{"title":"非半简 TQFT 的 QFT","authors":"Creutzig,Thomas, Dimofte,Tudor, Garner,Niklas, Geer,Nathan","doi":"10.4310/atmp.2024.v28.n1.a3","DOIUrl":null,"url":null,"abstract":"We construct a family of 3d quantum field theories ${\\mathcal T}_{n,k}^A$ that conjecturally provide a physical realization --- and derived generalization --- of non-semisimple mathematical TQFT's based on the modules for the quantum group $U_q(\\mathfrak{sl}_n)$ at an even root of unity $q=\\text{exp}(i\\pi/k)$. The theories ${\\mathcal T}_{n,k}^A$ are defined as topological twists of certain 3d ${\\mathcal N}=4$ Chern-Simons-matter theories, which also admit string/M-theory realizations. They may be thought of as $SU(n)_{k-n}$ Chern-Simons theories, coupled to a twisted ${\\mathcal N}=4$ matter sector (the source of non-semisimplicity). We show that ${\\mathcal T}_{n,k}^A$ admits holomorphic boundary conditions supporting two different logarithmic vertex operator algebras, one of which is an $\\mathfrak{sl}_n$-type Feigin-Tipunin algebra; and we conjecture that these two vertex operator algebras are related by a novel logarithmic level-rank duality. (We perform detailed computations to support the conjecture.) We thus relate the category of line operators in ${\\mathcal T}_{n,k}^A$ to the derived category of modules for a boundary Feigin-Tipunin algebra, and --- using a logarithmic Kazhdan-Lusztig-like correspondence that has been established for $n=2$ and expected for general $n$ --- to the derived category of $U_q(\\mathfrak{sl}_n)$ modules. We analyze many other key features of ${\\mathcal T}_{n,k}^A$ and match them from quantum-group and VOA perspectives, including deformations by flat $PGL(n,\\mathbb C)$ connections, one-form symmetries, and indices of (derived) genus-$g$ state spaces.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A QFT for non-semisimple TQFT\",\"authors\":\"Creutzig,Thomas, Dimofte,Tudor, Garner,Niklas, Geer,Nathan\",\"doi\":\"10.4310/atmp.2024.v28.n1.a3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We construct a family of 3d quantum field theories ${\\\\mathcal T}_{n,k}^A$ that conjecturally provide a physical realization --- and derived generalization --- of non-semisimple mathematical TQFT's based on the modules for the quantum group $U_q(\\\\mathfrak{sl}_n)$ at an even root of unity $q=\\\\text{exp}(i\\\\pi/k)$. The theories ${\\\\mathcal T}_{n,k}^A$ are defined as topological twists of certain 3d ${\\\\mathcal N}=4$ Chern-Simons-matter theories, which also admit string/M-theory realizations. They may be thought of as $SU(n)_{k-n}$ Chern-Simons theories, coupled to a twisted ${\\\\mathcal N}=4$ matter sector (the source of non-semisimplicity). We show that ${\\\\mathcal T}_{n,k}^A$ admits holomorphic boundary conditions supporting two different logarithmic vertex operator algebras, one of which is an $\\\\mathfrak{sl}_n$-type Feigin-Tipunin algebra; and we conjecture that these two vertex operator algebras are related by a novel logarithmic level-rank duality. (We perform detailed computations to support the conjecture.) We thus relate the category of line operators in ${\\\\mathcal T}_{n,k}^A$ to the derived category of modules for a boundary Feigin-Tipunin algebra, and --- using a logarithmic Kazhdan-Lusztig-like correspondence that has been established for $n=2$ and expected for general $n$ --- to the derived category of $U_q(\\\\mathfrak{sl}_n)$ modules. We analyze many other key features of ${\\\\mathcal T}_{n,k}^A$ and match them from quantum-group and VOA perspectives, including deformations by flat $PGL(n,\\\\mathbb C)$ connections, one-form symmetries, and indices of (derived) genus-$g$ state spaces.\",\"PeriodicalId\":50848,\"journal\":{\"name\":\"Advances in Theoretical and Mathematical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-07-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Theoretical and Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.4310/atmp.2024.v28.n1.a3\",\"RegionNum\":4,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Theoretical and Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.4310/atmp.2024.v28.n1.a3","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
We construct a family of 3d quantum field theories ${\mathcal T}_{n,k}^A$ that conjecturally provide a physical realization --- and derived generalization --- of non-semisimple mathematical TQFT's based on the modules for the quantum group $U_q(\mathfrak{sl}_n)$ at an even root of unity $q=\text{exp}(i\pi/k)$. The theories ${\mathcal T}_{n,k}^A$ are defined as topological twists of certain 3d ${\mathcal N}=4$ Chern-Simons-matter theories, which also admit string/M-theory realizations. They may be thought of as $SU(n)_{k-n}$ Chern-Simons theories, coupled to a twisted ${\mathcal N}=4$ matter sector (the source of non-semisimplicity). We show that ${\mathcal T}_{n,k}^A$ admits holomorphic boundary conditions supporting two different logarithmic vertex operator algebras, one of which is an $\mathfrak{sl}_n$-type Feigin-Tipunin algebra; and we conjecture that these two vertex operator algebras are related by a novel logarithmic level-rank duality. (We perform detailed computations to support the conjecture.) We thus relate the category of line operators in ${\mathcal T}_{n,k}^A$ to the derived category of modules for a boundary Feigin-Tipunin algebra, and --- using a logarithmic Kazhdan-Lusztig-like correspondence that has been established for $n=2$ and expected for general $n$ --- to the derived category of $U_q(\mathfrak{sl}_n)$ modules. We analyze many other key features of ${\mathcal T}_{n,k}^A$ and match them from quantum-group and VOA perspectives, including deformations by flat $PGL(n,\mathbb C)$ connections, one-form symmetries, and indices of (derived) genus-$g$ state spaces.
期刊介绍:
Advances in Theoretical and Mathematical Physics is a bimonthly publication of the International Press, publishing papers on all areas in which theoretical physics and mathematics interact with each other.