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

Joshua Lackman

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.

A Groupoid Approach to the Riemann Integral (and Path Integral Quantization of 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.
Paper Structure (11 sections, 7 theorems, 56 equations)

This paper contains 11 sections, 7 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Path Integrals and the Poisson Sigma Model
  3. Quantum Mechanics
  4. The van Est Map and Antisymmetric Cochains
  5. The van Est map
  6. The Main Result
  7. The Riemann Integral on a Closed Interval
  8. The General Case
  9. Lie Groupoids and Lie Algebroids, Nerves, Cochains and Forms
  10. The van Est Map
  11. The Pair Groupoid

Key Result

Theorem 1

Let $X$ be an oriented $n$-dimensional compact manifold (possibly with boundary), and let $\omega$ be an $n$-form. We define the Riemann integral by completing the following steps: This construction of the Riemann integral agrees with the usual integral.

Theorems & Definitions (35)

  • Theorem 1: Main
  • Remark 2
  • Lemma 3
  • Corollary 4
  • Remark 5
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Example 1.4
  • Definition 1.5
  • ...and 25 more