## Lie Group/Quantum Mathematics SeminarOrganizers Lisa Carbone, Yi-Zhi
Huang, Jim
Lepowsky and Siddhartha Sahi.
Starting from Spring, 2008, the Lie Group Seminar and Quantum Mathematics Seminar have merged together to a single seminar called the Lie Group/Quantum Mathematics Seminar. The information on seminar talks can also be found in the Seminars & Colloquia Calendar page in the department. For the Lie Group/Quantum Mathematics seminar in previous semesters, see this page. For talks in the Quantum Mathematics Seminar from Spring, 1998 to Fall, 2007, see this page. For a few years before 2008, the Quantum Mathematics Seminar shared the time and place with the Algebra Seminar. For talks in both the Algebra and Quantum Mathematics Seminars in these few semesters, see the page for the Previous Rutgers Algebra Seminars. For all the seminars and colloquia in the department, see the Seminars & Colloquia Calendar page.
## Fall, 2021Seminar talks in this semester will still be online using zoom until further notice. Zoom link: https://rutgers.zoom.us/j/93921465287 Meeting ID: 939 2146 5287 Passcode: 196884, the dimension of the weight 2 homogeneous subspace of the moonshine module Most of the talks will be recorded and will be placed in the YouTube channel Rutgers Lie Groups Quantum Math Seminar **Speaker**Lisa Carbone, Rutgers University**Title**Imaginary reflections and discrete symmetries in the Heterotic Monster**Time/place**9/17/2021, Friday, 11:00 am (Eastern Time), Zoom link above**Abstract**We give an interpretation of certain symmetries of the root system of the monster Lie algebra m, and more generally the infinite family of monstrous Lie algebras m_g, one for each g in the Monster finite simple group M, in terms of discrete symmetries of the heterotic string compactified to 0 + 1 dimensions. We show that a certain discrete symmetry of the monster Lie algebra can be extended to a discrete symmetry of a supersymmetric index Z which counts (with signs) the number of BPS-states. The imaginary reflection permutes two sets of BPS states labeled by imaginary roots up to a minus sign. This is joint work with Natalie Paquette.
**Speaker**Yi-Zhi Huang, Rutgers University**Title**Representation theory of vertex operator algebras and conformal field theory**Time/place**10/1/2021, Friday, 11:00 am (Eastern Time), Zoom link above**Abstract**A program to construct weakly and full conformal field theories in the sense of Segal using the representation theory of vertex operator algebras was started more than 30 years ago. After more than 30 years, major progresses have been made and many problems in this program have been solved. In this talk, I will review the history of and the major progresses in this program and will also discuss the current status of this program.**Slides**pdf file.**YouTube video**Representation theory of vertex operator algebras and conformal field theory
**Speaker**Cristian Lenart, SUNY Albany**Title**A combinatorial Chevalley formula for semi-infinite flag manifolds and related topics**Time/place**10/8/2021, Friday, 11:00 am (Eastern Time), Zoom link above**Abstract**I present a combinatorial Chevalley formula for an arbitrary weight in the equivariant K-theory of semi-infinite flag manifolds, which are certain affine versions of finite flag manifolds G/B. The formula is expressed in terms of the so-called quantum alcove model. One application is a Chevalley formula in the equivariant quantum K-theory of G/B. Another application is that the so-called quantum Grothendieck polynomials represent Schubert classes in the (non-equivariant) quantum K-theory of the type A flag manifold G/B. Both applications solve longstanding conjectures. Other results include the Chevalley formula for partial flag manifolds G/P and related combinatorics of the quantum alcove model. This is joint work with Takafumi Kouno, Satoshi Naito, and Daisuke Sagaki. The talk will be largely self-contained.**YouTube video**Braided rigidity for path algebras
**Speaker**Robert McRae, Tsinghua University**Title**On rigidity, non-degeneracy, and semisimplicity for vertex tensor categories coming from C_2-cofinite vertex operator algebras**Time/place**10/15/2021, Friday, 11:00 am (Eastern Time), Zoom link above**Abstract**One of the most important results in vertex operator algebras is Huang's result that if the module category of a positive-energy, self-contragredient, C_2-cofinite vertex operator algebra V is semisimple, then it is a modular tensor category. Huang also showed that if the V-module category is not necessarily semisimple, then it is at least a braided tensor category. In this talk, I will discuss my recent results on C_2-cofinite vertex operator algebras in the not-a priori-semisimple setting: First, if the modular S-transformation of the character of V is a linear combination of characters of V-modules, then the tensor category of V-modules is rigid, that is, contragredients of V-modules are duals in the tensor-categorical sense. Second, if the tensor category of V-modules is assumed to be rigid, then its braiding is non-degenerate, that is, the V-module category is a not-necessarily-semisimple analogue of a modular tensor category. Finally, if Zhu's associative algebra A(V) is semisimple, then the V-module category is in fact semisimple, so that Huang's theorem applies. Thanks to a recent Zhu algebra theorem of Arakawa and van Ekeren, this last result proves the conjecture of Kac-Wakimoto and Arakawa that all "exceptional" simple W-algebras have semisimple representation theory.**Slides**pdf file.**Archive paper**arXiv:2108.01898
**Speaker**Liang Kong, Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology**Title**A mathematical theory of gapless boundaries of 2+1D topological orders**Time/place**10/29/2021, Friday, 11:00 am (Eastern Time), Zoom link above**Abstract**It was well-known that the low energy effective field theory of a 2+1D topological order is a 2+1D topological field theory, which can be mathematically described by a modular tensor category C. It was also known as a forklore that a chiral gapless boundary of C is a "chiral conformal field theory" (CFT). In this talk, I will show that this boundary chiral CFT can be mathematically described by a pair (V,X), where V is a rational VOA and X is a fusion category enriched in the category Mod_V of V-modules. Moreover, the monoidal center of X is precisely C. If time permits, I will also discuss some applications of this result. This is a joint work with Hao Zheng.**Slides**pdf file.**Archive paper**arXiv:1705.01087, arXiv:1905.04924, arXiv:1912.01760**YouTube video**A mathematical theory of gapless boundaries of 2+1D topological orders
**Speaker**Maryam Khaqan, Stockholm University**Title**Moonshine and Elliptic Curves**Time/place**11/12/2021, Friday, 11:00 am (Eastern Time), Zoom link above**Abstract**The study of moonshine originated from a series of numerical coincidences connecting finite groups to modular forms. It has since evolved into a rich theory that sheds light on the underlying algebraic structures that these coincidences reflect.We prove the existence of one such structure, a module for Thompson's sporadic simple group, whose graded traces are specific half-integral weight weakly holomorphic modular forms. We then use this module to study the ranks of certain families of elliptic curves. In particular, this serves as an example of moonshine being used to answer questions in number theory.
