Pub Date : 2024-08-19DOI: 10.4310/atmp.2024.v28.n1.a4
Thomas Creutzig, Tudor Dimofte, Niklas Garner, Nathan Geer
$defTank{mathcal{T}^A_{n,k}}$$defUqsln{U_q(mathfrak{sl}_n)}$We construct a family of 3d quantum field theories $Tank$ that conjecturally provide a physical realization—and derived generalization—of non-semisimple mathematical TQFT’s based on the modules for the quantum group $Uqsln$ at an even root of unity $q = operatorname{exp}(i pi / k)$. The theories $Tank$ are defined as topological twists of certain 3d $mathcal{N=4}$ Chern–Simons-matter theories, which also admit string/M‑theory realizations. They may be thought of as $SU(n)_{k-n}$ Chern–Simons theories, coupled to a twisted $mathcal{N}=4$ matter sector (the source of non-semisimplicity). We show that $Tank$ admits holomorphic boundary conditions supporting two different logarithmic vertex operator algebras, one of which is an $mathfrak{sl}_n)$-type Feigin–Tipunin algebra; and we conjecture that these two vertex operator algebras are related by a novel logarithmic level-rank duality. (We perform detailed computations to support the conjecture.) We thus relate the category of line operators in $Tank$ to the derived category of modules for a boundary Feigin–Tipunin algebra, and—using a logarithmic Kazhdan–Lusztig-like correspondence that has been established for $n=2$ and expected for general $n$ — to the derived category of $Uqsln$ modules.We analyze many other key features of $Tank$ and match them from quantum-group and VOA perspectives, including deformations by flat $PGL(n,mathbb{C})$ connections, one-form symmetries, and indices of (derived) genus-$g$ state spaces.
{"title":"A QFT for non-semisimple TQFT","authors":"Thomas Creutzig, Tudor Dimofte, Niklas Garner, Nathan Geer","doi":"10.4310/atmp.2024.v28.n1.a4","DOIUrl":"https://doi.org/10.4310/atmp.2024.v28.n1.a4","url":null,"abstract":"$defTank{mathcal{T}^A_{n,k}}$$defUqsln{U_q(mathfrak{sl}_n)}$We construct a family of 3d quantum field theories $Tank$ that conjecturally provide a physical realization—and derived generalization—of non-semisimple mathematical TQFT’s based on the modules for the quantum group $Uqsln$ at an even root of unity $q = operatorname{exp}(i pi / k)$. The theories $Tank$ 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 $Tank$ 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 $Tank$ 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 $Uqsln$ modules.We analyze many other key features of $Tank$ 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":"74 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-14DOI: 10.4310/atmp.2023.v27.n8.a2
Eric Sharpe
In this paper we discuss noninvertible topological operators in the context of one-form symmetries and decomposition of twodimensional quantum field theories, focusing on two-dimensional orbifolds with and without discrete torsion. As one component of our analysis, we study the ring of dimension-zero operators in two-dimensional theories exhibiting decomposition. From a commutative algebra perspective, the rings are naturally associated to a finite number of points, one point for each universe in the decomposition. Each universe is canonically associated to a representation, which defines a projector, an idempotent in the ring of dimension-zero operators. We discuss how bulk Wilson lines act as defects bridging universes, and how Wilson lines on boundaries of two-dimensional theories decompose, and compute actions of projectors. We discuss one-form symmetries of the rings, and related properties. We also give general formulas for projection operators, which previously were computed on a case-by-case basis. Finally, we propose a characterization of noninvertible higher-form symmetries in this context in terms of representations. In that characterization, non-isomorphic universes appearing in decomposition are associated with noninvertible one-form symmetries.
