Relations on M̄g,n and the negative r-spin Witten conjecture

We construct and study various properties of a negative spin version of the Witten r r -spin class. By taking the top Chern class of a certain vector bundle on the moduli space of spin curves that parametrises r r -th roots of the anticanonical bundle, we construct a non-semisimple cohomological field theory (CohFT) that we call the Theta class Θ ʳ Θ 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 M̄g,n M ‾ g , n . Furthermore, we prove that the descendant potential of the Theta class is the unique solution to a set of W W -algebra constraints, which implies a recursive formula for all the descendant integrals. Using this result for $r = 2 $ r = 2 , we prove Norbury’s conjecture which states that the descendant potential of Θ ² Θ 2 coincides with the Brézin–Gross–Witten tau function of the KdV hierarchy. Furthermore, we conjecture that the descendant potential of Θ ʳ Θ r is the r r -BGW tau function of the r r -KdV hierarchy and prove the conjecture for $r = 3 $ r = 3 .