A Groupoid Approach to the Riemann Integral (and Path Integral Quantization of the Poisson Sigma Model)

TL;DR

This paper introduces a groupoid-based, coordinate-free method for defining Riemann sums on manifolds, connecting to the Poisson sigma model and emphasizing convergence to classical integrals.

Contribution

It presents a novel groupoid and van Est map framework for Riemann sums on manifolds, enabling lattice formulations of the Poisson sigma model.

Findings

Riemann sums converge to classical integrals over all triangulations

The van Est map encodes the n-jet of antisymmetric n-cochains

Framework facilitates lattice approaches to the Poisson sigma model

Abstract

We use groupoids and the van Est map to define Riemann sums on compact manifolds (with boundary), in a coordinate-free way. These Riemann sums converge to the usual integral after taking a limit over all triangulations of the manifold. We show that the van Est map determines the n-jet of antisymmetric n-cochains. We discuss using this Riemann sum construction to put the Poisson sigma model on a lattice.