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Representations of loop groups, Dirac operators on loop space, and modular forms. (English) Zbl 0715.22023

The author considers, “in a purely formal way”, the index of the Dirac operator on the loop space LM of a compact manifold M (i.e. the space of smooth maps \(S^ 1\to M)\) with coefficients in the vector bundle E over LM associated to a positive energy representation of (the universal central extension of) the loop group LSpin(d). The index of this operator may be formally written down, and the main result is that it is a modular form of weight k for a congruence subgroup of SL(2,Z), which depends on the level of the representation chosen. The proof uses the fact that the characters of such representations are Jacobi modular forms in several variables. The author also presents a conjecture relating representations of LSpin(d) with the elliptic cohomology theory of Landweber, Ravenel and Stong.
Reviewer: A.N.Pressley

MSC:

22E67 Loop groups and related constructions, group-theoretic treatment
58J22 Exotic index theories on manifolds
17B68 Virasoro and related algebras
11F06 Structure of modular groups and generalizations; arithmetic groups
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
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