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

Abstract

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 $.