论文标题
$ \叠加{\ Mathcal {m}} _ {g，n} $和负$ r $ -spin witten猜想的关系
Relations on $\overline{\mathcal{M}}_{g,n}$ and the negative $r$-spin Witten conjecture
论文作者
论文摘要
我们构建和研究Witten $ r $ spin类负面旋转版的各种属性。通过将特定向量捆绑包的顶级班级上的旋转曲线的模量空间划分为$ r $ $ th的根源，我们构建了一个非偏见的共同学场理论（cohft），我们称之为theta class $θ^r $。该cohft没有平坦的单位，其相关的dubrovin-弗罗贝尼乌斯歧管无处不在。尽管如此，我们还是构建了Theta类的半神经变形，并使用Teleman重建定理，我们在$ \ overline {\ Mathcal {m}}} _ {g，n} $上获得了重言式关系。我们进一步考虑了Theta类的后代潜力，并证明它是一组$ \ Mathcal {W} $ -Algebra约束的独特解决方案，这意味着后代积分的递归公式。使用此结果$ r = 2 $，我们证明了诺伯里的猜想，该猜想指出，$θ^2 $的后代潜力与KDV层次结构的Brézin-gross- gross-gross-witten tau功能相吻合。此外，我们猜想$θ^r $的后代潜力是$ r $ -bgw tau函数的$ r $ -kdv层次结构的功能，并证明了$ r = 3 $的猜想。
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 $ \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 $ Θ^2 $ 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 $.
