Mathematical Physics

 In the 90s Turaev, Viro, Barrett and Westbury constructed a (2+1)D state sum TQFT associated to any fusion category. The associated phases of matter were popularized by Kiteav, Levin and Wen and are now central  examples in condensed matter physics and quantum information theory.

Despite the importance of these phases, many of the computational techniques for working with fusion categories have not percolated into condensed matter physics. Many of these techniques are "folk theorems"
and have not appeared in the literature in a digestible form.

Jacob Bridgeman and myself have spent the last two years extracting these computational techniques from Corey Jones, and have been using them to understand defects in Levin-Wen phases. Our papers 1806.01279,
1810.09469, 1901.08069, 1907.06692 document the development of our understanding, and demonstrate how to do physically relevant fusion category computations.

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. In arbitrary type, we can establish an infinite family of cases where it holds. Time permitting, we'll indicate how the new conjecture relates to P = W phenomena in nonabelian Hodge theory.

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. The precise values of these integrals are currently unknown.  I will describe recent joint work with Banks and Panzer, in which we develop a theory of integration on these moduli spaces via suitable sheaves of polylogarithms, and use it to prove that Kontsevich's integrals evaluate to integer-linear combinations of special transcendental constants called multiple zeta values, yielding the first algorithm for their calculation.

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. This talk will mainly be based on joint work with A.B. Goncharov.

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. In the simple case of Kähler quantisation of a symplectic vector space, one may consider the space of all Kähler structures, and organise the corresponding Hilbert spaces to form an infinite-rank vector bundle on it, so the quantisation of symplectic transformations may be understood in terms of their action on this bundle.
In this talk I shall consider a variation of this construction for the case of a hyper-Kähler vector space V, in which case the symplectic form and the pre-quantum line bundle are allowed to change together with the complex structure. In a joint work with Andersen and Rembado, we construct a bundle of Hilbert spaces on the sphere of Kähler structures, on which a flat connection is naturally defined. The group of isometries of V which preserve this sphere globally also acts on the bundle, and the construction determines a quantisation of this group. Time permitting, I shall discuss our main application of this construction—the quantisation of the circle action on the moduli space of reductive Higgs bundles over an elliptic curve.

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. Time permitting, I will sketch how this approach is roughly dual to AQFT, and/or how to define vacua in terms of the DLR equations. I am also hoping for feedback on how naive this approach really is in the light of renormalization.

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 is joint work with Jon Brundan and Ben Webster.

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. I investigate 3d $\cN=4$ mirror symmetry in this context, which implies an equality of enumerative invariants associated to mirror pairs of Higgs branches under the exchange of equivariant and degree counting parameters. I will conclude with some remarks about how holomorphic Morse theory can be used to access the spaces of supersymmetric ground states in limits where $\cN=4$ supersymmetry is fully restored. These spaces of ground states may be related to the spaces of conformal blocks for the VOAs introduced by Costello and Gaiotto.

I will review the geometric approach to the description of Coulomb branches and Chern-Simons terms of gauge theories coming from compactifications of M-theory on elliptically fibered Calabi-Yau threefolds. Mathematically, this involves finding all the crepant resolutions of a given Weierstrass model and understanding the network of flops connecting them together with computing certain topological invariants. I will further check that the uplifted theory in 6d is anomaly-free using Green-Schwartz mechanism. I will give examples on the theory with semi-simple gauge groups such as SU(2)xG2, which plays a major role in the classification of 6d superconformal theories, and SU(2)xSU(3), which describes the non-abelian sector of the standard model.