Spring 2017

Apr. 27, 2017

Speaker: Joel Clingempeel

Title: Introduction to Sigma Models

Abstract: A bosonic sigma model is a quantum field theory which describes the dynamics of a string propagating on a target Riemannian manifold and can be used to probe the geometry of the target. By incorporating fermions in the right way, one can make such a model supersymmetric, and in the case of a Kahler manifold, one can further construct a theory with extended supersymmetry which has rather stringent constraints, often enabling one to make exact computations that would otherwise be intractable. A powerful tool for studying such models introduced by Witten in '93 is the gauged linear sigma model (with superpotential) which at low energies reduces to (a hypersurface in) the symplectic reduction of an appropriate momentum map, thus yielding a supersymmetric sigma model with a toric (hypersurface) target. Such a theory undergoes “phase transitions” as one varies parameters across certain walls in a moduli space, enabling one to move between different sigma models as well as related theories ((orbifold) LG models / hybrid models). Though the walls naively appear to be codimension one, by considering theta angles which couple to background electric fields, one effectively complexifies the moduli space which leads to complex codimension one walls, thus enabling one to interpolate between chambers. Time permitting, further topics may be discussed depending on audience interest such as topological twisting, (homological) mirror symmetry, superconformal field theories, BPS solitons and Picard-Lefschetz theory, etc.

Apr. 20, 2017

Speaker: Alejandro Ginory

Title: Generalized Theta Functions and VOA's

Abstract: Theta functions arise in a multitude of mathematical fields, such as number theory, algebraic geometry, and representation theory. In this talk, I will discuss their role in the representation theory of vertex operator algebras and present a generalization of these functions.

Apr. 13, 2017

Speaker: Haijun Tan

Title: Some non-graded modules for Virasoro algebra and affine Lie algebras

Abstract: In this talk, we will introduce several classes of non-graded modules for the Virasoro algebra and affine Lie algebras. Non-graded modules include some restricted modules, which are in some sense the most important modules for the representation theory of vertex algebras, and non-restricted modules. We will describe the structures and the irreducible conditions of these modules.

Mar. 23, 2017

Speaker: Fei Qi

Title: Rationality of iterates of vertex operators in a MOSVA

Abstract: I'll use complex analysis I talked about before to give a proof to the rationality of iterates.

Mar. 2, 2017

Speaker: Johannes Flake

Title: Deligne's category Rep(S_t)

Abstract: The representations of S_n form a semisimple tensor category Rep(S_n) for any natural number n. Deligne constructed interesting tensor categories Rep(S_t) for, say, all complex numbers t which interpolate / extrapolate from these in a certain precise way. Understanding the construction will be a very original and hands-on exercise in representation theory, combinatorics and tensor categories for us.

References: J. Comes, V. Ostrik. On blocks of Deligne's category Rep(S_t). https://arxiv.org/pdf/0910.5695.pdf

P. Deligne. La Cat'egorie des Repr'esentations du Groupe Sym'etrique S_t, lorsque t n'est pas un Entier Naturel. http://www.math.ias.edu/files/deligne/Symetrique.pdf

P. Etingof. Deligne categories. (lecture notes). http://math.mit.edu/~innaento/DeligneCatSeminar/Pasha_notes_Feb_2013_talk.pdf

A. Mathew. Deligne's category Rep(S_t) for t not necessarily an integer. (blog entry). http://wp.me/pIbuP-no

D. Speyer. Deligne's ``La Categorie des Representations du Groupe Symetrique S_t, lorsque t n'est pas un Entier Naturel.'' (blog entry). http://wp.me/p56JZ-nZ

Feb. 23, 2017

Speaker: Robert Laugwitz

Title: Dijkgraaf--Witten theory for finite groups and the Drinfeld double

Abstract: In this expository talk, I will attempt to explain how the Drinfeld double of a finite group (with a 3-cocycle) naturally appears in the work of Dijkgraaf--Witten to construct a topological gauge field theory via Chern--Simons theory. This way, one obtains a 3,2,1-extended topological field theory associating to the 1-sphere the modular tensor category of finite-dimensional representations over this Drinfeld double.

References: Dijkgraaf, Robbert; Witten, Edward. Topological gauge theories and group cohomology. Comm. Math. Phys. 129 (1990), no. 2, 393--429

Fuchs, Jürgen; Schweigert, Christoph. Symmetries and defects in three-dimensional topological field theory. String-Math 2014, 21--40, Proc. Sympos. Pure Math., 93, Amer. Math. Soc., Providence, RI, 2016

Witten, Edward. Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121 (1989), no. 3, 351--399.

Feb. 16, 2017

Speaker: Fei Qi

Title: Complex analysis in the study of MOSVA

Abstract: In order to prove the rationality of iterates of any numbers of vertex operators, we need some theorems in several complex variable functions as preparation. We shall discuss them in great detail.

Feb. 9, 2017

CANCELLED DUE TO SEVERE WEATHER

Speaker: Fei Qi

Title: Complex analysis in the study of MOSVA

Abstract: In order to prove the rationality of iterates of any numbers of vertex operators, we need some theorems in several complex variable functions as preparation. We shall discuss them in great detail.

Feb. 2, 2017

Speaker: Fei Qi

Title: Vertex operators in Meromorphic Open String Vertex Algebras

Abstract: I'll recall the definition of MOSVA. Then I'll talk about how to interpret the vertex operator as \overline{V}-valued maps and how to deal with the subtle issues arising in the interpretation.

Jan. 26, 2017

Speaker: Alejandro Ginory

Title: Kazhdan-Lusztig Tensor Products of Affine Lie Algebra Modules

Abstract: In this talk, I will present the so-called "double dual" construction of the Kazhdan- Lusztig tensor product of affine Lie algebra modules of level (or central charge) k-h where k is NOT a non-negative rational (and h is the Coxeter number). This construction is analogous to the tensor product structure of modules of certain classes of VOA's.

Reference: D. Kazhdan and G. Lusztig. Tensor Structures arising from Affine Lie Algebras I

Spring 2016, Spring 2015, Fall 2014 (Organized by Fei Qi)

Spring 2013, Fall 2012 (Organized by Shashank Kanade)