Fall, 2016

Fall, 2016

  • Speaker Sven Möller, Technische Universität Darmstadt
    • Title A Cyclic Orbifold Theory for Holomorphic VOAs and Applications
    • Time/place 9/23/2016, Friday, 12:00 in Hill 705
    • Abstract We develop an orbifold theory for a finite, cyclic group G acting on a suitably regular, holomorphic VOA V. To this end we describe the fusion algebra of the fixed-point VOA V^G and show that V^G has group-like fusion. Then we solve the extension problem for VOAs with group-like fusion.

      We also show that Schellekens' classification of V_1-structures of "meromorphic conformal field theories" of central charge 24 is a theorem on VOAs.

      Finally, we use these results to construct some new holomorphic VOAs of central charge 24 as lattice orbifolds. Together with the work of other authors we arrive at a complete classification of the V_1-structures of suitably regular, holomorphic VOAs of central charge 24.

      Moreover, some progress towards a classification of these VOAs themselves has been made recently by us and others.

      This is joint work with Nils Scheithauer (Darmstadt) and Jethro van Ekeren (IMPA, Rio de Janeiro) and is partly based on my Ph.D. thesis.

  • Speaker Henrik Gustafsson, Chalmers University of Technology
    • Title Eisenstein series attached to small automorphic representations
    • Time/place 9/30/2016, Friday, 12:00 in Hill 005 (in the basement of Hill Center; note the special room)
    • Abstract In this talk we study certain Fourier coefficients of Eisenstein series attached to small automorphic representations motivated by open problems in string theory. Whittaker coefficients on the Borel subgroup can be computed using the Casselman-Shalika formula and Langlands' constant term formula, but no similar tools are available for Fourier coefficients on other parabolic subgroups. We therefore demonstrate a method to compute parabolic Fourier coefficients of Eisenstein series in terms of known Whittaker coefficients on the Borel and show how they simplify for small automorphic representations. This is joint work with Dmitry Gourevitch, Axel Kleinschmidt, Daniel Persson and Siddhartha Sahi.
    • Slides pdf file.

  • Speaker Darlayne Addabbo, University of Illinois at Urbana-Champaign
    • Title Q-systems and Generalizations in Representation Theory
    • Time/place 10/7/2016, Friday, 12:00 in Hill 705
    • Abstract Tau-functions given as matrix elements for the action of the loop group, \widehat{GL_2}, on two-component Fermionic Fock space are solutions to the A_{\infty/2} Q-system. Since Q-systems are of interest in representation theory and in combinatorics, it is interesting to ask what sort of relations are satisfied by analogous tau-functions, given as matrix elements for the action of \widehat{GL_3} on three-component Fermionic Fock space. In this talk, we will define these new \widehat{GL_3} tau-functions and derive the system of discrete equations that they satisfy, and will discuss the progress we have made in analyzing this new system of equations. We will present a conjecture for relations satisfied by \widehat{GL_n} tau-functions.

      Since loop groups can be embedded in infinite matrix groups, we can generalize the above by calculating tau-functions given as matrix elements for infinite matrix groups, which will satisfy more general systems of equations than those mentioned above. This is analogous to the way in which the KP and KdV hierarchies are related. We have begun to calculate some of these new systems of equations and will present preliminary results and conjectures in this direction.

      This is joint work with Maarten Bergvelt.

  • Speaker Robert McRae, Vanderbilt University
    • Title Vertex algebraic intertwining operators among generalized Verma modules for affine Lie algebras
    • Time/place 10/14/2016, Friday, 12:00 in Hill 705
    • Abstract Intertwining operators among modules for a vertex operator algebra are essential for understanding tensor products of modules for vertex operator algebras. In this talk I will discuss techniques for constructing intertwining operators among modules for non-rational affine Lie algebra vertex operator algebras, in particular for constructing intertwining operators among generalized Verma modules. These results generalize earlier joint work with Jinwei Yang.

