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 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 prove the conjecture for $r=3$.

48 pages, comments welcome, v2: fixed typos, clarified the $n=0$ case; v3: version accepted in Invent. Math.;

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