{"title":"Topological operators, noninvertible symmetries and decomposition","authors":"Eric Sharpe","doi":"10.4310/atmp.2023.v27.n8.a2","DOIUrl":"https://doi.org/10.4310/atmp.2023.v27.n8.a2","url":null,"abstract":"In this paper we discuss noninvertible topological operators in the context of one-form symmetries and decomposition of twodimensional quantum field theories, focusing on two-dimensional orbifolds with and without discrete torsion. As one component of our analysis, we study the ring of dimension-zero operators in two-dimensional theories exhibiting decomposition. From a commutative algebra perspective, the rings are naturally associated to a finite number of points, one point for each universe in the decomposition. Each universe is canonically associated to a representation, which defines a projector, an idempotent in the ring of dimension-zero operators. We discuss how bulk Wilson lines act as defects bridging universes, and how Wilson lines on boundaries of two-dimensional theories decompose, and compute actions of projectors. We discuss one-form symmetries of the rings, and related properties. We also give general formulas for projection operators, which previously were computed on a case-by-case basis. Finally, we propose a characterization of noninvertible higher-form symmetries in this context in terms of representations. In that characterization, non-isomorphic universes appearing in decomposition are associated with noninvertible one-form symmetries.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"15 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-14DOI: 10.4310/atmp.2023.v27.n8.a4
Sergio Cecotti
The classical theory of Fuchsian differential equations is largely equivalent to the theory of Seiberg dualities for quiver SUSY gauge theories. In particular: all known integral representations of solutions, and their connection formulae, are immediate consequences of (analytically continued) Seiberg duality in view of the dictionary between linear ODEs and gauge theories with $4$ supersymmetries. The purpose of this divertissement is to explain “physically” this remarkable relation in the spirit of Physical Mathematics. The connection goes through a “mirror-theoretic” identification of irreducible logarithmic connections on $mathbb{P}^1$ with would-be BPS dyons of 4d $mathcal{N} = 2 : SU(2)$ SYM coupled to a certain Argyres–Douglas “matter”. When the underlying bundle is trivial, i.e. the log‑connection is a Fuchs system, the world-line theory of the dyon simplifies and the action of Seiberg duality on the Fuchsian ODEs becomes quite explicit. The duality action is best described in terms of Representation Theory of Kac–Moody Lie algebras (and their affinizations).
{"title":"Fuchsian ODEs as Seiberg dualities","authors":"Sergio Cecotti","doi":"10.4310/atmp.2023.v27.n8.a4","DOIUrl":"https://doi.org/10.4310/atmp.2023.v27.n8.a4","url":null,"abstract":"The classical theory of Fuchsian differential equations is largely equivalent to the theory of Seiberg dualities for quiver SUSY gauge theories. In particular: <i>all</i> known integral representations of solutions, and their connection formulae, are immediate consequences of (analytically continued) Seiberg duality in view of the dictionary between linear ODEs and gauge theories with $4$ supersymmetries. The purpose of this divertissement is to explain “physically” this remarkable relation in the spirit of Physical Mathematics. The connection goes through a “mirror-theoretic” identification of irreducible logarithmic connections on $mathbb{P}^1$ with would-be BPS dyons of 4d $mathcal{N} = 2 : SU(2)$ SYM coupled to a certain Argyres–Douglas “matter”. When the underlying bundle is trivial, i.e. the log‑connection is a Fuchs system, the world-line theory of the dyon simplifies and the action of Seiberg duality on the Fuchsian ODEs becomes quite explicit. The duality action is best described in terms of Representation Theory of Kac–Moody Lie algebras (and their affinizations).","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"20 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216885","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-14DOI: 10.4310/atmp.2023.v27.n8.a3
Vicent Gimeno i Garcia, Steen Markvorsen
In this paper, we establish geometric and topological upper bounds on the first energy level gap of a particle confined to move on a compact surface in $3$-space. Our main contribution is proving that the first gap in the energy spectrum of a confined particle (a physical property) is bounded above by the Willmore energy of the confining surface (a geometric property). Furthermore, we demonstrate that the only surfaces that permit a confined particle with a stationary and uniformly distributed wave function are surfaces with constant skew curvature.
{"title":"Energy spectrum of a constrained quantum particle and the Willmore energy of the constraining surface","authors":"Vicent Gimeno i Garcia, Steen Markvorsen","doi":"10.4310/atmp.2023.v27.n8.a3","DOIUrl":"https://doi.org/10.4310/atmp.2023.v27.n8.a3","url":null,"abstract":"In this paper, we establish geometric and topological upper bounds on the first energy level gap of a particle confined to move on a compact surface in $3$-space. Our main contribution is proving that the first gap in the energy spectrum of a confined particle (a physical property) is bounded above by the Willmore energy of the confining surface (a geometric property). Furthermore, we demonstrate that the only surfaces that permit a confined particle with a stationary and uniformly distributed wave function are surfaces with constant skew curvature.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"36 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-14DOI: 10.4310/atmp.2023.v27.n8.a1
Charles F. Doran, Matt Kerr, Soumya Sinha Babu
A 2015 conjecture of Codesido-Grassi-Mariño in topological string theory relates the enumerative invariants of toric CY $3$-folds to the spectra of operators attached to their mirror curves. We deduce two consequences of this conjecture for the integral regulators of $K_2$-classes on these curves, and then prove both of them; the results thus give evidence for the CGM conjecture. (While the conjecture and the deduction process both entail forms of local mirror symmetry, the consequences/theorems do not: they only involve the curves themselves.) Our first theorem relates zeroes of the higher normal function to the spectra of the operators for curves of genus one, and suggests a new link between analysis and arithmetic geometry. The second theorem provides dilogarithm formulas for limits of regulator periods at the maximal conifold point in moduli of the curves.
