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

Abstract

We construct a countable number of differential operators $$\widehat{L}_n$$ that annihilate a generating function for intersection numbers of $$\kappa $$ classes on $$\overline{{\mathscr {M}}}_g$$ (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 intersection 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 $$\widehat{L}_n$$ .