On SL2(Z) and Topology

FP (g) = trgKerP − trgCokerP ∈ R(S) where R(S1) is the character ring of the S1-modules. We say that P is rigid with respect to this S1-action, if FP (g) is independent of g. Two well-known examples of rigid elliptic operators are the signature operator ds and the Dirac operator D [AH]. Now let L̃Spin(2l) denote the central extension of the loop group LSpin(2l) and E be a positive energy representation of it. Then under the rotation action of the loop, E has decomposition ⊕n≥0En where En’s are finite dimensional representations of Spin(2l). Given a real rank 2l spin vector bundle V on M , let P be its frame bundle and Ẽn be the bundle associated to P and En. We define