{"title":"$K_2$ and quantum curves","authors":"Charles F. Doran, Matt Kerr, Soumya Sinha Babu","doi":"10.4310/atmp.2023.v27.n8.a1","DOIUrl":"https://doi.org/10.4310/atmp.2023.v27.n8.a1","url":null,"abstract":"A 2015 conjecture of Codesido-Grassi-Mariño in topological string theory relates the enumerative invariants of toric CY $3$-folds to the spectra of operators attached to their mirror curves. We deduce two consequences of this conjecture for the integral regulators of $K_2$-classes on these curves, and then prove both of them; the results thus give evidence for the CGM conjecture. (While the conjecture and the deduction process both entail forms of local mirror symmetry, the consequences/theorems do not: they only involve the curves themselves.) Our first theorem relates zeroes of the higher normal function to the spectra of the operators for curves of genus one, and suggests a new link between analysis and arithmetic geometry. The second theorem provides dilogarithm formulas for limits of regulator periods at the maximal conifold point in moduli of the curves.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"8 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"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 $q=text{exp}(ipi/k)$. 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.
{"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":"https://doi.org/10.4310/atmp.2024.v28.n1.a3","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 $q=text{exp}(ipi/k)$. 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":"263 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-29DOI: 10.4310/atmp.2024.v28.n1.a1
Sopenko,Nikita
We propose a class of pure states of two-dimensional lattice systems realizing topological order associated with unitary rational vertex operator algebras. We show that the states are well-defined in the thermodynamic limit and have exponential decay of correlations. The construction provides a natural way to insert anyons and compute certain topological invariants. It also gives candidates for bosonic states in non-trivial invertible phases, including the $E_8$ phase.
{"title":"Chiral topologically ordered states on a lattice from vertex operator algebras","authors":"Sopenko,Nikita","doi":"10.4310/atmp.2024.v28.n1.a1","DOIUrl":"https://doi.org/10.4310/atmp.2024.v28.n1.a1","url":null,"abstract":"We propose a class of pure states of two-dimensional lattice systems realizing topological order associated with unitary rational vertex operator algebras. We show that the states are well-defined in the thermodynamic limit and have exponential decay of correlations. The construction provides a natural way to insert anyons and compute certain topological invariants. It also gives candidates for bosonic states in non-trivial invertible phases, including the $E_8$ phase.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"134 2 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-29DOI: 10.4310/atmp.2024.v28.n1.a2
Kimura,Yusuke
We study aspects of an equivalent relation of the charge completeness in six-dimensional (6D) $mathcal{N}=(1,0)$ supergravity theory and a standard assumption on the global structure of the gauge group involving F-theory geometry, recently proved by Morrison and Taylor. We constructed and analyzed a novel 6D supergravity theory, realized as F-theory, on an elliptically fibered Calabi--Yau 3-fold. Our construction yields a novel 6D theory with Mordell--Weil torsion $mathbb{Z}_4oplusmathbb{Z}_4$. Furthermore, we deduce the gauge group and matter fields arising in the 6D F-theory model on the constructed elliptically fibered Calabi--Yau 3-fold. We also discuss the relations of the 6D F-theory model constructed in this study to stable degeneration and the dual heterotic string.
{"title":"Jacobian Calabi--Yau 3-fold and charge completeness in six-dimensional theory","authors":"Kimura,Yusuke","doi":"10.4310/atmp.2024.v28.n1.a2","DOIUrl":"https://doi.org/10.4310/atmp.2024.v28.n1.a2","url":null,"abstract":"We study aspects of an equivalent relation of the charge completeness in six-dimensional (6D) $mathcal{N}=(1,0)$ supergravity theory and a standard assumption on the global structure of the gauge group involving F-theory geometry, recently proved by Morrison and Taylor. We constructed and analyzed a novel 6D supergravity theory, realized as F-theory, on an elliptically fibered Calabi--Yau 3-fold. Our construction yields a novel 6D theory with Mordell--Weil torsion $mathbb{Z}_4oplusmathbb{Z}_4$. Furthermore, we deduce the gauge group and matter fields arising in the 6D F-theory model on the constructed elliptically fibered Calabi--Yau 3-fold. We also discuss the relations of the 6D F-theory model constructed in this study to stable degeneration and the dual heterotic string.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"67 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141866421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-16DOI: 10.4310/atmp.2023.v27.n6.a3
Zhi-Cong Ong, Meng-Chwan Tan
We revisit Vafa–Witten theory in the more general setting whereby the underlying moduli space is not that of instantons, but of the full Vafa–Witten equations. We physically derive (i) a novel Vafa–Witten four-manifold invariant associated with this moduli space, (ii) their relation to Gromov–Witten invariants, (iii) a novel Vafa–Witten Floer homology assigned to three-manifold boundaries, (iv) a novel Vafa–Witten Atiyah–Floer correspondence, (v) a proof and generalization of a conjecture by Abouzaid–Manolescu in $href{https://doi.org/10.4171/jems/994}{[2]}$ about the hypercohomology of a perverse sheaf of vanishing cycles, (vi) a Langlands duality of these invariants, Floer homologies and hypercohomology, and (vii) a quantum geometric Langlands correspondence with purely imaginary parameter that specializes to the classical correspondence in the zero-coupling limit, where Higgs bundles feature in (ii), (iv), (vi) and (vii). We also explain how these invariants and homologies will be categorified in the process, and discuss their higher categorification. We thereby relate differential and enumerative geometry, topology and geometric representation theory in mathematics, via a maximally-supersymmetric topological quantum field theory with electric-magnetic duality in physics.
