Relations on Mg,n-spin Witten conjecture

• 成果类型: Article
• 署名作者: Chidambaram, Nitin Kumar; Garcia-Failde, Elba; Giacchetto, Alessandro
• 署名单位: Max Planck Society; University of Edinburgh; Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite; Universitat Politecnica de Catalunya; Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); CNRS - Institute of Physics (INP); CEA; Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01351-y
• 发表日期: 2025
• 页码: 929-997
• 关键词: topological recursion; moduli space; tautological relations; enumerative geometry; twisted curves; external-field; INVARIANTS; COHOMOLOGY; unitary

摘要

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 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 Mg,n. Furthermore, we prove that the descendant potential of the Theta class is the unique solution to a set of W-algebra constraints, which implies a recursive formula for all 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 Brezin-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.