Abstract: Given a polarized hyperplane arrangement, Braden-Licata-Proudfoot-Webster defined two convolution algebras: A & B. We show that both of them can be realized as Fukaya categories of hypertoric varieties. This proves a conjecture of Braden-Licata-Phan-Proudfoot-Webster in 2009 and gives a geometric interpretation of the Koszul duality between A & B. The proof relies on the construction of non-commutative vector fields … Read More
Abstract: In 2013, Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees constructed a correspondence between four dimensional N=2 SCFTs, a certain kind of supersymmetric quantum field theory, and vertex algebras. When applied to the theories of class S, one obtains a rich family of vertex algebras which furnish novel representations of critical level, affine Kac-Moody algebras. Moreover, these vertex algebras satisfy an intricate set of gluing … Read More
Abstract: By now it is known that many interesting phenomena in geometry and representation theory can be understood as aspects of mirror symmetry of 3d N=4 SUSY QFTs. Such a QFT is associated to a hyper-Kähler manifold X equipped with a hyper-Hamiltonian action of a compact Lie group G and admits two topological twists. The first twist, which is known as … Read More
We study the quantum connection in positive characteristic for conical symplectic resolutions. We conjecture the equivalence of the p-curvature of such connections with (equivariant generalizations of) quantum Steenrod operations of Fukaya and Wilkins. The conjecture is verified in a wide range of examples, including the Springer resolution, thereby providing a geometric interpretation of the p-curvature and a full computation of … Read More
Abstract: Two-dimensional chiral/holomorphic quantum field theories serve as an arena where exact algebraic techniques offer a great deal of control over the spectrum of the theory. The underlying algebraic structure, known as a vertex algebra or vertex operator algebra, has seen countless applications ranging from superstring theory and pure mathematics to statistical and condensed matter physics. In this talk I … Read More
Abstract: Fixed point Floer cohomology of a symplectic automorphism categorifies the Lefschetz trace formula. On the categorical side, twisted Hochschild homology of an automorphism on an A infinity category can be understood as a categorical trace. I will explain how the twisted open-closed maps are used to relate these two invariants in the setting of Fukaya categories of Landau-Ginzburg models. … Read More
ABSTRACT: In its simplest incarnation, the geometric Langlands program was defined by Beilinson and Drinfeld in the 90’s as relating, on one side, a flat connection on a Riemann surface, and on the other side, a more sophisticated structure known as a D-module. A recent generalization of the correspondence, due to Aganagic-Frenkel-Okounkov, establishes an isomorphism between q-deformed versions of conformal … Read More
Abstract: Like many physical objects, under smallperturbations, black holes possess the property of vibrating at discretecharacteristic frequencies (known as QNM frequencies). They are complex anddepend on the type of black hole and the boundary conditions imposed on theperturbation. As we anticipate “hearing” these frequenciesthrough new experiments more distinctly, it becomes essential to understandtheir mathematical structure better or, in other words, … Read More
Abstract: We discuss joint work with Tom Gannon, showing that the algebra $D(SL_n/U)$ of differential operators on the base affine space of $SL_n$ is the quantized Coulomb branch of a certain 3d $\mathcal{N} = 4$ quiver gauge theory. In the semiclassical limit this confirms a conjecture of Dancer-Hanany-Kirwan on the universal hyperk\”ahler implosion of $SL_n$. In fact, we prove a … Read More
Abstract: Mirror symmetry for 3d N=4 SUSY QFTs has recently received much attention in geometry and representation theory. Theories within this class give rise to interesting moduli spaces of vacua, whose most relevant components are called the Higgs and Coulomb branches. Nakajima initiated the mathematical study of Higgs branches in the 90s; since then, their geometry has been pivotal in … Read More