The Gromov–Witten Potential of A Point, Hurwitz Numbers, and Hodge Integrals
The Gromov–Witten Potential of A Point, Hurwitz Numbers, and Hodge Integrals
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A 点的 Gromov-Witten 势、Hurwitz 数和 Hodge 积分
DOI：
10.1112/plms/83.3.563
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发表时间：
1999
影响因子：
1.8
通讯作者：
R. Vakil
中科院分区：
文献类型：
--
作者：
I. Goulden;D. Jackson;R. Vakil
一个多世纪以来，人们对赫维茨数进行了广泛的研究，它计算的是投影线的某些覆盖(或者，等价地，将排列分解为置换)。点的Gromov-Witten势F是曲线的模空间上的下降积分的生成级数，是Gromov-Witten理论的中心研究对象。我们定义了一个略微丰富的Gromov-Witten势G(包括涉及一个‘λ类’的积分)，并证明了在变量的非平凡变化后，G=H在正亏格中，其中H是Hurwitz数的生成级数。我们证明了Goulden和Jackson关于高亏格Hurwitz数的一个猜想，它类似于Itzykson和Zuber的亏格扩展ansatz。作为结果，我们得到了关于F的新的组合约束，并且更直接地证明了Itzykson和Zuber的Ansatz。
Hurwitz numbers, which count certain covers of the projective line (or, equivalently, factorizations of permutations into transpositions), have been extensively studied for over a century. The Gromov‐Witten potential F of a point, the generating series for descendent integrals on the moduli space of curves, is a central object of study in Gromov‐Witten theory. We define a slightly enriched Gromov‐Witten potential G (including integrals involving one ‘λ‐class’), and show that, after a non‐trivial change of variables, G = H in positive genus, where H is a generating series for Hurwitz numbers. We prove a conjecture of Goulden and Jackson on higher genus Hurwitz numbers, which turns out to be an analogue of a genus expansion ansatz of Itzykson and Zuber. As consequences, we have new combinatorial constraints on F, and a much more direct proof of the ansatz of Itzykson and Zuber.