Witten's conjecture and recursions for $\kappa$ classes

TL;DR

This paper develops a set of differential operators that generate recursive relations for intersection numbers of  classes on moduli spaces, extending the Virasoro constraints to  classes and enabling computation from initial data.

Contribution

It constructs explicit differential operators that produce recursions for  class intersection numbers, generalizing Virasoro constraints to these classes.

Findings

Derived a countable set of differential operators  annihilating the -potential.

Established explicit recursions for intersection numbers of  classes.

Connected  class intersection theory with Virasoro constraints and Gromov-Witten potentials.

Abstract

We construct a countable number of differential operators $\hat{L}_n$ that annihilate a generating function for intersection numbers of $\kappa$ classes on $\Moduli_g$ (the $\kappa$-potential). This produces recursions among intersection numbers of $\kappa$ classes which determine all such numbers from a single initial condition. The starting point of the work is a combinatorial formula relating intersecion numbers of $\psi$ and $\kappa$ classes. Such a formula produces an exponential differential operator acting on the Gromov-Witten potential to produce the $\kappa$-potential; after restricting to a hyperplane, we have an explicit change of variables relating the two generating functions, and we conjugate the "classical" Virasoro operators to obtain the operators $\hat{L}_n$.