  • Speaker Martin Cederwall, Chalmers University of Technology
    • Title Extended geometry
    • Time/place 10/21/2016, Friday, 12:00 in Hill 705
    • Abstract Extended (double or exceptional) geometry is a framework where diffeomorphisms and certain tensor gauge transformations are united in order to reproduce duality symmetries in string theory / M-theory. I will review the basics of the formalism, focussing on local symmetries and geometric concepts, and make the connection to duality symmetries precise. If time allows, generalisation to supergeometry will be discussed, as well as situations where "dual gravity" becomes relevant.

  • Speaker Jakob Palmkvist, Texas A&M University
    • Title Generalized Cartan superalgebras
    • Time/place 10/28/2016, Friday, 12:00 in Hill 705
    • Abstract In Kac's classification of finite-dimensional Lie superalgebras, the contragredient ones can be constructed from Dynkin diagrams similar to those of the simple finite-dimensional Lie algebras, but with additional types of nodes. For example, A(0,n)=sl(1|n+1) can be constructed by adding a "gray'' node to the Dynkin diagram of A_n=sl(n+1), corresponding to an odd null root. The Cartan superalgebras constitute a different class, where the simplest example is W(n), the derivation algebra of the differential algebra in n dimensions, which in turn is generated by the differentials dx^m under the wedge product. I will in my talk relate A(0,n) and W(n) to each other, and show that also W(n) can be constructed from the Dynkin diagram of A(0,n). I will then aim to generalize the result and consider the Kac-Moody algebras E_n, each of which can be extended to an infinite-dimensional Borcherds superalgebra, in the same way as A_n can be extended to A(0,n). At least for n \le 8 there is also a different but related infinite-dimensional Lie superalgebra, in the same way as W(n) is related to A(0,n), which has been named tensor hierarchy algebra because of its applications to gauged supergravity. I will explain these applications, as well as connections to so called exceptional geometry.

  • Speaker Johannes Flake, Rutgers University
    • Title Dirac cohomology, Hopf-Hecke algebras and infinitesimal Cherednik algebras
    • Time/place 11/4/2016, Friday, 12:00 in Hill 705
    • Abstract I will present some results from joint work with Siddhartha Sahi: Motivated by the success of Dirac cohomology in the study of representations of connected semisimple Lie groups and degenerate affine Hecke algebras, we study a generalization suggested by Dan Barbasch and Siddhartha Sahi.

      The natural candidates to consider are PBW deformations satisfying an orthogonality condition, which we call Hopf-Hecke algebras. Besides the mentioned instances, they include infinitesimal Cherednik algebras as a new example.

      We will discuss a result relating Dirac cohomology with infinitesimal characters in the general situation, partial results on the classification of Hopf-Hecke algebras, and concrete computations of the Dirac cohomology for infinitesimal Cherednik algebras of the general linear group.

  • Speaker Jinwei Yang, University of Notre Dame
    • Title Logarithmic twisted modules for vertex algebras associated to affine Lie algebras.
    • Time/place 11/11/2016, Friday, 12:00 in Hill 705
    • Abstract We construct and classify lower bounded (logarithmic) twisted modules for vertex operator algebras associated to affine Lie algebras, using twisted Zhu's algebra tools, developed by Huang and myself, for a vertex operator algebra and a general automorphism.

  • Speaker Chris Sadowski, Ursinus College
    • Title Principal subspaces of twisted modules for lattice vertex algebras
    • Time/place 11/18/2016, Friday, 12:00 in Hill 705
    • Abstract Principal subspaces of modules for affine Lie algebras and lattice vertex algebras have received considerable attention in recent years due to their connection with the Rogers-Ramanujan and Andrews-Gordon partition identities and other related q-series. Vertex-algebraic techniques have been especially powerful in the study of principal subspaces, leading to the construction of exact sequences among these structures which naturally give their multigraded dimensions through the Euler-Poincare principle. In this talk, we discuss recent results on extending these techniques to principal subspaces of twisted modules for lattice vertex algebras, with emphasis on certain special cases.