesses its own N = 1/2 supersymmetric charge do, the states with /:0 ¢ g will be projected out of the trace. This will happen in the case of periodic Ramond boundary conditions and will in a particular case give the Euler number of M.

We will by the subscript 0 (respectively 1, 2, 3) denote the Ramond periodic boundary conditions (Ramond anti-periodic,Neveu-Schwarz periodic, Neveu-Schwarz anti-periodic). In this notation we will obtain unified expression for the four indices. Gauge or gravitational background fields can be coupled to this sector. In the latter case, the only one treated explicitly here, the condition pl (M) = 0 is not required anymore. The treatment of the external gauge field coupling could be done as easily and would require Tr R 2 - Tr F 2 -- 0. Of the four indices constructed, two of them have a

simple geometrical interpretation. They correspond respectively to the extension to loop space of the Euler number and the Hirzebruch signature that we treated briefly in section 2. To construct the Euler and the Hirzebruch indices we use the isomorphism between the exterior algebra and the tensor product of the spinor algebra with itself. This correspondence is well understood in finite dimensions and there seems to be a formal infinite dimensional analogue. Let ¢ be a spinor and let p be a dual spinor. The tensor product Cp is a bispinor which may be decomposed into a complete series of gamma matrices and their antisymmetric products:

Cp tr(p¢) +

+ E ~/'0 'vtr(pT"~v¢) + " " • (4.1)

This is the standardcorrespondence between spinors and the exterior algebra via the association of the wedge products d x 1 A d x ~ A . . . A d x k with the antisymmetric products of gamma matrices.


This observation is used in the following way. The canonical anticom- mutat ion relations for the fermions in the N -- 1/2 right moving sector are {¢u(a), ¢~(a ')} o(gU~'6(a-a ' ) . This means tha t the ¢ ' s are the gamma matrices in loop space. The carrier of the representation space for the gamma matrices are the spinors on loop space. We now introduce a right moving fermionic Ramond superfield which transforms as a covector

N u = r/u + 0tou. (4.2)

The canonical commutat ion relations will tell us tha t the ~?'s are dual to the 'gamma' matrices and therefore the carrier space for the representations are the dual spinors. The state created by the combination of ¢ 's and ~?' will be spinors tensored with dual spinors. These are just the exterior algebra. One can make this identification more explicit by writing down the tangent space in terms of oscillators and comparing it with the Hilbert space generated by looking at the oscillator content of the fermions ¢ and 7/. The fermion parity in the right moving sector ( -1 ) FR may be interpreted as multiplication of the bispinor by "Y5 on the left. Likewise, the fermion parity in the left moving sector ( - 1 ) EL may be interpreted as multiplication of the bispinor by "Y5 on the right.

The Euler characteristic compares forms of even degree with odd degree, therefore it may be computed heuristically as

X = T r ( - 1 ) fL+fn (4.3)

In the supersymmetry approach one calculates the above by having both the left and right moving sectors satisfying periodic boundary condition in the temporal direction.

In finite dimensions the Hirzebruch signature compares self dual forms with anti-self dual forms. In Dirac language, the duality transformation is implemented by multiplication by 75 on the left. The associated index is formally

T = Tr ( -1 ) FR . (4.4)

Wha t we are then led to calculate is the pa th integral with periodic temporal boundary conditions for both ¢ and 7/ in the case of the Euler number, and with periodic ¢ and anti-periodic 7/in the case of the Hirze- bruch signature [1,21]. In analogy with (3.24) these pa th integrals compute respectively:

I× ---- T r ( -1 ) FL+FRqL°+~clt°+~, (4.5)

