Spring, 2022

• Speaker Sven Möller, University of Hamburg
  • Title BRST Construction of 10 Borcherds-Kac-Moody Algebras
  • Time/place 2/11/2022, Friday, 12:10 pm (Eastern Time), Zoom link above
  • Abstract Borcherds-Kac-Moody algebras are natural generalisations of finite-dimensional simple Lie algebras. There are exactly 10 Borcherds-Kac-Moody algebras whose denominator identities are completely reflective automorphic products of singular weight on lattices of squarefree level (classified by Scheithauer). These belong to a larger class of Borcherds-Kac-Moody (super)algebras obtained by Borcherds by twisting the denominator identity of the Fake Monster Lie algebra. For the 10 Lie algebras we prove a conjecture by Borcherds that they can be realised uniformly as the physical states of bosonic strings moving on suitable spacetimes. This amounts to applying the BRST formalism to certain vertex algebras of central charge 26 obtained as graded tensor products of abelian intertwining algebras.
  • Slides pdf file.

• Speaker Siddhartha Sahi, Rutgers University
  • Title A Stone-von Neumann equivalence of categories for smooth representations of the Heisenberg group
  • Time/place 2/25, Friday, 12:10 pm (Eastern Time), Zoom link above
  • Abstract Let H_n be the 2n+1 dimensional Heisenberg group and let \chi be a non-trivial unitary character of its center Z. The celebrated Stone von-Neuman theorem says that there is a unique irreducible unitary representation of H_n with central character \chi. This uniqueness result has many applications -- in particular it plays a crucial role in the construction of the oscillator representation for the metaplectic group.
    We give an extension of this result to non-unitary and non-irreducible representations. Our main result is that there is an equivalence of categories between non-degenerate representations of Z, suitably defined, and those of H_n. We give both algebraic and analytic versions of this equivalence. The algebraic version, which is easier, is closely related to Kashiwara's lemma from the theory of D-modules. The analytic version, which is much more delicate, is an equivalence in the setting of smooth nuclear Frechet representations of moderate growth. Finally, we show how to extend the oscillator representation to the smooth setting, and we give an application to degenerate Whittaker models for representations of reductive groups.
    This is joint work with Raul Gomez (UANL, Mexico) and Dmitry Gourevitch (Weizmann, Israel).
  • Archive paper arXiv:2201.12638
  • YouTube video A Stone-von Neumann equivalence of categories for smooth representations of the Heisenberg group

• Speaker Angela Gibney, University of Pennsylvania
  • Title Towards vector bundles on the moduli space of curves from strongly finite VOAs
  • Time/place 3/25, Friday, 12:10 pm (Eastern Time), Zoom link above
  • Abstract Given any vertex operator algebra V, Zhu defined an associative algebra A(V), and showed that to any A(V)-module, one can associate an admissible V-module. This ultimately gives rise to a functor taking n-tuples of finite dimensional A(V)-modules to a sheaf of coinvariants (and its dual sheaf of conformal blocks) on the moduli space of stable n-pointed curves of genus g. If V is rational and C_2-cofinite, so A(V) is finite and semi-simple, much is known about these sheaves, including that they are coherent (fibers are finite dimensional) and satisfy a factorization property. Factorization ultimately allows one to show they are vector bundles. In this talk I will describe a program in which we are aiming for analogous results after removing the assumption of rationality, and weakening C_2-cofiniteness. As a first step, we replace the standard factorization formula with an inductive one that holds for sheaves defined by modules over any VOA of CFT-type. As an application, we show that if V is of CFT-type and A(V) is finite, then sheaves of coinvariants and conformal blocks are coherent. This is a preliminary description of new and ongoing joint work with Daniel Krashen and Chiara Damiolini.

• Speaker Robert Penna, IAS, School of Natural Sciences
  • Title General Relativity and Integrability
  • Time/place 4/1, Friday, 12:10 pm (Eastern Time), Zoom link above
  • Abstract A remarkable fact about general relativity is that it admits many exact solutions. This fact can be explained (at least in many cases) by the fact that the dimensional reduction of general relativity to two spacetime dimensions is an integrable system. The 2d theory has an infinite dimensional hidden symmetry called the Geroch group that underlies its integrability. I will review this story and then describe a new exact solution for gravitational pulse wave scattering. The pulses maintain their shapes after scattering, as in ordinary soliton scattering. In the last part of the talk, I will describe evidence for a further symmetry enhancement in one dimension. This enhanced symmetry is apparently a hyperbolic Kac-Moody algebra, which is an interesting and mysterious mathematical object in its own right.
  • Slides pdf file.
  • Archive paper arXiv:2112.05661 and arXiv:2201.07662

