Path Integration in Two-Dimensional Toplogical Quantum Field Theory

TL;DR

This paper constructs a diffeomorphism-invariant measure on 2D manifolds' metrics and introduces actions that produce 2D topological quantum field theories, advancing the mathematical foundation of TQFTs.

Contribution

It develops a generalized measure on 2D metrics and presents a family of actions that generate 2D TQFTs, linking measure theory with quantum field theory.

Findings

Constructed a positive, diffeomorphism-invariant measure on 2D metrics.

Presented a family of actions producing 2D TQFTs.

Connected measure theory with the axiomatic framework of 2D TQFTs.

Abstract

A positive, diffeomorphism-invariant generalized measure on the space of metrics of a two-dimensional smooth manifold is constructed. We use the term generalized measure analogously with the generalized measures of Ashtekar and Lewandowski and of Baez. A family of actions is presented which, when integrated against this measure, give the two-dimensional axiomatic topological quantum field theories, or TQFT's, in terms of which Durhuus and Jonsson decompose every two-dimensional unitary TQFT as a direct sum.