I~ = Tr ( -1 ) FRqL°+~qL°+~, (4.6)


where now L0 + e and L0 + ~ contain contributions from the full system of two dimensional fields X, ¢ and ~? 4. As in the supersymmetric quantum mechanical case, the particular boundary conditions chosen to calculate the Euler number give rise to a left handed N = 1/2 supersymmetry [21]. By expressing L0 + g as the square of this new supersymmetry, we can infer as before tha t only states with L0 + g = 0 contribute to the trace. So altogether only the zero modes should enter the expression for I x . By explicitly carrying out the functional integration we obtain:

i x _ (27r) d/21 f ddXo dd¢o [det,(_O~+7~)]_l/2[det,(~)]U2[det(_O_7~)]U2

(4.7) This expression directly simplifies into

1 Ix -- (27r)d/2 / ddX0 dd~b0 [det (-7~)] '/2 , (4.8)

which as we promised only contains the contribution from the zero modes. Integrating over the fermionic zero modes ¢0 leaves us with the Euler characteristic of £:(M), which coincides with the Euler number of M itself, ac- cording to the Lefshetz fixed-point theorem:

Ix = x ( M ) -~ fM I!. (--xj). (4.9) 2

We remark tha t in this case we do not need to impose pl(M) = 0 in order to have a well defined index: the Euler number of a manifold is defined even if the manifold does not admit a spin structure (or a string structure in the case of the loop space). Possible inconsistencies due to the use of spinors to obtain the result should cancel out in the final expression, as they do.

Performing the pa th integral for the Hirzebruch signature gives rise to the same formula as in (4.7), but the determinant arising from the left handed fermions is to be evaluated over states with anti-periodic boundary conditions in ~-. The details of the computat ion for this case as well as the next two can be found in the Appendix. One obtains for the Hirzebruch signature of £ (M) :

(4.1o) O(ixk/21r, ~) k

There is a simple interpretat ion of this result in terms of the Atiyah-Bott character index theorem. It will be given elsewhere [15]. Once again the manifold M is not required to satisfy Pl (M) --- 0 in order for this index to be

4We have eliminated the auxiliary fields ~ ,


defined (cf. appendix). Using the notations introduced in the appendix one can see tha t the two topological invariants corresponding to the boundary conditions (NS,P) and (NS,A) are given by:

I,~ = fM II ixk. 9(ixk/27r,~) a=(2,3) (4.11) k

As with the generalization of the A-genus there is a simple interpretation of these results in terms of a generating function for the the indices of the ordinary Dirac operator twisted by the different bundles corresponding to spectrum of the appropriate sector. However one would like to obtain a nice geometrical interpretat ion for the last two. It is stiU missing for the moment 5.

As for the case of A(~') the topological invariants I~ have simple transformations under the modular group or more precisely under some subgroup. This is easily understood since the different boundary conditions are pre- served by some subgroup of F. Only some phase anomalies appear under such a modular transformation. Then one has a modular invariant result only if the dimension is a multiple of eight. The generalized Hirzebruch signature is a modular invariant in dimension d = 8k under the subgroup

r F(2) generated by T --+ ~" + 2 and ~- --* - - . In this case it is easy to see

1 - 2~- tha t the value at q = 0 is jus t the ordinary Hirzebruch signature of M. I2 is also invariant under F(2). Also in d = 8k we find tha t /3 is a invariant under F0 generated by T --+ --1/T and T --+ T + 2. Note tha t it is possible to rewrite the four indices in terms of Jacobi eRiptic functions. The necessary definitions are listed in the Appendix. It provides an immediate connection with the definition of the elliptic genus given in the work of Peter Landweber and Robert Stong [10,22] and Serge Ochanine [11,23].

I1 = f , II " V/"~sn(wlixk /2) 1

I2 = fM IIk v sn( lixk/2)

dn(wxz~) I3 = [I . Z-Zsn( z )

01(0, ) where Wl -- 93~(0, ~), v/~ -- 93(0, ~) and ~ -- - -

5see W i t t e n ' s work [14] for a use of 13.

02(0, e) 03(0,

(4.12)

(4.13)

(4.14)

(4.15)

It is then very easy

210 O. Alvarez et aL / The Dirac-Rarnond operator in string theory

to find the expression for the parameters e and 5 defined in [11] in terms of )~(¢) = n2 __ 1 - (n,)2. In particular it is important to notice tha t ),(¢) is invariant under the subgroup of the modular group F(2) defined above and vanishes st q = 0.

