Modular Invariance and Characteristic Numbers

We prove that a general miraculous cancellation formula, the divisibility of certain characteristic numbers, and some other topological results related to the generalized Rochlin invariant, the η-invariant and the holonomies of certain determinant line bundles, are consequences of the modular invariance of elliptic operators on loop space. 1. Motivations In [AW], a gravitational anomaly cancellation formula, which they called the miraculous cancellation formula, was derived from very nontrivial computations. See also [GS] and [GSW], pp. 347–361. This is essentially a formula relating the L-class to the Â-class and a twisted Âclass of a 12-dimensional manifold. More precisely, let M be a smooth manifold of dimension 12, then this miraculous cancellation formula is L(M) = 8Â(M,T )− 32Â(M) where T = TM denotes the tangent bundle of M and the equality holds at the top degree of each differential form. Here recall that, if we use {±xj} to denote the formal Chern roots of TM ⊗ C, then