This series consists of talks in the area of Mathematical Physics.
In 2012, Maulik proved a conjecture of Oblomkov-Shende relating: (1) the Hilbert schemes of a plane curve (alternatively, its compactified Jacobian), (2) the HOMFLY polynomials of the links of its singularities. We recast his theorem from the viewpoint of representation theory. For a split semisimple group G with Weyl group W, we state a stronger conjecture relating two virtual modules over Lusztig's graded affine Hecke algebra, constructed from: (1) fibers of a parabolic Hitchin map, (2) generalized Bott-Samelson spaces attached to conjugacy classes in the braid group of W.
Strominger-Yau-Zaslow explained mirror symmetry via duality between tori. There have been a lot of recent developments in the SYZ program, focusing on the non-equivariant setting. In this talk, I explain an equivariant construction and apply it to toric Calabi-Yau manifolds. It has a close relation to the equivariant open GW invariants found by Aganagic-Klemm-Vafa and studied by Katz-Liu, Graber-Zaslow and many others.
In 1997, Kontsevich gave a universal solution to the "deformation quantization" problem in mathematical physics: starting from any Poisson manifold (the classical phase space), it produces a noncommutative algebra of quantum observables by deforming the ordinary multiplication of functions. His formula is a Feynma expansion, involving an infinite sum over graphs, weighted by volume integrals on the moduli space of marked holomorphic disks.
Let G be a split semi-simple algebraic group over Q. We introduce a natural cluster structure on moduli spaces of G-local systems over surfaces with marked points. As a consequence, the moduli spaces of G-local systems admit natural Poisson structures, and can be further quantized. We will study the principal series representations of such quantum spaces. It will recover many classical topics, such as the q-deformed Toda systems, quantum groups, as well as the modular functor conjecture for such representations, which should lead to new quantum invariants of threefolds.
In the context of geometric quantisation, one starts with the data of a symplectic manifold together with a pre-quantum line bundle, and obtains a quantum Hilbert space by means of the auxiliary structure of a polarisation, i.e. typically a Lagrangian foliation or a Kähler structure. One common and widely studied problem is that of quantising Hamiltonian flows which do not preserve it.
This is practice for a talk at Berkeley, so it will involve explaining stuff many people here probably already know. I'll try to summarize what I've learned about 3 dimensional N=4 supersymmetric quantum field theories, their twists and how these manifest in terms of interesting objects in mathematics. If nothing else, hopefully there will be some comedy value in my attempt.
In a synthetic approach to geometry and physics, one attempts to formulate an axiomatic system in purely logical terms, abstracting away from irrelevant "implementation details". In this talk, I will explain how intuitionistic logic and topos theory provide a synthetic theory of space, and then consider a (naive version of) Euclidean QFTs in terms of a conjectural elegant synthetic reformulation: a Euclidean QFT is nothing but a probability space in intuitionistic logic, extended by suitable modalities formalizing compact regions of space.
The Heisenberg algebra plays a vital role in many areas of mathematics and physics. We will describe a family of quantum Heisenberg categories, depending on a choice of central charge, that categorify this algebra. When the central charge is nonzero, these categories act on modules for cyclotomic quotients of the affine Hecke algebra. In central charge zero, we obtain an affinization of the HOMFLY-PT skein category, which acts on modules for $U_q(\mathfrak{gl}_n)$. We will also discuss how the categories can be generalized by adding a Frobenius superalgebra into the construction. This
Principal bundles and their moduli have been important in various aspects of physics and geometry for many decades. It is perhaps not so well-known that a substantial portion of the original motivation for studying them came from number theory, namely the study of Diophantine equations. I will describe a bit of this history and some recent developments.
I discuss a geometric interpretation of the twisted indexes of 3d (softly broken) $\cN=4$ gauge theories on $S^1 \times \Sigma$ where $\Sigma$ is a closed genus $g$ Riemann surface, mainly focussing on quivers with unitary gauge groups. The path integral localises to a moduli space of solutions to generalised vortex equations on $\Sigma$, which can be understood algebraically as quasi-maps to the Higgs branch. I demonstrate that the twisted indexes computed in previous work reproduce the virtual Euler characteristic of the moduli spaces of twisted quasi-maps.