In conclusion, we have shown how to use two-dimensional N = 1/2 supersymmetric theories to probe the topology of the loop space I:(M) of compact Riemannian manifolds M without boundaries. The existence of a group ac- t ion on £ (M ) makes it possible to sprit the infinite dimensional kernels that appear in the definitions of various indices into finite dimensional represen- tat ions of the group action. The results tha t we obtain suggest tha t the finite dimensional character-valued index theorems hold true when naively applied to the infinite dimensional manifold £ (M) , provided certain topological restrictions are satisfied by M in the case of certain indices. Finally, let us point out tha t we could have coupled the Dirac-Ramond operator to other bundles like gauge fields by simply modifying the left-moving sector, introducing additional fermi fields and coupling them to external gauge fields, just as we did in the point particle case in section 2. As in all the cases studied here, the structure of the final result would be A(~-) multiplied by the corresponding generalized Chern character. It is clear tha t for us the fundamental result is the expression of A(T) obtained in section 3. All the other topological invariants being specializations of it.

Acknowledgements P.W would like to thank the organizers of the Irvine Conference on Non-

Perturbative methods in Field Theory for the opportuni ty to present this work. He would also like to thank Serge Ochanine and Peter Landweber for informing him about their work. O.A. would like to acknowledge general discussions with Iz Singer about index theorems and elliptic genera. O.A. would like to thank the A.P. Sloan Foundation for their generous support.

A Appendix

We collect in this appendix most of the computations needed to obtain the results of the preceeding sections. For the sake of completeness there is some overlap with section 3. Two good general references for this appendix are Whit taker and Watson [20] and Elliptic functions by K. Chandrasekharan [24]. We have to evaluate a series of determinants which are all of the same type, the only difference being the boundary conditions. The four bound-

O. Alvarez et aL / The Dirac-Ramond operator in string theory 211

axy conditions axe Ramond or Neveu-Schwaxz,i.e.periodic or anti-periodic in space, and periodic or anti-periodic in time. They are symbolically noted (R,P), (R,A), (NS,P) and (NS,A). In the following computations the last three will be labelled by a single index a = (1, 2, 3) while occasionally the first one will be denoted by the subscript 0. We introduce the following notations: w = m+n'r with Imp- > 0 and m, n integers. A shift by one of the half- period of the lattice F = {w} will be denoted by wa = (1/2, ~-/2, 1/2 + T/2). We give below a list of the definitions to be used later, where the products and sums run over the entire lattice F except when a prime denotes the omission of the origin:

~(z) - z2 + (z - ~)2 5~

( 1 1 3 ) 1 + y], z w w -- - - - + - - + (A.2) ¢ ( z ) -

o-(z) z I l ' ( l - ; ) exp ( z z 2 ) = - - + (A.3)

a.(z) -- a(z + w.) exp (-r / .z) (A.4) ¢(~°) ( (z2)

= IX 1 w--wa ~ exp Aa- -~ -ea , (A.5)

z 1 ( ~ ) ~ where p(wa) = ea and ~(w~) = r]~ and A. - (w - w~) + 2 w - w. " These

are respectively the Weierstrass p and ~ functions and the four sigma functions which are in one to one correspondence with the four the ta functions through

O(z, ~) ~,~ a(z) - o--T-~,v) e (A.6)

Oa(Z''T) c'lZ2 ( 1 . 7 ) o - ° ( ~ ) _ o ° ( o , - , - )

In the notat ion of Mumford [25] the four the ta functions axe respectively denoted -011,810,801 and 800. For the convenience of the reader we also give their s tandard product expansions:

oO 8 ( z , ' r ) o - - 8 ( z , z ) = c ' q 1/s2sin~rz" 1-I (1-q'~e2'~'z)(1-q '~e-2~'~) (A.8)

oO 81(Z,T) : d q 1/Selriz" H (1 +qne2'~'z)(1 -}-qn--le--2~riz), (A.9)


oo

O~(z, r) -- c' I I (1 - q~-l/2e2~iz)(1 - qn-1/2e-2'~iz), (A.10) n = l

oo

03(z,'r) = c' II (1 q- q'*-l/2e2"i~)(1 + qn-1/2e-2=iz) , (A.11) n----1

o o

with c' = I I (1 - q~). (A.12) n = l

Using the above one easily evaluates the general class of determinants [26] 1

~ e t ~ ) J. = IIIIk 1 w - - wa (A.13)

