Organized by Mina Aganagic, Semeon Artamonov, Miroslav Rapcak, and Vivek Shende

Mondays 2:00-3:00 PM at 402 Le Conte Hall

*Special Day/Time.

This is a research seminar, intended for mathematicians and physicists. For the speaker to successfully reach the audience in both fields, it is important to explain, as clearly as possible: the motivations for the work, questions addressed, key ideas. The audience may fail to appreciate the glory of the result, otherwise.

In the recent papers by Braverman-Finkelberg-Nakajima a mathematical construction of the Coulomb branches of 3d N=4 quiver gauge theories was proposed, whose quantization is conjecturally described via the so-called shifted Yangians and shifted quantum affine algebras.

The goal of this talk is to explain how both of these shifted algebras provide a new insight towards integrable systems via the RTT realization. In particular, the study of Bethe subalgebras associated to the antidominantly shifted Yangians of sl(n) provides an interesting plethora of integrable systems generalizing the famous Toda and DST systems. As another interesting application, the shifted quantum affine algebras in the simplest case of sl(2) give rise to a new family of 3^{n-2} q-Toda systems of sl(n), generalizing the well-known one due to Etingof and Sevostyanov. Time permitted, I will also explain how one can generalize the latter construction to produce exactly 3^{rk(g)-1} modified q-Toda systems for any semisimple Lie algebra g.

These talks are based on the joint works with M. Finkelberg, R. Gonin and a current project with R. Frassek, V. Pestun.

The nonabelian Hodge correspondence gives a diffeomorphism between the moduli of flat connections and the moduli of higgs bundles on a smooth Riemann surface. The two moduli, however, are completely different as algebraic varieties. Thus natural structures on one side of the correspondence, such as Hitchin's integrable system, become rather mysterious on the other side.

I will discuss joint work with Zsuzsanna Dancso and Vivek Shende, which defines a 'microlocal model' of the correspondence in a neighborhood of a singular fiber of the Hitchin system. In particular, we prove a conjecture of Hausel and Proudfoot relating the weight and perverse filtrations on either side of this microlocal correspondence.

In this talk I will present some of the results and recent developments in 1811.10592, where a technique for constructing Bridgeland stability conditions on Fukaya categories associated to marked surfaces. I will introduce the results of that paper and give examples; time allowing I will also mention some recent developments.

We discuss a class of vertex operator algebras \mathcal{W}_{m|n\times \infty} generated by a super-matrix of fields for each integral spin 1,2,3,\dots. The algebras admit a large family of truncations that are in correspondence with holomorphic functions on the Calabi-Yau singularity given by solutions to xy=z^mw^n. We propose a free-field realization of such truncations generalizing the Miura transformation for \mathcal{W}_N algebras. Relations in the ring of holomorphic functions lead to bosonization-like relations between different free-field realizations. The discussion provides a concrete example of a non-trivial interplay between vertex operator algebras, algebraic geometry and gauge theory.

Recently, Costello proposed how to systematically apply Omega deformation on string theory and M-theory. Upon the deformation, one can discuss the exact topological holography of both M2 and M5 branes.

The goal of the talk is to review Costello's formalism and to discuss a combined system of the M2 and M5 branes, which is the joint work with D. Gaiotto. The M2 brane worldvolume theory is 3d N=4 ADHM gauge theory, which is a self-mirror. The relevant degree of freedom associated to the M5 brane is W_{1+\infty} Vertex Operator Algebra. I will explain how two systems interact each other and point out the connection to Y-algebra configuration, introduced by Gaiotto and Rapcak.

3d N=4 gauge theories admit two distinct topological twists, 'A' and 'B', analogous to the well-studied A and B twists of 2d N=(2,2) theories, and exchanged by 3d mirror symmetry. Unlike their 2d analogues, much of the categorical structure of the 3d twists is still being worked out. I will discuss some aspects of the category of line operators in the A and B of gauge theories, and an application to knot homology, deriving the A-twisted mirror of a construction of Oblomkov-Rozansky. Joint work with Niklas Garner, Justin Hilburn, Alexei Oblomkov, and Lev Rozansky.

2d mirror symmetry relates the algebraic and symplectic geometry of a pair of Kahler manifolds obtained by dualizing a Lagrangian torus fibration. We will discuss some known and expected results in the case when the spaces involved are actually hyperkahler, arising as moduli spaces of 4-dimensional supersymmetric gauge theories (for instance, by a K-theoretic version of the Braverman-Finkelberg-Nakajima construction), and the torus fibration is a Hitchin system. In this case, the relation between the dual spaces can be seen as a Langlands duality, and many structures of interest in representation theory appear.

I will give an overview and some of the highlights of a research program initiated in 2013 with Gadde and Putrov, the main subject of which is the study of 2d-3d combined systems that involve 3d N=2 theories with 2d N=(0,2) boundary conditions. We will see how holomorphic-topological twists of such systems lead to boundary chiral algebras, somewhat similar to Beem-Rastelli chiral algebras of 4d N=2 theories. In particular, just like the characters of Beem-Rastelli chiral algebras are computed by the so-called Schur index of 4d N=2 theories, characters of our boundary chiral algebras are computed by the "half-index" of 2d-3d combined systems, also introduced in the above-mentioned work with Gadde and Putrov. It was originally motivated by applications to topology, which are the subject of the math colloquium on Thursday.

The Affine Grassmannian is an ind-scheme associated to a reductive group G. It has a cell structure similar to the one in the usual Grassmannian. Transversal slices to these cells give an interesting family of Poisson varieties. Some of them admit a smooth symplectic resolution and have an interesting geometry related to the representation theory of the Langlands dual group. We will focus on equivariant cohomology of such resolutions and will show how the trigonometric Knizhnik-Zamolodchikov equation arises as a quantum differential equation in this setting.

I will give an overview of some new invariants of 3- and 4-manifolds that arise naturally in the study of compactifications of M-theory. I will discuss homological invariants of 3-manifolds and their "decategorifications" that take the form of q-series with integer coefficients, and, if time permits, homotopy-theoretic invariants of 4-manifolds that generalize the more familiar Donaldson and Seiberg-Witten invariants.

G2 manifolds constitute a class of Einstein seven-manifolds and are of substantial interest both in Riemannian geometry and theoretical physics. At present a vast number of compact G2 manifolds is known to exist. In this talk I will discuss a gauge-theoretic approach to the construction of invariants of compact G2 manifolds. I will focus on an interplay between gauge theories in dimensions 7 and 3 and how this can be used for the construction of the invariants.