Probabilistic conformal blocks for Liouville CFT on the torus
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arXiv
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| Format: | Preprint |
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2020
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| author | Ghosal, Promit Remy, Guillaume Sun, Xin Sun, Yi |
| author_facet | Ghosal, Promit Remy, Guillaume Sun, Xin Sun, Yi |
| contents | Virasoro conformal blocks are a family of important functions defined as power series via the Virasoro algebra. They are a fundamental input to the conformal bootstrap program for 2D conformal field theory (CFT) and are closely related to four dimensional supersymmetric gauge theory through the Alday-Gaiotto-Tachikawa correspondence. The present work provides a probabilistic construction of the 1-point toric Virasoro conformal block for central change greater than 25. More precisely, we construct an analytic function using a probabilistic tool called Gaussian multiplicative chaos (GMC) and prove that its power series expansion coincides with the 1-point toric Virasoro conformal block. The range $(25,\infty)$ of central charges corresponds to Liouville CFT, an important CFT originating from 2D quantum gravity and bosonic string theory. Our work reveals a new integrable structure underlying GMC and opens the door to the study of non-perturbative properties of Virasoro conformal blocks such as their analytic continuation and modular symmetry. Our proof combines an analysis of GMC with tools from CFT such as Belavin-Polyakov-Zamolodchikov differential equations, operator product expansions, and Dotsenko-Fateev type integrals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2003_03802 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Probabilistic conformal blocks for Liouville CFT on the torus Ghosal, Promit Remy, Guillaume Sun, Xin Sun, Yi Probability Mathematical Physics 81T40, 60D99 Virasoro conformal blocks are a family of important functions defined as power series via the Virasoro algebra. They are a fundamental input to the conformal bootstrap program for 2D conformal field theory (CFT) and are closely related to four dimensional supersymmetric gauge theory through the Alday-Gaiotto-Tachikawa correspondence. The present work provides a probabilistic construction of the 1-point toric Virasoro conformal block for central change greater than 25. More precisely, we construct an analytic function using a probabilistic tool called Gaussian multiplicative chaos (GMC) and prove that its power series expansion coincides with the 1-point toric Virasoro conformal block. The range $(25,\infty)$ of central charges corresponds to Liouville CFT, an important CFT originating from 2D quantum gravity and bosonic string theory. Our work reveals a new integrable structure underlying GMC and opens the door to the study of non-perturbative properties of Virasoro conformal blocks such as their analytic continuation and modular symmetry. Our proof combines an analysis of GMC with tools from CFT such as Belavin-Polyakov-Zamolodchikov differential equations, operator product expansions, and Dotsenko-Fateev type integrals. |
| title | Probabilistic conformal blocks for Liouville CFT on the torus |
| topic | Probability Mathematical Physics 81T40, 60D99 |
| url | https://arxiv.org/abs/2003.03802 |