- II oa(z~)exp e ° . e x p ( - E A ° ( z ~ ) , (1.14) k

where zk = 2ir2xk and xk axe the formal eigenvalues of the curvature matr ix R/2~'. Note the presence of the regulaxization factor N,,k = exp - (E A~(zk)) which in section 3 resulted in the condition Pl (M) = 0. The determinants in flat background can be evaluated as in section 3 and give

d , (0a(0,,))~ (detcg)~ = [ ~ - ~ . (A .15)

o o

with rl(q) = q~ 1-I(1 - q~). (A.16) 1

We then have 0.(~k,~) z 2

cha(z) -- det(a + R) i = II N,,k .exp (~iz~ + 2 e , ) . (A.17) ~(q)

Recalling the contribution coming from the boson fields and the right moving fermions:

~--[det'(O+R)]-½ = [H r2ak(zk) O'(Zk, T)] -1 (A.18) A(~) ~ z ~ " ~ ) 1

k L "1-2 " O ( z k , ' r ) . e x p ~ ' A ( z k ) - e x p r h z k

W e h a v e d e f i n e d A = z 1 ( ~ ) 2. -- + We can now combine A.(T) with any

of the Cha(T). The regulaxization factors of both terms will cancel using - E A~ + E'A = -x2e~/2. This results in

A(~). ch°(~) = II ~z~ 0°(~, ~) k ~~ O(z~,.,-) (A19)

If the left moving fermions have (R,P) boundary conditions 0~ is simply replaced by 0 in the above, resulting in the Euler number. Finally we give


the re la t ion wi th Jacobi elliptic functions. They are ob ta ined by solving the equa t ion

(y'(u)) 2 = (1 - 26y2(u) + ey4(u)) (A.20)

In t roduc ing the modu lus ~, the simplest of t h e m is given by e = ~2 and 26 = ~2 + 1. It corresponds to the Jacobi s n ( u ) funct ion which has the following expression in terms of t he t a funct ions

1 0(z, ~-) s n ( u ) = x~ ~ • 02(z, "r) ' (A.21)

wi th v ~ = 01(0, ~)/03(0, T) and z = u/(rrO~(O, r ) ) . The other basic functions are

an(u) =

d n ( u ) (--

The following relat ions are satisfied

c n 2 + s n 2

d n 2 + ~2 s n 2

t¢ 2 + i¢ ~2

-~__~ , 01(Z , T) 02(Z,T) (A.22)

03(z, ) v/-~.) O2(z, ~_ ) . (A.23)

---- 1 (A.24)

= 1 (A.25)

= 1 , (A.26)

wi th ~ = 02(0, T)/O3(O, T). It is easy to see t h a t (26, e) ---- (~2 _ 2, ~,2) corresponds to s n / c n and (26, e) ---- (1 - 2~ 2, _~2~,2) to s n / d n . It is impor t an t

to not ice t h a t the modu la r propert ies of all these funct ions are governed by A(T) = ~2 as can be seen from their defining equation. It is well known tha t A(T) is a modu la r funct ion invariant unde r the subgroup of the modula r group genera ted by ~- --+ T + 2 and ~- --~ ~-/(1 - 2T) or equivalently, if a, b, c, d

. , aT + b, )~(~-) if and only if b are integers such t h a t a d - bc = 1, t ha t A ( ~ ) =

and c are even, so t h a t a and d are odd.

R e f e r e n c e s

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[5] M.B. Green and J.H. Schwarz. Anomaly cancellation in supersymmetric D = 10 gauge theory require SO(32). Phys. Lett. 149B, 117 (1984).

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[23] S. Ochanine. Genres elliptiques 6quivariants. In P.S. Landweber, editor, Elliptic Curves and Modular Forms in Algebraic Topology, Institute for Advanced Study, to appear.

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The Dirac-Ramond operator in string theory and loop space index theorems