• Speaker Shashank Kanade, University of Denver
  • Title Completing the A_2 Andrews-Schilling-Warnaar identities
  • Time/place 4/15, Friday, 12:10 pm (Eastern Time), Zoom link above
  • Abstract Since the pioneering works of Lepowsky-Milne and Lepowsky-Wilson, affine Lie algebras have been an incredibly rich source of Rogers-Ramanujan-type identities. In a groundbreaking paper in 1999, Andrews-Schilling-Warnaar invented an A_2 generalization of the (A_1) Bailey lemma and discovered sum-side representations of principal characters for some (but not all!) standard A_2^{(1)} -modules. Recently, in a joint work with Matthew C. Russell, we have been able to complete their picture by providing conjectures encompassing all standard modules. We have proved our conjectures in some low levels, and proved some new identities in levels divisible by 3 (= dual Coxeter number of A_2^{(1)}). For this talk, my plan is to explain this picture and its relations to cylindric partitions, principal W-algebras, etc.
  • Slides pdf file
  • Archive paper arXiv:2112.05661
  • YouTube video Completing the A_2 Andrews-Schilling-Warnaar identities

• Speaker Martin Andler, University of Versailles Saint-Quentin
  • Title Equivariant cohomology in a tensor category
  • Time/place 4/22, Friday, 12:10 pm (Eastern Time), Zoom link above
  • Abstract In this talk, I will present some joint work with Siddhartha Sahi. Motivated by Cartan and Weil's work in differential geometry and Lie theory, specifically equivariant cohomology of a manifold with a Lie group action, reinterpreted by Guillemin-Sternberg in terms of a supersymmetric extension of a Lie algebra, we introduce the notion of a tensor category endowed with an odd unit, and prove, in that context, some generalisations of well know theorems like the Koszul resolution in Lie algebra cohomology. Our results apply not only to standard tensor categories, like the category of modules over a Lie algebra, but to more exotic categories, like the category of sheaves over a supermanifold or supervariety, the indizations of the Deligne interpolation categories and related categories constructed by Knop, Serganova and others.
  • YouTube video Equivariant cohomology in a tensor category

• Speaker Darlayne Addabbo, University of Arizona
  • Title Higher Level Zhu Algebras and their Applications
  • Time/place 4/29, Friday, 12:10 pm (Eastern Time), Zoom link above
  • Abstract Higher level Zhu algebras are associative algebras that are important in the study of vertex operator algebras. In this talk, I will discuss motivation for the study of these algebras and results on determining their structure. Some of this talk is based on joint work with Barron and some is based on joint work with Barron, Batistelli, Orosz-Hunziker, Pedi\'{c}, and Yamskulna.
  • Archive paper arXiv:2110.07671

• Speaker Colin Guillarmou (Universite Paris-Saclay)
  • Joint with Mathematical Physics Seminar
  • Title Conformal bootstrap and Segal axioms for Liouville Conformal Field Theory
  • Time/place 5/3, Tuesday, 2:00 pm (Eastern Time), Hill 705 and broadcasted online using the Zoom link above (note the special time,)
  • Abstract In this talk, I will review recent works done in collaboration with Kupiainen, Rhodes and Vargas on the Liouville conformal field theory (CFT). This two dimensional CFT is a theory of random Riemannian surfaces introduced and intensely studied in physics since the 80s. We use a probabilistic approach to define the path integral and we combine these probabilistic tools with scattering theory to give a mathematical proof of the so called "conformal bootstrap". This allows us to express the n-point correlation functions on all surfaces in terms of conformal blocks and structure constants. This approach also gives a rigourous definition of the conformal blocks as converging series in the moduli parameters for this CFT.
  • Archive paper arXiv:2112.14859, arXiv:2005.11530

• Speaker Daniel Tan, Rutgers University
  • Title LG/CFT correspondence
  • Time/place 5/20, Friday, 12:10 pm, Hill 705
  • Abstract N = 2 supersymmetric conformal field theories reveal a connection between Landau-Ginzburg (LG) theories and CFTs. Mathematically, this suggests equivalences between certain tensor categories of matrix factorizations and certain tensor categories of VOA modules. In this talk, I will explain some notions required to understand this conjecture and outline some results that I have learned.

• Speaker Genkai Zhang, Chalmers/Gothenburg
  • Title Representations of Hermitian Lie groups induced from Heisenberg parabolics
  • Time/place 7/15/2022, Friday, 12:10 pm, Hill 705
  • Abstract We study representations Ind_P^G(e^\nu) induced from a character of a Heisenberg parabolic subgroup P of a Hermitian Lie group G. We find the composition factors and the complementary series.

Some additional talks by members of this seminar

• Speaker Yi-Zhi Huang, Rutgers University
  • Title Convergence in conformal field theory
  • Event School on Representation theory, Vertex and Chiral Algebras, 3/14/2022 to 3/18/2022, IMPA, Rio de Janeiro, Brazil
  • Time 3/14, 3/15, 3/18, 12:00 to 1:30 pm (Eastern Time) or 1:00 to 2:30 pm (Brasília Time)
  • Location This minicourse will be online.
  • Abstract We review some main convergence results, conjectures and problems in the construction and study of conformal field theories using the representation theory of vertex operator algebras. We also discuss the analytic extension results obtained together with the convergence results. We discuss the convergence and analytic extensions of products of (logarithmic) intertwining operators (chiral conformal fields), the convergence and analytic extensions of q-traces and pseudo-q-traces of products of (logarithmic) intertwining operators. We also discuss the convergence results related to the Virasoro algebra and a higher-genus convergence result. We then explain the convergence and analytic extension conjectures in orbifold conformal field theory and the convergence problems in the cohomology theory of vertex operator algebras.