On the extension of a TCFT to the boundary of the moduli space

TL;DR

This paper extends a construction related to topological conformal field theories to the boundary of the moduli space of Riemann surfaces, using finite-dimensional Frobenius algebras, and shows its equivalence to Kontsevich's dual construction.

Contribution

It introduces an extension of Costello's construction to a compactified moduli space using finite-dimensional Frobenius algebras, linking it to Kontsevich's dual approach.

Findings

Construction extends to compactified moduli space

No ultra-violet divergences in finite-dimensional case

Equivalence to Kontsevich's dual construction

Abstract

The purpose of this paper is to describe an analogue of a construction of Costello in the context of finite-dimensional differential graded Frobenius algebras which produces closed forms on the decorated moduli space of Riemann surfaces. We show that this construction extends to a certain natural compactification of the moduli space which is associated to the modular closure of the associative operad, due to the absence of ultra-violet divergences in the finite-dimensional case. We demonstrate that this construction is equivalent to the "dual construction" of Kontsevich.