ABSTRACT: I will explain how to use Gromov-Witten theory to classify all rational curves with primitive curve class on the Fano variety of lines in a very general cubic fourfold. Joint work with Junliang Shen and Qizheng Yin.
Gukov, Putrov and Vafa postulated the existence of some 3-manifold invariants, obtained by counting BPS states in the 3d N=2 theory T[M_3]. The GPV invariants take the form of power series converging in the unit disk, and whose radial limits at the roots of unity give the Witten-Reshetikhin-Turaev invariants. Furthermore, these power series have integer coefficients, and should admit a … Read More
We explain how quantum affine algebras can be used to systematically construct “exotic” t-structures. One of the application is to obtain exotic t-structures on certain convolution varieties defined using affine Grassmannians (these varieties play an important role in the geometric Langlands program, knot homology constructions, the coherent Satake category etc.) As a special case we also recover the exotic t-structures … Read More
Abstract: Several deep mathematical and physical results such as Kontsevich’s deformation-quantization, Drinfeld’s associators, and the Deligne hypothesis are controlled by the vanishing of certain obstruction classes in the theory of differential graded operads. I will talk about a way to obtain such vanishing results, as well as higher-genus analogues, using a weight theory implied by a new motivic point of … Read More
Abstract: A conjecture of Dunfield-Gukov-Rasmussen predicts a family of differentials on reduced HOMFLYPT homology, indexed by the integers, that give rise to a corresponding family of reduced link homologies. We’ll discuss a variant of this conjecture, constructing an unreduced link homology theory categorifying the quantum gl_n link invariant for all non-zero values of n (including negative values!). To do so, … Read More
ABSTRACT: A conjecture of Gorsky-Negut-Rasmussen asserts the existence of a pair of adjoint functors relating the Hecke category for symmetric groups and the Hilbert scheme of points in the plane. One topological consequence of this conjecture is the prediction of a deformation of the triply graded Khovanov-Rozansky link homology which restores the missing q—>tq^{-1} symmetry of KR homology for links. … Read More
ABSTRACT: I will describe a construction which, for a given 4D N=2 Argyres-Douglas SCFT, seems to produce a three-dimensional TQFT, whose underlying modular tensor category coincides with that of a 2d chiral algebra of the parent 4d N=2 theory.
ABSTRACT: We revisit Donaldson-Witten theory, that is the N=2 topologically twisted super Yang-Mills theory with gauge group SU(2) or SO(3) on compact 4-manifolds. We study the effective action in the Coulomb branch of the theory and by considering a specific Q-exact deformation to the theory we find interesting connections to mock modular forms. A specific operator of this theory computes … Read More
Abstract: I will present the rank N magnificent four theory, which is the supersymmetric localization of U(N) super-Yang-Mills theory with matter on a Calabi-Yau fourfold, and conjecture an explicit formula for the partition function Z: it has a free-field representation, and surprisingly it depends on Coulomb and mass parameters in a simple way. Based on joint work with Nikita Nekrasov.
Abstract: In my talk I will consider a quantum integrable Hamiltonian system with two generic complex parameters q,t whose classical phase space is the moduli space of flat SL(2,C) connections on a genus two surface. This system and its eigenfunctions provide genus two generalization of the trigonometric Ruijsenaars-Schneider model and Macdonald polynomials, respectively. I will show that the Mapping Class … Read More