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

Abstract

We study the intersection theory of the ${\Theta }^{r,s}$-classes, where $r⩾2$ and $1⩽s⩽r-1$, 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 ${\Theta }^{r,s}$-classes. As a consequence, we deduce that the descendant potential of the ${\Theta }^{r,s}$-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 $W$-constraints satisfied by the descendant potential, obtained as differential representations of the $W\left({\mathrm{gl}}_r\right)$-algebra at self-dual level. This work extends previously known results on the Witten $r$-spin class, the $r$-spin $\Theta$-classes (the case $s=r-1$), and the Norbury $\Theta$-classes (the special case $r=2$, $s=1$).