Posted October 4, 2021

Last modified November 5, 2021

Mathematical Physics and Representation Theory Seminar

3:30 pm - 4:20 pm Zoom: https://lsu.zoom.us/j/7376728101
Colleen Delaney, Indiana University Bloomington

Zesting and Witten-Reshetikhin-Turaev invariants

I’ll discuss the ribbon zesting construction on pre-modular categories from a diagrammatic point of view and show that Witten-Reshetikhin-Turaev invariants of framed knots and links decouple under zesting. As an application, I will explain how the Mignard-Schauenburg ``modular isotopes” can be understood through zesting. This talk is based on joint work with Cesar Galindo, Julia Plavnik, Eric Rowell, and Qing Zhang as well as Sung Kim.

Posted October 3, 2021

Last modified November 5, 2021

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm Lockett 233
Gurbir Dhillon, Yale University

Kazhdan--Lusztig theory for affine Lie algebras at critical level

Formulas for simple characters have a long and rich history in representation theory, and our main result is one more such formula, originally conjectured by Feigin--Frenkel. However, in the majority of the talk, we will provide a general survey of such results for non-specialists. After recalling Weyl's character formula for highest weight modules for simple algebraic groups, we will discuss the Kazhdan--Lusztig character formula for highest weight modules for simple Lie algebras. In particular, we will convey some of the striking ideas involved in its proof via localization, due to Beilinson--Bernstein and Brylinski-Kashiwara, which birthed the subject of geometric representation theory. Moving beyond simple Lie algebras and groups, we will recall that associated to each simple Lie algebra is a one parameter family of infinite dimensional Lie algebras, the affine Lie algebras, which appear repeatedly in algebraic geometry and mathematical physics. By work of Kashiwara--Tanisaki, the highest weight characters at all points in the family save one were understood by the mid 1990s. At this remaining point, the critical level, the representation theory of affine Lie algebras undergoes a phase transition, and the remarkable phenomena present at this point have deep connections to the geometric Langlands program. An analog of the Kazhdan--Lusztig conjecture for affine Lie algebras at critical level was proposed by Feigin--Frenkel in the early 1990s. We have proven this conjecture in forthcoming work joint with David Yang, using localization theory at critical level as developed by Beilinson--Drinfeld and Frenkel--Gaitsgory. The main emphasis throughout will be on basic ideas and simple examples, and we will not presume familiarity with any of these subjects beyond the finite dimensional representations of SL2.

Posted September 28, 2021

Last modified November 14, 2021

Mathematical Physics and Representation Theory Seminar

3:30 pm - 4:20 pm Lockett 233
Mee Seong Im, United States Naval Academy

Iterated wreath products and foams, with applications

I will explain a new perspective of foams with connections to the representation theory of iterated wreath products. If I have time, I will discuss the connections of foams to field extensions, Sylvester sums, and matrix factorizations. This is joint work with Mikhail Khovanov, with Appendix joint with Lev Rozansky.

Posted September 29, 2021

Last modified January 30, 2022

Mathematical Physics and Representation Theory Seminar

3:30 pm - 4:20 pm Zoom: https://lsu.zoom.us/j/98489192227, Lockett 233
Iva Halacheva, Northeastern University

Welded tangles and the Kashiwara-Vergne group

Welded or w-tangles are a higher dimensional analogue of classical tangles, which admit a yet further generalization to welded foams, or w-trivalent graphs, a class of knotted tubes in 4-dimensional space. Welded foams can be presented algebraically as a circuit algebra. Together with Dancso and Robertson we show that their automorphisms can be realized in Lie theory as the Kashiwara-Vergne group, which plays a key role in the setting of the Baker-Campbell-Hausdorff series. In the process, we use a result of Bar-Natan and Dancso which identifies homomorphic expansions for welded foams, a class of powerful knot invariants, with solutions to the Kashiwara-Vergne equations.

