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

2022/05/31 by Chidambaram, Nitin Kumar, Garcia-Failde, Elba, Giacchetto, Alessandro
#14H10 #14H70 (Primary) 37K20 #81R12 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2205.15621

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 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 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 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 Θ² coincides with the Brézin--Gross--Witten tau function of the KdV hierarchy. Furthermore, we conjecture that the descendant potential of Θ^r is the r -BGW tau function of the r -KdV hierarchy and prove the conjecture for r = 3 .

Related

• Relations on Mbar_g,n via 3-spin structures
• Tautological relations via r-spin structures
• Theta classes: generalized topological recursion, integrability and W-constraints
• Tensor products of Frobenius manifolds and moduli spaces of higher spin curves
• A generalization of the Witten conjecture through spectral curve