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 stable twisted spin curves, we construct a non-semisimple cohomological field theory 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 the moduli space of stable curves. We further consider the descendant potential of \(\Theta^r\) 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 the Theta class coincides with the Brézin–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 we prove the conjecture for \(r = 3\).