论文标题
$ \ mathbb {z} _2 $ hurwitz-hodge积分的多项式
Polynomiality of $\mathbb{Z}_2$ Hurwitz-Hodge Integrals
论文作者
论文摘要
使用Atiyah-Bott在稳定地图上的本地化与堆栈商的空间进行本地化，$ [\ MATHBB {p}^1/\ MATHBB {Z} _2] $，我们发现确定所有hodge积分，这些递归与降低插入的所有hodge积分在一个标记的点上，在hyperelliptic locus $ $ \ mathline { 2} \ subseteq \ overline {\ mathcal {m}} _ {g，2g + 2} $。我们递归所需的初始条件是一个标记点的重力后裔，已知为$ \ frac {1} {2} $。我们发现了一个有关这些交叉数字的新结构：对于$λ$ - 类的固定单元，由此产生的高ellip虫杂货积分家族在$ g $中是多项式。我们制定了有关这些多项式系数的对数洞穴的猜想。最后，我们将递归转变为非线性偏微分方程的非线性系统，以生成过椭圆形霍奇积分的生成函数。
Using Atiyah-Bott localization on the space of stable maps to the stack quotient $[\mathbb{P}^1/\mathbb{Z}_2]$, we find recursions that determine all Hodge integrals with descendent insertions at one marked point on the hyperelliptic locus $\overline{\mathcal{H}}_{g, 2g + 2} \subseteq \overline{\mathcal{M}}_{g, 2g + 2}$. The initial conditions required for our recursions are gravitational descendents at one marked point, which are known to be $\frac{1}{2}$. We discover a new structure concerning these intersection numbers: for a fixed monomial of $λ$-classes, the resulting family of hyperelliptic Hodge integrals is polynomial in $g$. We formulate a conjecture concerning the log-concavity of the coefficients of these polynomials. Lastly, we turn our recursions into a non-linear system of partial differential equations for the generating functions of hyperelliptic Hodge integrals.