Posted January 31, 2022

Last modified March 6, 2022

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm Lockett 233, https://lsu.zoom.us/j/91730960984
Peter Koroteev, UC Berkeley

DAHA Representations and Branes

I will describe our recent geometric representation theory construction for the double affine Hecke algebra (DAHA) of rank one. The spherical subalgebra of DAHA can be understood as flat one-parameter deformation (geometric quantization) of the SL(2, C) character variety X of a one-punctured torus. This variety for rank one DAHA is described by an affine cubic surface which is an elliptic fibration of Kodaira type I_0^*. Our main result provides an equivalence between the Fukaya category of X and the category of finite-dimensional modules of DAHA. Upon this correspondence, compact Lagrangian submanifolds of X are related to finite-dimensional representations of DAHA. This is a work in progress with S. Gukov, S. Nawata, D. Pei, and I. Saberi.

Posted February 8, 2022

Last modified March 16, 2022

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm Lockett 233
Akos Nagy, UC Santa Barbara

BPS monopoles with arbitrary symmetry breaking

Magnetic monopole solutions of Maxwell's equations have been known since Dirac's famous paper in 1931. In the mid-1970s, Bogomolny and, independently, Prasad and Sommerfeld, gave nonabelian generalizations to these solutions in Yang-Mills theory, which are now called Bogomolny-Prasad-Sommerfeld (BPS) monopoles. These are gauge theoretic field configurations over 3-dimensional backgrounds. In this talk I will introduce BPS monopoles both from the mathematical and physical points of view and recall some of the most important results about them, with an emphasis on BPS monopoles over the euclidean 3-space. In particular, I will introduce the concept of symmetry breaking for these fields. While a lot is known about monopoles with maximal symmetry breaking, the general case has been much less understood. After the general introduction, I will present my recent results on the construction of monopoles with arbitrary, nonmaximal symmetry breaking. This is achieved by understanding the analytic behavior of harmonic spinors associated to Dirac operators twisted by monopoles. This is a joint work with Benoit Charbonneau.

Posted January 30, 2022

Last modified April 3, 2022

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm Lockett 233
Tudor Padurariu, Columbia University

Categorical Hall algebras in Donaldson-Thomas theory

Kontsevich-Soibelman defined the cohomological Hall algebra (CoHA) of a quiver with potential. By a result of Davison-Meinhardt, CoHAs are deformations of the universal enveloping algebra of the BPS Lie algebra of the quiver with potential. One can also define categorical and K-theoretic Hall algebras of a quiver with potential. Examples of such Hall algebras are (positive parts of) quantum affine algebras. I will introduce the categorical and K-theoretic replacements of the BPS spaces and explain how to prove analogues of the Davison-Meinhardt theorem in these contexts. These results have applications in Donaldson-Thomas theory and in the study of Hall algebras of surfaces.

Posted February 8, 2022

Last modified April 16, 2022

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm Lockett 233
Vasily Krylov, MIT

Symplectic duality and equivariant Hikita-Nakajima conjecture for ADHM spaces

We will discuss the general notion of symplectic duality between symplectic resolutions of singularities and give examples. Equivariant Hikita-Nakajima conjecture is a general conjecture about the relation between the geometry of symplectically dual varieties. We will consider the example of the Hilbert scheme of points on the affine plane and briefly discuss the proof of the equivariant Hikita-Nakajima conjecture in this particular case. We will also briefly discuss the generalization of this proof to the case of ADHM spaces (moduli spaces of instantons on R^4). Time permitting we will say about the possible approach towards the proof of Hikita-Nakajima conjecture for other symplectically dual pairs (such as Higgs and Coulomb branches of quiver gauge theories). The talk is based on the joint work with Pavel Shlykov arXiv:2202.09934.

Posted February 6, 2022

Last modified April 25, 2022

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm Lockett 233
Eugene Rabinovich, University of Notre Dame

Classical Bulk-Boundary Correspondences via Factorization Algebras

A factorization algebra is a cosheaf-like local-to-global object which is meant to model the structure present in the observables of classical and quantum field theories. In the Batalin-Vilkovisky (BV) formalism, one finds that a factorization algebra of classical observables possesses, in addition to its factorization-algebraic structure, a compatible Poisson bracket of cohomological degree +1. Given a ``sufficiently nice'' such factorization algebra on a manifold $N$, one may associate to it a factorization algebra on $N\times \mathbb{R}_{\geq 0}$. The aim of the talk is to explain the sense in which the latter factorization algebra ``knows all the classical data'' of the former. This is the bulk-boundary correspondence of the title. Time permitting, we will describe how such a correspondence appears in the deformation quantization of Poisson manifolds.