Lie Group/Quantum Mathematics Seminar

Time Friday, 12:10 pm to 1:10 pm (Eastern time).

Place Hill 705 or online via Zoom (see below for the Zoom link and passcode).

YouTube channel Rutgers Lie Groups Quantum Math Seminar.

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.

Spring, 2024

In this semester, the seminar will be mostly in person. Occasionally there might be online talks using zoom. See the information below on each talk. For online talks, here is the information for the zoom meeting:

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

Some of the talks will be recorded and will be placed in the YouTube Channel for the seminar.

Upcoming talks

• Speaker Semyon Alesker, Tel Aviv University
  • Title The Kaehler package on the space of valuations on convex sets
  • Time/place 12/13/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
  • Abstract A valuation is a finitely additive functional on convex compacts subsets of a finite dimensional space. Over the past 20 years a number of structures on the space of continuous translation invariant valuations has been discovered, in particular a multiplicative structure. These developments have culminated in a construction of the Kaehler package on the algebra of valuations: Poincaré duality, hard Lefschetz theorem, Hodge-Riemann relations.

All scheduled talks

• Speaker Liang Kong, Southern University of Science and Technology
  • Title VOAs and full field algebras in anomalous CFTs
  • Time/place 9/6/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
  • Abstract A vertex operator algebra (VOA) and its non-chiral analogue full field algebra (FFA) play important roles in the mathematical theory of gapless boundaries of 2+1D topological order (or TQFT), where a gapless boundary can be viewed as an anomalous 1+1D CFT. In this talk, I will review this theory with an emphasize on the applications of VOA and FFA. I will explain some open questions of VOA and FFA arisen from this theory.

• Speaker Fei Qi, University of Denver
  • Title First-order deformations of freely generated vertex algebras
  • Time/place 10/18/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
  • Abstract We solve the problem of how to classify the first-order vertex-algebraic deformations for any grading-restricted vertex algebra V that is freely generated by homogeneous elements of positive weights. We approach by computing the second cohomology H^2_{1/2}(V, V) constructed by Yi-Zhi Huang. We start with the cocycle on two generators and show that its cohomology class is completely determined by its singular part. To extend the cocycle to any pair of elements in $V$, we take a generating function approach, formulate the cocycle equation, and show that all the complementary solutions are coboundaries. Then, we use a very general procedure to construct a particular solution. Using these results, we explicitly determine the first-order deformations of the universal Virasoro VOA Vir_c, universal affine VOA V^l(\g), Heisenberg VOA V^l(\h), and the universal Zamolodchikov VOA W_3^c. This work is joint with Vladimir Kovalchuk.

• Speaker Baptiste Cerclé, École Polytechnique Fédérale de Lausanne (EPFL)
  • Title Around the algebraic and probabilistic perspectives for the free-field and Toda theories
  • Time/place 11/1/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
  • Abstract In the context of two-dimensional conformal field theory (CFT), the free-field ranks among the most simple though fundamental models and serves as a key building block in the understanding of other CFTs such as Liouville and Toda theories.

    There are several perspectives describing the notion of a free-field, should it be for instance in probability theory with the notion of Gaussian Free Field or in the realm of Vertex Operator Algebra via the Heisenberg vertex algebra.

    The goal of this talk is two-fold: first of all we will explain how these different approaches can be unified within the same framework. We will then discuss some implications of this correspondence, on the probabilistic side for the correlation functions of Liouville and Toda CFTs (e.g. with the rigorous computations of some structure constants) and on the algebraic side for the representation theory of W-algebras.

• Speaker Yi-Zhi Huang, Rutgers University
  • Title Orbifold conformal field theory: General conjectures, open probelms and initial results
  • Time/place 11/8/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
  • Abstract I will discuss the general conjectures and open problems on orbifold conformal field theories formulated in 2020. I will also discuss or mention some initial results, including in particular, the construction of twisted modules, the formulation of the most general twisted intertwining operators (jointly with Jishen Du), the construction of tensor product of twisted modules (jointly with Jishen Du) and differential equations for products of suitable twisted intertwining operators (Daniel Tan).
  • Slides pdf file.

• Speaker Daniel Tan, Rutgers University
  • Title Differential equations for twisted intertwining operators among mostly untwisted modules
  • Time/place 11/15/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
  • Abstract Modules for a vertex operator algebra can be twisted by an automorphism of the vertex operator algebra. Intertwining operators among twisted modules describe how a twisted module can act on another twisted module, analogous to how the vertex operator algebra acts on itself.

    It is believed that products of twisted intertwining operators should converge, under suitably nice conditions, as they describe 4-point correlation functions in orbifold conformal field theory. Following Yi-Zhi Huang’s method for proving convergence of products of (untwisted) intertwining operators, we obtain new differential equations for products of certain types of twisted intertwining operators after deriving a new Jacobi identity for them. Physically, the correlation functions we have proved to converge describe how chiral fields from the untwisted sector act on a single chiral twisted sector.

• Speaker Siddhartha Sahi, Rutgers University
  • Title Basic orthogonal polynomials of Jacobi type
  • Time/place 11/22/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
  • Abstract Many important classical families of orthogonal polynomials, such as Jacobi, Laguerre, Legendre, etc., arise via specialization from some hypergeometric series or from a q-analog (the so-called basic series). The Askey scheme is a hierarchical list of some 48 such families, and it is natural to ask if this list is complete.
    In joint work with Joseph Bernstein and Dmitry Gourevitch (https://arxiv.org/abs/2401.14715), we have axiomatized some part of the Askey scheme, which we refer to as families of Jacobi type, and discovered 2 new families.
    In this lecture I will describe the q-analog of the above result, namely the definition and classification of polynomials of basic Jacobi type. It turns out that in this setting there are 4 new families beyond the Askey scheme.
    We are in the process of extending our results to obtain a complete classification of hypergeometric and basic orthogonal polynomials. Our expectation is that we will recover the full Askey scheme along with a small number of new families.

• Speaker Colton Griffin, University of Pennsylvania
  • Title Cohomological Vertex Algebras
  • Time/place 12/6/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
  • Abstract Vertex operator algebras have seen numerous applications in algebraic geometry, particularly in the study of moduli spaces of curves. It is a natural question as to whether there exists a generalization of vertex algebras that plays a similar role in the study of moduli spaces of higher-dimensional varieties. We provide such a higher-dimensional generalization, which we call cohomological vertex algebras. We develop the theory from basic definitions, discussing analogues of the Jacobi identity, reconstruction theorem, and more. Moreover we provide an analogue of the Witt algebra and its universal central extension, the Virasoro algebra, giving examples of cohomological vertex operator algebras such as the Witt vertex algebra and the bosonic ghost vertex algebra.

• Speaker Semyon Alesker, Tel Aviv University
  • Title The Kaehler package on the space of valuations on convex sets
  • Time/place 12/13/2024, Friday, 12:10 pm (Eastern Time), Hill 705 (in person)
  • Abstract A valuation is a finitely additive functional on convex compacts subsets of a finite dimensional space. Over the past 20 years a number of structures on the space of continuous translation invariant valuations has been discovered, in particular a multiplicative structure. These developments have culminated in a construction of the Kaehler package on the algebra of valuations: Poincaré duality, hard Lefschetz theorem, Hodge-Riemann relations.

Previous semesters