{"title":"Vafa-Witten theory: invariants, Floer homologies, Higgs bundles, a geometric Langlands correspondence, and categorification","authors":"Zhi-Cong Ong, Meng-Chwan Tan","doi":"10.4310/atmp.2023.v27.n6.a3","DOIUrl":"https://doi.org/10.4310/atmp.2023.v27.n6.a3","url":null,"abstract":"We revisit Vafa–Witten theory in the more general setting whereby the underlying moduli space is not that of instantons, but of the full Vafa–Witten equations. We physically derive (i) a novel Vafa–Witten four-manifold invariant associated with this moduli space, (ii) their relation to Gromov–Witten invariants, (iii) a novel Vafa–Witten Floer homology assigned to three-manifold boundaries, (iv) a novel Vafa–Witten Atiyah–Floer correspondence, (v) a proof and generalization of a conjecture by Abouzaid–Manolescu in $href{https://doi.org/10.4171/jems/994}{[2]}$ about the hypercohomology of a perverse sheaf of vanishing cycles, (vi) a Langlands duality of these invariants, Floer homologies and hypercohomology, and (vii) a quantum geometric Langlands correspondence with purely imaginary parameter that specializes to the classical correspondence in the zero-coupling limit, where Higgs bundles feature in (ii), (iv), (vi) and (vii). We also explain how these invariants and homologies will be categorified in the process, and discuss their higher categorification. We thereby relate differential and enumerative geometry, topology and geometric representation theory in mathematics, via a maximally-supersymmetric topological quantum field theory with electric-magnetic duality in physics.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"52 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-16DOI: 10.4310/atmp.2023.v27.n6.a2
Richard J. Szabo, Michelangelo Tirelli
We study rank $r$ cohomological Donaldson–Thomas theory on a toric Calabi–Yau orbifold of $mathbb{C}^4$ by a finite abelian subgroup $Gamma$ of $mathsf{SU}(4)$, from the perspective of instanton counting in cohomological gauge theory on a noncommutative crepant resolution of the quotient singularity. We describe the moduli space of noncommutative instantons on $mathbb{C}^4 / Gamma$ and its generalized ADHM parametrization. Using toric localization, we compute the orbifold instanton partition function as a combinatorial series over $r$-vectors of $Gamma$-coloured solid partitions. When the $Gamma$-action fixes an affine line in $mathbb{C}^4$, we exhibit the dimensional reduction to rank $r$ Donaldson–Thomas theory on the toric Kähler three-orbifold $mathbb{C}^3 / Gamma$. Based on this reduction and explicit calculations, we conjecture closed infinite product formulas, in terms of generalized MacMahon functions, for the instanton partition functions on the orbifolds $mathbb{C}^2 / mathbb{Z}_n times mathbb{C}^2$ and $mathbb{C}^3 / (mathbb{Z}_2 times mathbb{Z}_2) times mathbb{C}$, finding perfect agreement with new mathematical results of Cao, Kool and Monavari.
我们从共计量理论中的瞬子计数角度出发，研究了$mathsf{SU}(4)$的有限无性子群$Gamma$的$mathbb{C}^4$的环状卡拉比-约轨道上的秩$r$共计量唐纳森-托马斯理论。我们描述了 $mathbb{C}^4 / Gamma$ 上的非交换瞬子模态空间及其广义 ADHM 参数化。利用环定位，我们计算了作为$r$-向量的$Gamma$彩色实体分区的组合数列的轨道瞬子分区函数。当$Gamma$作用固定了$mathbb{C}^4$中的仿射线时，我们展示了环状凯勒三轨道$mathbb{C}^3 / Gamma$上秩为$r$的唐纳森-托马斯（Donaldson-Thomas）理论的维度还原。基于这种还原和显式计算，我们用广义麦克马洪函数来猜想封闭的无限乘积公式、和 $mathbb{C}^3 / (mathbb{Z}_2 times mathbb{Z}_2) times mathbb{C}$上的瞬子分区函数，发现与曹（Cao）、库尔（Kool）和莫纳瓦里（Monavari）的新数学结果完全一致。
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