Representations of loop groups, Dirac operators on loop space, and modular forms

Abstract

FOR M a compact Spin manifold of dimension d= 2k, the unicersal elliptic genus p(M), in the sense of D. V. Chudnovsky, G. V. Chudnovsky, P. Landweber, S. Ochanine, and R. E. Stong, is a modular form of weight k for a level 2 congruence subgroup of SL(2, Z) [SJ, [17], [19], [26], and [28]. E. Witten showed that p(M) could be interpreted as the index of a sort of twisted Dirac operator on the loop manifold LM [26] and [27]. He actually considered several sorts of twistings of this Dirac operator; all of them give rise to modular forms for level 2 congruence subgroups. Similar results were obtained by Alvarez, Killingback, Mangano, and Windey [l] and by Schellekens and Warner [23]. In this article we consider, in a purely formal way, the index of the Dirac operator on LM, with coefficients in the vector bundle Reassociated to a “positive energy” representation of the universal central extension e Spin(d) of the loop group LSpin(d). We do not claim to understand anything about this operator, except that its index may be formally written down. The main result (Theorem 2.2) is that this index is a modular form of weight k for a congruence subgroup of SL(2, Z), which depends on the level of the representation of zSpin(d). The proof uses the results of V. KaE and D. Peterson [ll] and [12] on the characters of such representations, in particular the fact they are Jacobi modular forms in several variables. We present a conjecture relating the representations of tSpin(d) with elliptic cohomology, the theory of Landweber, Ravenel, and Stong [17], [19]. More precisely, we extend elliptic cohomology to higher levels (i.e. for each integer N, we construct a homology theory Ellz such that Ellt(pt) is the ring of modular forms for the principal congruence subgroup F(N), whose Fourier expansions at every cusp have coefficients in Z [ e2ni’N]). The point of this construction is that Ellz(pt) is a faithfully flat module over Ell,(pt); so we do not claim that Ellt(pt) is a really new homology theory, from the topological veiwpoint. From the differential-geometric viewpoint, we show that the vector bundle E, mentioned above, is equivariant under the Virasoro algebra, which acts on LM through its quotient vecr(S’). We think it might be useful to view the index of the Dirac operator on LM as a virtual representation of the Virasoro algebra, rather than merely the group of rotations of S’.