Abstract: 3d mirror symmetry predicts equivalences between topological twists of dual 3d N=4 theories, which we would like to understand as equivalences between their 2-categories of boundary conditions. Unfortunately, it is not known how to describe these 2-categories mathematically, although the B-side has been partially understood from work of Kapustin-Rozansky-Saulina and Arinkin. For abelian gauge theories, we propose that perverse … Read More
Abstract: I will describe a physically-motivated conjectural approach (developed with Shamit Kachru and Arnav Tripathy) to the construction of hyperkahler deformations of $(\mathbb{R}^{4-r} \times T^r)/\Gamma$ with $0\le r \le 4$, where $\Gamma$ is a finite group, as well as recent work with Arnav Tripathy which proves some of these conjectures. This generalizes Kronheimer’s construction of ALE manifolds in the $r=0$ … Read More
Abstract: \hat{Z} is a 3d TQFT whose existence was predicted by S. Gukov, D. Pei, P. Putrov, and C. Vafa in 2017 using 3d/3d correspondence. To each 3-manifold equipped with a spin^c structure, \hat{Z} is supposed to assign a q-series with integer coefficients that is categorifiable and provides an analytic continuation of Witten-Reshetikhin-Turaev invariants. In 2019, S. Gukov and C. Manolescu initiated … Read More
Abstract: [Joint work with Arvin Shahbazi-Moghaddam.] We prove a singularity theorem based on the covariant entropy bound. The theorem connects singularities to quantum information, and it eliminates the assumption of noncompactness from Penrose’s 1965 theorem. A quantum extension of our theorem further eliminates the null energy condition. It can be applied to an evaporating black hole, where it leads to … Read More
Abstract: In almost all situations, the twist of a supersymmetric QFT has the structure of a holomorphic QFT. I’ll review general aspects of holomorphic QFT while drawing parallels to the familiar situation of chiral CFT. I will then define a holomorphic model which we propose describes the minimal twist of the six-dimensional superconformal theory associated to the Lie algebra sl(2). … Read More
Abstract: In this talk I will explain how 3d mirror symmetry predicts an equivalence between 2-categories associated to dual pairs of hyperkahler quotients. The first 2-category is of an algebro-geometric flavor and was described by Kapustin/Rozansky/Saulina. The second category depends on symplectic topology and has a conjectural description in terms of the 3d generalized Seiberg-Witten equations. Both of these 2-categories are expected … Read More
ABSTRACT: It was shown by M.Aganagic and A.Okounkov that the relative insertions in the quantum K-theory of Nakajima varieties are equivalent to non-singular descendent insertions. Among other things, this result leads to integral representation for solutions of qKZ equations and explicit description of the Bethe vectors. From representation theoretic viewpoint, the result of Aganagic-Okoukov deals with the “slope” zero … Read More
3d N=4 gauge theories admit two topological twists, “A” and “B”, which are expected to give rise to extended TQFT’s. Aspects of these TQFT’s are starting to be defined mathematically. My goal in this talk is to discuss recent developments in understanding line operators (the value of the TQFT’s on a circle), which are expected to form a dg braided … Read More
Abstract: The goal of this talk will be to describe the relation between generators of Lagrangian Floer cohomology on a surface and functions on its mirror — both locally on building blocks such as cylinders, pairs of pants, and mirrors of pairs of pants, and globally on elliptic curves, higher genus surfaces, and their mirrors. The common theme throughout will be … Read More
Abstract: The spectral decomposition of the Hilbert space of automorphic functions is a very old and central topic in number theory, and mathematics in general. In particular, the Eisenstein series produce automorphic functions on a group G from automorphic functions on its Levi subgroups and one is interested in spectrally decomposing them. I will review some classical as well as … Read More