{"title":"三维引力中的壳外分部函数","authors":"Lorenz Eberhardt","doi":"10.1007/s00220-024-04963-2","DOIUrl":null,"url":null,"abstract":"<p>We explore three-dimensional gravity with negative cosmological constant via canonical quantization. We focus on chiral gravity which is related to a single copy of <span>\\(\\text {PSL}(2,\\mathbb {R})\\)</span> Chern-Simons theory and is simpler to treat in canonical quantization. Its phase space for an initial value surface <span>\\(\\Sigma \\)</span> is given by the appropriate moduli space of Riemann surfaces. We use geometric quantization to compute partition functions of chiral gravity on three-manifolds of the form <span>\\(\\Sigma \\times {{\\,\\textrm{S}\\,}}^1\\)</span>, where <span>\\(\\Sigma \\)</span> can have asymptotic boundaries. Most of these topologies do not admit a classical solution and are thus not amenable to a direct semiclassical path integral computation. We use an index theorem that expresses the partition function as an integral of characteristic classes over phase space. In the presence of <i>n</i> asymptotic boundaries, we use techniques from equivariant cohomology to localize the integral to a finite-dimensional integral over <span>\\(\\overline{\\mathcal {M}}_{g,n}\\)</span>, which we evaluate in low genus cases. Higher genus partition functions quickly become complicated since they depend in an oscillatory way on Newton’s constant. There is a precise sense in which one can isolate the non-oscillatory part which we call the fake partition function. We establish that there is a topological recursion that computes the fake partition functions for arbitrary Riemann surfaces <span>\\(\\Sigma \\)</span>. There is a scaling limit in which the model reduces to JT gravity and our methods give a novel way to compute JT partition functions via equivariant localization.</p>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Off-shell Partition Functions in 3d Gravity\",\"authors\":\"Lorenz Eberhardt\",\"doi\":\"10.1007/s00220-024-04963-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We explore three-dimensional gravity with negative cosmological constant via canonical quantization. We focus on chiral gravity which is related to a single copy of <span>\\\\(\\\\text {PSL}(2,\\\\mathbb {R})\\\\)</span> Chern-Simons theory and is simpler to treat in canonical quantization. Its phase space for an initial value surface <span>\\\\(\\\\Sigma \\\\)</span> is given by the appropriate moduli space of Riemann surfaces. We use geometric quantization to compute partition functions of chiral gravity on three-manifolds of the form <span>\\\\(\\\\Sigma \\\\times {{\\\\,\\\\textrm{S}\\\\,}}^1\\\\)</span>, where <span>\\\\(\\\\Sigma \\\\)</span> can have asymptotic boundaries. Most of these topologies do not admit a classical solution and are thus not amenable to a direct semiclassical path integral computation. We use an index theorem that expresses the partition function as an integral of characteristic classes over phase space. In the presence of <i>n</i> asymptotic boundaries, we use techniques from equivariant cohomology to localize the integral to a finite-dimensional integral over <span>\\\\(\\\\overline{\\\\mathcal {M}}_{g,n}\\\\)</span>, which we evaluate in low genus cases. Higher genus partition functions quickly become complicated since they depend in an oscillatory way on Newton’s constant. There is a precise sense in which one can isolate the non-oscillatory part which we call the fake partition function. We establish that there is a topological recursion that computes the fake partition functions for arbitrary Riemann surfaces <span>\\\\(\\\\Sigma \\\\)</span>. There is a scaling limit in which the model reduces to JT gravity and our methods give a novel way to compute JT partition functions via equivariant localization.</p>\",\"PeriodicalId\":522,\"journal\":{\"name\":\"Communications in Mathematical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.2000,\"publicationDate\":\"2024-03-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1007/s00220-024-04963-2\",\"RegionNum\":1,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s00220-024-04963-2","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
摘要
我们通过规范量子化来探索负宇宙学常数的三维引力。我们把重点放在手性引力上,它与(text {PSL}(2,\mathbb {R})\) 的单份Chern-Simons 理论有关,在规范量子化中处理起来比较简单。它的初值曲面的相空间由黎曼曲面的适当模空间给出。我们使用几何量子化来计算手性引力在形式为\(\Sigma \times {{\,\textrm{S}\,}^1\)的三芒星上的分割函数,其中\(\Sigma \)可以有渐近边界。这些拓扑结构中的大多数都没有经典解,因此无法直接进行半经典路径积分计算。我们使用一个索引定理,将分割函数表示为相空间上特征类的积分。在存在 n 个渐近边界的情况下,我们使用等变同调技术将积分局部化为对 \(\overline{\mathcal {M}}_{g,n}\) 的有限维积分,并在低属的情况下对其进行评估。高属划分函数很快变得复杂起来,因为它们以振荡的方式依赖于牛顿常数。有一种精确的方法可以分离出非振荡部分,我们称之为假分割函数。我们建立了一个拓扑递归,它可以计算任意黎曼曲面 \(\Sigma \)的假分割函数。我们的方法提供了一种通过等变局部化计算 JT 分区函数的新方法。
We explore three-dimensional gravity with negative cosmological constant via canonical quantization. We focus on chiral gravity which is related to a single copy of \(\text {PSL}(2,\mathbb {R})\) Chern-Simons theory and is simpler to treat in canonical quantization. Its phase space for an initial value surface \(\Sigma \) is given by the appropriate moduli space of Riemann surfaces. We use geometric quantization to compute partition functions of chiral gravity on three-manifolds of the form \(\Sigma \times {{\,\textrm{S}\,}}^1\), where \(\Sigma \) can have asymptotic boundaries. Most of these topologies do not admit a classical solution and are thus not amenable to a direct semiclassical path integral computation. We use an index theorem that expresses the partition function as an integral of characteristic classes over phase space. In the presence of n asymptotic boundaries, we use techniques from equivariant cohomology to localize the integral to a finite-dimensional integral over \(\overline{\mathcal {M}}_{g,n}\), which we evaluate in low genus cases. Higher genus partition functions quickly become complicated since they depend in an oscillatory way on Newton’s constant. There is a precise sense in which one can isolate the non-oscillatory part which we call the fake partition function. We establish that there is a topological recursion that computes the fake partition functions for arbitrary Riemann surfaces \(\Sigma \). There is a scaling limit in which the model reduces to JT gravity and our methods give a novel way to compute JT partition functions via equivariant localization.
期刊介绍:
The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.