Topological recursion of the Weil–Petersson volumes of hyperbolic surfaces with tight boundaries

IF 1.2 3区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Journal of Mathematical Physics Pub Date : 2024-09-11 DOI:10.1063/5.0192711
Timothy Budd, Bart Zonneveld
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Abstract

The Weil–Petersson volumes of moduli spaces of hyperbolic surfaces with geodesic boundaries are known to be given by polynomials in the boundary lengths. These polynomials satisfy Mirzakhani’s recursion formula, which fits into the general framework of topological recursion. We generalize the recursion to hyperbolic surfaces with any number of special geodesic boundaries that are required to be tight. A special boundary is tight if it has minimal length among all curves that separate it from the other special boundaries. The Weil–Petersson volume of this restricted family of hyperbolic surfaces is shown again to be polynomial in the boundary lengths. This remains true when we allow conical defects in the surface with cone angles in (0, π) in addition to geodesic boundaries. Moreover, the generating function of Weil–Petersson volumes with fixed genus and a fixed number of special boundaries is polynomial as well, and satisfies a topological recursion that generalizes Mirzakhani’s formula. This work is largely inspired by recent works by Bouttier, Guitter, and Miermont [Ann. Henri Lebesgue 5, 1035–1110 (2022)] on the enumeration of planar maps with tight boundaries. Our proof relies on the equivalence of Mirzakhani’s recursion formula to a sequence of partial differential equations (known as the Virasoro constraints) on the generating function of intersection numbers. Finally, we discuss a connection with Jackiw–Teitelboim (JT) gravity. We show that the multi-boundary correlators of JT gravity with defects are expressible in the tight Weil–Petersson volume generating functions, using a tight generalization of the JT trumpet partition function.
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具有紧密边界的双曲面的魏尔-彼得森体积的拓扑递归
众所周知,具有大地边界的双曲面的模空间的魏尔-彼得森体积是由边界长度的多项式给出的。这些多项式满足米尔扎哈尼递推公式,符合拓扑递推的一般框架。我们将递推公式推广到具有任意数量的特殊测地线边界的双曲面,这些边界必须是紧密的。如果一个特殊边界在所有将其与其他特殊边界分开的曲线中长度最小,那么这个边界就是紧密的。这个受限双曲面族的魏尔-彼得森体积再次被证明是边界长度的多项式。除了测地线边界之外,当我们允许曲面上存在锥角在 (0, π) 范围内的锥形缺陷时,情况依然如此。此外,具有固定种属和固定数量特殊边界的 Weil-Petersson 体积的生成函数也是多项式的,并且满足拓扑递归,概括了 Mirzakhani 公式。这项工作的灵感主要来自布蒂埃、吉特和米尔蒙最近关于枚举具有紧边界的平面映射的工作[Ann. Henri Lebesgue 5, 1035-1110 (2022)]。我们的证明依赖于米尔扎哈尼递推公式与交点数生成函数上的偏微分方程序列(称为维拉索罗约束)的等价性。最后,我们讨论了与 Jackiw-Teitelboim (JT) 引力的联系。我们利用 JT 小号分区函数的严密广义化,证明有缺陷的 JT 引力的多边界相关因子可以用严密的魏尔-彼得森体生成函数来表达。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of Mathematical Physics
Journal of Mathematical Physics 物理-物理:数学物理
CiteScore
2.20
自引率
15.40%
发文量
396
审稿时长
4.3 months
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