Quadratic functions in geometry, topology,and M-theory

TL;DR

This paper interprets the Kervaire invariant of certain manifolds via holomorphic line bundles on abelian varieties, inspired by M-theory, and develops a refined algebraic topology framework using differential functions.

Contribution

It introduces a new approach linking the Kervaire invariant to holomorphic line bundles, extending differential characters with a theory of differential functions for mathematical physics.

Findings

Relates Kervaire invariant to holomorphic line bundles

Develops a refined algebraic topology framework

Extends differential characters with differential functions

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.