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

We study the intersection theory of the \(\Theta^{r,s}\)-classes, where \(r \geq 2\) and \(1 \leq s \leq r - 1\), which are cohomological field theories arising as the top-degree parts of Chiodo classes. We show that generalised topological recursion on the \((r,s)\) spectral curves computes the descendant integrals of the \(\Theta\)-classes. Consequently, we prove that the descendant potential of the \(\Theta\)-classes is a tau function of the \(r\)-KdV hierarchy, generalising 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, given by 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 \(\Theta\)-classes (corresponding to \(s = r - 1\)), and the Norbury \(\Theta\)-classes (the case \(r = 2, s = 1\)).