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

We study the intersection theory of the Θ^{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 Θ^{𝑟, 𝑠} -classes. As a consequence, we deduce that the descendant potential of the Θ^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⁡(𝔤𝔩𝑟)-algebra at self-dual level. This work extends previously known results on the Witten 𝑟-spin class, the 𝑟-spin Θ-classes (the case 𝑠 =𝑟 −1), and the Norbury Θ-classes (the special case 𝑟 =2, 𝑠 =1).