## Some additional talks by members of this seminar**Speaker**Yi-Zhi Huang, Rutgers University**Title**Representation theory of vertex operator algebras and conformal field theory**Event**The Mathematics of Conformal Field Theory II, Mathematical Sciences Institute, Australian National University**Time**7/6/2021, Tuesday, 9:15 am (Australian Eastern Standard Time) or 7/5/2021, Monday, 7:15 pm (Eastern time)**Location**The conference is entirely online. Registered participants can access the Zoom links through the SharePoint page. Registration is free for online participants**Abstract**A program to construct weakly and full conformal field theories in the sense of Segal using the representation theory of vertex operator algebras was started more than 30 years ago. After more than 30 years, major progresses have been made and many problems in this program have been solved. In this talk, I will review the history of and the major progresses in this program and will also discuss the current status of this program.
**Speaker**Yi-Zhi Huang, Rutgers University**Title**Associative algebras and intertwining operators**Event**Subfactors, Vertex Operator Algebras, and Tensor Categories, the Institute for Advanced Study in Mathematics, Zhejiang University, Hangzhou, China**Time**9/22/2021, Wednesday, 8:00 pm (Eastern Time) or 9/23/2021, Thursday, 8:00 pm (Beijing time)**Location**The conference is entirely online. The talks are being live-streamed and the videos of most of the talks will be posted in the bottom of the page above after the talks.**Abstract**The connection between intertwining operators and A(V)-module maps (where A(V) is the Zhu algebra associated to a vertex operator algebra V) is needed in the last step of my proof of the Moore-Seiberg conjecture on the modular invariance of intertwining operators for rational conformal field theories. In this talk, I will discuss my recent work proving that the spaces of (logarithmic) intertwining operators are isomorphic to suitable spaces of module maps between suitable modules for the associative algebras A^∞(V) and A^N(V) that I introduced last year. The main motivation of this work is to give the last step of the proof of the modular invariance of (logarithmic) intertwining operators when V is C_2-cofinite but might not be reductive.
**Speaker**Siddhartha Sahi, Rutgers University**Title**On the extension of the FKG inequality to n functions**Event**Mathematical Physics Seminar, Department of Mathematics, Rutgers University**Time/Place**11/4/2021, Thursday, 1:20 pm in Hill 705**Abstract**The 1971 Fortuin-Kasteleyn-Ginibre (FKG) correlation inequality for two monotone functions on a distributive lattice is well known and has seen many applications in statistical mechanics, combinatorics, statistics, probability, and other fields of mathematics.In 2008 the speaker conjectured an extended version of this inequality for all n>2 monotone functions on a distributive lattice. This reveals an intriguing connection with the representation theory of the symmetric group. We give a proof of the conjecture for two special cases: for monotone functions on the unit square in R^k whose (upper) level sets are k-dimensional rectangles, and, more significantly, for *arbitrary*monotone functions on the unit square R^2. The general case for R^k, k>2 remains open.This is joint work with Elliott Lieb.
**Speaker**Yi-Zhi Huang, Rutgers University**Title**Vertex operator algebras and tensor categories**Event**MIT Infinite Dimensional Algebra Seminar, Department of Mathematics, MIT**Time/Place**11/5/2021, Friday, 3 pm in Room 2-135, Simons Building (Building 2)**Abstract**In 1988, based on the fundamental conjectures on operator product expansion and modular invariance, Moore and Seiberg observed that there should be tensor categories with additional structures associated to rational conformal field theories. Since then, tensor category structures from conformal field theories have been constructed, studied and applied to solve mathematical problems. Mathematically, conformal field theories can be constructed and studied using the representation theory of vertex operator algebras. In this talk, I will give a survey on the constructions and studies of various tensor category structures on module categories for vertex operator algebras.
## A conference with talks given by members of this seminar**Conference Name**Subfactors, Vertex Operator Algebras, and Tensor Categories (online), an IASM-BIRS Workshop at the Institute for Advanced Study in Mathematics, Zhejiang University, Hangzhou, China**Organizers**Robert McRae (Tsinghua University), Terry Gannon (University of Alberta), Vaughan Jones (Vanderbilt University), Jinwei Yang (Shanghai Jiao Tong University)**Time**September 19 to September 24, 2021, starting at 8:50 pm (Eastern Time) on September 19**Web page**Subfactors, Vertex Operator Algebras, and Tensor Categories . The talks are being live-streamed and videos of most of the talks will be posted in the bottom of this page after the talks.**Schedule**Schedule for: 21w5247 - Subfactors, Vertex Operator Algebras, and Tensor Categories (Online).
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