Quadratic functions in geometry, topology,and M-theory

Abstract

We describe an interpretation of the Kervaire invariant of a Riemannian manifold of dimension $4k+2$ in terms of a holomorphic line bundle on the abelian variety $H^{2k+1}(M)\otimes R/Z$. Our results are inspired by work of Witten on the fivebrane partition function in $M$-theory (hep-th/9610234, hep-th/9609122). Our construction requires a refinement of the algebraic topology of smooth manifolds better suited to the needs of mathematical physics, and is based on our theory of "differential functions." These differential functions generalize the differential characters of Cheeger-Simons, and the bulk of this paper is devoted to their study.

To appear in the Journal of Differential Geometry. This version addresses several issues brought up by the referee. We have also added an appendix describing stable exponential characteristic class for Spin bundles, taking values in cohomology with integer coefficients, whose mod 2 reduction is the total Wu class. LaTeX. 99 pages

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