Weil-Petersson体积的边界加法恒等式及其超对称推广
A boundary-addition identity for Weil-Petersson volumes and their supersymmetric generalizations
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中文总结 AI 辅助
本文推导出Weil-Petersson体积及其超对称推广的边界加法恒等式，将其与KdV层级关联，统一并推广了已知递推关系，并揭示了N=2和小N=4情形的相交理论描述。
中文摘要 AI 辅助
我们考虑$V_{g,n}(\{b_i\})$，即具有$n$个长度为$b_i$（$i=1,...,n$）的测地边界、亏格为$g$的黎曼曲面模空间的Weil-Petersson体积，以及它们的各种$N\geq 1$超对称推广，采用一个框架，在此框架中它们是受可积KdV层级自然支配的更大一族量的特例。我们推导出一个精确的边界加法恒等式，该恒等式直接将任意$b$的$V_{g,n+1}(\{b_1,b_2,...,b_n,b\})$与$V_{g,n}(\{b_1,b_2,...,b_n\})$所基于的相应$n$边界KdV数据联系起来。对于普通Weil-Petersson体积，在特殊可移除锥值$b=2\pi i$处展开重现了Do和Norbury的三个已知结果，将它们组织为完整层级的第一层。更高层自然由额外的KdV数据构建，我们在相交理论中将其几何解释为一系列$\kappa$装饰体积。文献中的若干其他递推关系被该恒等式阐明，并推导出各种新关系。$N=1,2$和（小）$N=4$超对称情形也被该恒等式覆盖，我们展示了每种情形中出现的一些显著特殊特征和结果。在众多应用中，我们利用该恒等式揭示了$N=2$和小$N=4$ Weil-Petersson体积的相交理论描述。
英文摘要
We consider $V_{g,n}(\{b_i\})$, the Weil-Petersson volumes of the moduli space of Riemann surfaces of genus $g$ with $n$ geodesic boundaries of lengths $b_i$, $(i=1,...,n)$, and various ${ N}{\geq}1$ supersymmetric generalizations of them, using a framework within which they are specializations of a larger family of quantities naturally governed by the integrable KdV hierarchy. An exact boundary-addition identity is derived that directly relates $V_{g,n+1}(\{b_1,b_2,...,b_n,b\})$, for any $b$, to the corresponding $n$-boundary KdV data underlying $V_{g,n}(\{b_1,b_2,...,b_n\})$. For ordinary Weil-Petersson volumes, expansion about the special removable-cone value $b=2πi$ reproduces three known results of Do and Norbury, organizing them as the first levels of a complete hierarchy. The higher levels are naturally built from additional KdV data, which we interpret geometrically in intersection theory as a tower of $κ$-decorated volumes. Several other recursive relations in the literature are illuminated by the identity, and various new ones are derived. The cases with ${ N}=1,2$ and (small) ${N}=4$ supersymmetry are also covered by the identity, and we exhibit some of the striking special features and results arising in each case. Among many applications presented, we use the identity to uncover the intersection theory descriptions of ${N}=2$ and small ${ N}=4$ Weil-Petersson volumes.
发表机构
- University of California, Santa Barbara（加州大学圣塔芭芭拉分校）
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