Teleman's classification of 2D semisimple cohomological field theories

TL;DR

This paper reviews Teleman's 2011 classification of 2D semisimple cohomological field theories, showing they are uniquely determined by their smooth part, and provides a detailed, self-contained exposition of the proof.

Contribution

It offers a comprehensive, self-contained presentation of Teleman's classification proof, emphasizing clarity and completeness without introducing new results.

Findings

Cohomological field theories are uniquely determined by their restriction to smooth curves.

Semisimple Frobenius algebras classify these theories.

The classification simplifies understanding of 2D cohomological field theories.

Abstract

In his 2011 paper, Teleman proved that a cohomological field theory on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable complex curves is uniquely determined by its restriction to the smooth part $\mathcal{M}_{g,n}$, provided that the underlying Frobenius algebra is semisimple. This leads to a classification of all semisimple cohomological field theories. The present paper, the outcome of the author's master's thesis, presents Teleman's proof following his original paper. The author claims no originality: the main motivation has been to keep the exposition as complete and self-contained as possible.