Quadratic functions in geometry, topology,and M-theory
M. J. Hopkins, I. M. Singer
TL;DR
We develop a higher-dimensional analogue of Riemann’s quadratic function by introducing differential integral Wu-structures and a generalized determinant κ that yields a quadratic refinement q_{E/S}^{λ} over oriented families E→S of relative dimension 4k−i (i≤2). The central result is a functor κ_{E/S} from twisted differential Wu-structures to differential cocycles, with normalization, symmetry, base-change, and transitivity properties, enabling q_{E/S}^{λ}(x)=κ(λ−2x) as a refinement of the cup product ∫_{E/S}x∪y. This framework connects to the fivebrane partition function by producing a holomorphic line bundle on the Jacobian J=H^3(M)⊗R/Z, with monodromy matching Witten’s formula, thereby unifying topology, differential geometry, and physics in the study of anomalies and loop-space indices. The paper expands Cheeger–Simons cohomology to generalized differential cohomology via differential function spaces and spectra, providing a robust language for fields and action functionals in mathematical physics. Overall, it offers a comprehensive toolkit for quadratic refinements in higher dimensions, differential refinements of characteristic classes, and applications to M-theory partition 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.