Relations on $\overline{\mathcal{M}}_{g,n}$ and the negative $r$-spin Witten conjecture

We construct and study various properties of a negative spin version of the Witten $r$-spin class. By taking the top Chern class of a certain vector bundle on the moduli space of twisted spin curves that parametrises $r$-th roots of the anticanonical bundle, we construct a non-semisimple cohomological field theory (CohFT) that we call the Theta class $\Theta^r$. This CohFT does not have a flat unit and its associated Dubrovin--Frobenius manifold is nowhere semisimple. Despite this, we construct a semisimple deformation of the Theta class, and using the Teleman reconstruction theorem, we obtain tautological relations on $\overline{\mathcal{M}}_{g,n}$. We further consider the descendant potential of the Theta class and prove that it is the unique solution to a set of $\mathcal{W}$-algebra constraints, which implies a recursive formula for the descendant integrals. Using this result for $r = 2$, we prove Norbury's conjecture which states that the descendant potential of $\Theta^2$ coincides with the Br\'ezin--Gross--Witten tau function of the KdV hierarchy. Furthermore, we conjecture that the descendant potential of $\Theta^r$ is the $r$-BGW tau function of the $r$-KdV hierarchy and prove the conjecture for $r = 3$.