Weekly Bulletin

I will talk about a sequence of cohomology classes Theta^{r,s}_{g,n} on the moduli space of curves, which depend on two parameters (r,s) where 1 ≤ s ≤ r–1. Defined using the moduli space of r-th roots, Theta^{r,s} defines a non semi-simple CohFT of rank r–1 that can be viewed as an analogue of the Witten r-spin class (which corresponds to the case s = r+1). Theta^{r,s} satisfies various nice properties — the descendant potential is a tau function of the r-KdV hierarchy (i.e., a version of the Witten r-spin conjecture), satisfies W-algebra constraints, can be calculated via “determinantal formulas” and satisfies a generalised Eynard–Orantin style topological recursion.