Fall, 2019

• Speaker Sven Moeller, Rutgers University
  • Title Bounds for the constant terms of weight-0 vector-valued modular forms and generalized deep holes of the Leech lattice vertex operator algebra
  • Time/place 9/20/2019, Friday, 12:00 in Hill 705
  • Abstract We derive an upper bound for (the sum corresponding to an isotropic subgroup of) the constant terms of a weight-0 vector-valued modular form that transforms under the Weil representation, based on a pairing argument with weight-2 vector-valued Eisenstein series.

    As an application we prove a vertex algebra analogue of the remarkable result by Conway, Parker and Sloane (and Borcherds) that there is a bijection between the deep holes of the Leech lattice and the 23 Niemeier lattices with roots.

• Speaker Yi-Zhi Huang, Rutgers University
  • Title Existence of twisted modules for a grading-restricted vertex (super)algebra
  • Time/place 9/27/2019, Friday, 12:00 in Hill 705
  • Abstract We give a linearly independent subset of the universal lower-bounded g-twisted modules for a grading-restricted vertex superalgebra V and an automorphism g of V. In particular, such a lower-bounded g-twisted V-module is not 0. We then prove the existence of irreducible lower-bounded generalized g-twisted V-modules generated by one-dimensional spaces. Assuming that V is a M\"{o}bius vertex superalgebra and suitable additional natural conditions hold, we prove that there exists an irreducible grading-restricted generalized g-twisted V-module, which is in fact an irreducible ordinary g-twisted V-module when g is of finite order. We also prove that every lower-bounded generalized module with an action of g for the fixed-point subalgebra V^{g} of V under g can be extended to a lower-bounded generalized g-twisted V-module.

• Speaker Hadi Salmasian, University of Ottawa
  • Title Spherical superharmonics, singular Capelli operators, and the Dougall-Ramanujan identity
  • Time/place 10/14/2019, Monday, 12:00 in Hill 313 (Note the special time and place)
  • Abstract Given a multiplicity-free action V of a simple Lie (super)algebra g, one can define a distinguished "Capelli" basis for the algebra of g-invariant differential operators on V. The problem of computing the eigenvalues of this basis was first proposed by Kostant and Sahi, and has led to the theory of interpolation polynomials and their generalizations. In this talk, we consider an example associated to the orthosymplectic Lie superalgebras, which leads to "singular" Capelli operators, and we obtain two formulas for their eigenvalues. Along the way, the Dougall-Ramanujan identity appears in an unexpected fashion. This talk is based on joint work with Siddhartha Sahi and Vera Serganova.

• Speaker Jinwei Yang, University of Alberta
  • Title Tensor categories from Virasoro algebra
  • Time/place 10/14/2019, Monday, 3:20 pm (Note the special time) in Hill 705
  • Abstract We construct tensor category structures on the lower bounded C_1-cofinite modules for the Virasoro algebra of arbitrary central charges. We first show that this category is the same as the category of finite length modules with C_1-cofinite simple composition factors. Then we verify that all the conditions needed in the logarithmic tensor category theory of Huang, Lepowksy and Zhang hold. We also prove rigidity of this category for the generic central charges.

• Speaker Si Li, Tsinghua University and IAS
  • Title Factorization algebra from Quantum Field Theory and Index Theorem
  • Time/place 11/8/2019, Friday, 12:00 in Hill 705
  • Abstract We explain the general idea of using factorization algebras to represent structures of observables in quantum field theory. As an application, we show how such structures can be used to prove and derive various index type theorems.

• Speaker Abid Ali, MacEwan University
  • Title Gindikin-Karpelevich Finiteness for Local Kac-Moody Groups
  • Time/place 11/15/2019, Friday, 12:00 in Hill 705
  • Abstract One of the main difficulties in extending Macdonald’s theory of spherical functions from p-adic Chevalley groups to p-adic Kac-Moody groups is the absence of Haar measure in the infinite dimensional case. Related to this problem is the question of how to generalize the integral defining Harish-Chandra’s c-function to the p-adic Kac-Moody setting. These questions are answered by proving the Approximation Theorem, which is a formal analogue of Harish-Chanda's limit relating spherical and c-fucntion, and two certain finiteness’. We call these finiteness results the Gindikin-Karpelevich Finiteness and the Spherical Finiteness. In this talk, we will present the proofs of these results for infinite dimensional local Kac-Moody groups.

    This work was completed under the supervision of Manish M. Patnaik.

• Speaker Angela Gibney, Rutgers University
  • Title On factorization and vector bundles of conformal blocks from vertex algebras
  • Time/place 11/22/2019, Friday, 12:00 in Hill 705
  • Abstract Modules over conformal vertex algebras give rise to sheaves of coinvariants and conformal blocks on moduli of stable pointed curves. We show that under certain natural hypotheses, these sheaves satisfy the factorization property, a reflection of their inherent combinatorial nature. As an application, we prove they are vector bundles. These provide a generalization of vector bundles defined by integrable modules over affine Lie algebras at a fixed level. Satisfying factorization is essential to a recursive formulation of invariants, like ranks and Chern classes, and to produce new constructions of rational conformal field theories.

• Speaker Shashank Kanade, University of Denver
  • Title Tensor structure on relaxed categories at admissible levels
  • Time/place 12/6/2019, Friday, 12:00 in Hill 705
  • Abstract Representation theory of vertex operator algebras based on affine Lie algebras at admissible (yet non-integral) levels is quite rich. Here, the underlying VOAs are non-rational. A culmination of various deep results of Arakawa, Creutzig--Huang--Yang and Creutzig is that the sub-category of ordinary modules is finite, vertex tensor, rigid (at least in the simply-laced case) and often (but not always) a modular category. However, many considerations necessitate looking at a (much) bigger non-finite category containing the so-called relaxed highest-weight modules. In an ongoing joint work with David Ridout, we are looking at the vertex tensor structure on these relaxed categories. I will present a few preliminary results obtained (only for sl_2!) in this direction.

• Speaker Vidya Venkateswaran, Center for Communications Research at Princeton
  • Title Metaplectic representations of Hecke algebras and a new family of polynomials
  • Time/place 12/6/2019, Friday, 10:30 am in Hill 705
  • Abstract In this talk, we will discuss some recent joint work with Siddhartha Sahi and Jasper Stokman. We introduce a new "metaplectic" action of the double affine Hecke algebra on polynomials. Next, we show how one can obtain the Chinta-Gunnells Weyl group action (a key ingredient in their construction of Weyl group multiple Dirichlet series) via localization. Finally, we show that there exist families of metaplectic polynomials indexed by the weight lattice, and depending on additional parameters, which are eigenfunctions of metaplectic variants of Cherednik's Y -operators. These polynomials satisfy various nice properties, and special cases connect with well-studied objects. In particular, they reduce to nonsymmetric Macdonald polynomials at n = 1, and metaplectic Iwahori-Whittaker functions can be obtained by taking a limit in the q-parameter.