Cut-and-join operators in cohomological field theory and topological recursion

TL;DR

This paper develops a cubic cut-and-join operator framework for topological recursion partition functions, connecting algebraic structures with Virasoro constraints and cohomological field theories.

Contribution

It introduces a novel cubic cut-and-join operator approach for topological recursion partition functions, unifying algebraic and geometric perspectives.

Findings

Constructed a cubic cut-and-join operator for topological recursion

Derived Virasoro constraints for the partition functions

Proved the equivalence of constraints and the cut-and-join description

Abstract

We construct a cubic cut-and-join operator description for the partition function of the Chekhov-Eynard-Orantin topological recursion for a local spectral curve with simple ramification points. In particular, this class contains partition functions of all semi-simple cohomological field theories. The cut-and-join description leads to an algebraic version of topological recursion. For the same partition functions we also derive N families of the Virasoro constraints and prove that these constraints, supplemented by a deformed dimension constraint, imply the cut-and-join description.