Twisted de Rham cohomology, homological definition of the integral and "Physics over a ring"

TL;DR

This paper introduces twisted de Rham cohomology to define integrals over arbitrary rings, explores homological properties, and applies these concepts to p-adic cohomology and topological quantum field theories.

Contribution

It provides a novel homological framework for defining integrals over rings and connects this to p-adic cohomology and quantum field theories.

Findings

Homological definition of integrals over rings

Application to Frobenius map on p-adic cohomology

Extension to topological quantum field theories

Abstract

We define the twisted de Rham cohomology and show how to use it to define the notion of an integral of the form $\int g(x) e^{f(x)}dx$ over an arbitrary ring. We discuss also a definition of a family of integrals and some properties of the homological definition of integral. We show how to use the twisted de Rham cohomology in order to define the Frobenius map on the p-adic cohomology. Finally, we consider two-dimensional topological quantum field theories with general coefficients.