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

Understand

We study the intersection theory of the $\Theta^{r,s}$-classes, where $r \geq 2$ 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 $\Theta^{r,s}$-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 Br\'ezin--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.