Theta classes: generalized topological recursion, integrability and $\mathcal{W}$-constraints

Abstract

We study the intersection theory of the $\hat{I}{˜}^{r,s}$-classes, where $r\ge 2$ and $1\le s\le r-1$, which are cohomological field theories obtained as the top degrees of Chiodo classes. We show that the recently introduced generalized topological recursion on the $(r,s)$ spectral curves computes the descendant integrals of the $\hat{I}{˜}^{r,s}$-classes. As a consequence, we deduce that the descendant potential of the $\hat{I}{˜}^{r,s}$-classes is a tau function of the $r$-KdV hierarchy, generalizing the Brézin--Gross--Witten tau function (the special case $r=2$, $s=1$). We also explicitly compute the $\mathcal{W}$-constraints satisfied by the descendant potential, obtained as differential representations of the $\mathcal{W}({\mathfrak{gl}}_r)$-algebra at self-dual level. This work extends previously known results on the Witten $r$-spin class, the $r$-spin $\hat{I}˜$-classes (the case $s=r-1$), and the Norbury $\hat{I}˜$-classes (the special case $r=2$, $s=1$).

36 pages, comments welcome!