Mathematical Physics

In this talk I will present an update on my work with Ben Gammage and Aaron Mazel-Gee on the 2-categories of boundary conditions in the A and B-twists. In particular I will explain how 2-categorical 3d mirror symmetry decategorifies to the Koszul duality of hypertoric categories O discovered by Braden-Licata-Proudfoot-Webster.

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One way of constructing mirror partners to Fano varieties is via toric degenerations. The case in which this is best understood is the Grassmannian, using the well-known SAGBI basis of the Plucker coordinate ring indexed by semi-standard Young tableaux (SSYT). The mirror construction goes back to work of Eguchi—Hori—Xiong, however its geometry and combinatorics still plays an important role in current mirror constructions. In this talk, I will give an overview of this story, then turn to the question of what can be generalized for Kronecker moduli spaces. Like Grassmannians (which they generalize), Kronecker moduli spaces are high Fano index Picard rank 1 smooth Fano varieties. I will introduce linked SSYT pairs, which play the analogous role of SSYT for Grassmannians in understanding the coordinate ring of the Kronecker moduli space. This is joint work with Liana Heuberger.

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After briefly reviewing the dictionary between branes of Type IIB string theory and representations of the simplest quantum toroidal algebra, I will present a new class of brane setups which I call spiralling branes. Partition functions of these remarkable configurations reproduce K-theoretic vertex partition function, while on the algebraic side they correspond to Drinfeld twists of R-matrices. Time permitting, I will also discuss other applications of spiralling branes.

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The prototypical example of a symmetric tensor category is Rep(G) for G a compact group. The other main source of symmetric tensor categories are Deligne's interpolation categories (S_t, GL_t, and O_t) which extend a family of group representation categories defined at positive integers to all complex numbers. Harman and Snowden have developed a theory of symmetric tensor categories coming from Oligomorphic permutation groups together with a well-behaved measure on G-sets. This includes Deligne's S_t which comes from the infinite symmetric group together with a measure where the usual permutation G-set has volume t. The simplest new example of their theory is the group of order-preserving bijections of the real line with the measure given by Euler characteristic. In joint work with Harman and Snowden (arXiv:2211.15392) we give a detailed description of this new symmetric tensor category, which has a number of novel properties. We call this category the Delannoy category because the dimensions of Hom spaces are given by Delannoy numbers. In this talk I'll outline our main results, including the classification of simple objects, the tensor product rules, and a combinatorial model for the category using Delannoy paths.

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This talk explains the theory of "factorisation" or "vertex algebra" analogues of the theory of quantum groups, (2312.07274).

We will first recap the theory of quantum groups and their connections to wider mathematics, due to Drinfeld, Etingof, Kazhdan, Jimbo, Reshetikhin, Turaev, and many others. We will then explain how to give a natural definition of "factorisation" version of these objects, give the basic structure theory about their categories of representations, and derive an explicit model for them. This relates to previous work of Etingof-Kazhdan and Frenkel-Reshetikhin.

We will sketch some physics heuristics for the above theory; briefly, our structures should appear on the category of line operators in 4d theories due to Costello, Witten and Yamazaki, e.g. on the representations Rep U_q(g^) of affine quantum groups.

We will then give some examples. The last part discusses the conjectures that naturally suggest themselves, for instance by analogy to the connections referred to above.

Prerequisites on chiral and vertex algebras will be given at a talk earlier in the week. (1-2pm Tuesday, Bob Room)

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Brane Webs in Type IIB String Theory can be used to engineer 5d N=1 gauge theories and SCFTs. These brane webs can also be used to obtain so called magnetic quivers, whose 3d Coulomb branches gives the Higgs branch of the 5d theory in question. We will see how to obtain these magnetic quivers for brane webs possibly in the presence of orientifold planes. Then we will see how 3d Coulomb branch technology (such as quiver subtraction) can be used to understand Higgsings and mass deformations of 5d SCFTs.

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The Hitchin systems are a remarkable family of integrable models associated to the moduli space of principal bundles on a compact Riemann surface. In this talk, I explain how the loop group uniformization of this moduli space can be used to construct an r-matrix for the Hitchin systems. This r-matrix has been previously used in the description of the Friedan-Schenker connection on the space of conformal blocks.

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For a noncommutative algebra $A$ and an antilinear automorphism $\rho$ of $A$ there is a notion of positive trace. On the physics side, positive traces are related to quantizations of superconformal field theories. On the mathematical side, positive traces are connected to spherical unitary representations of complex Lie groups. We will discuss the classification of positive traces in the case when the algebra $A$ is a deformation or a $q$-deformation of a Kleinian singularity of type A.

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The moduli of Higgs bundles and the Hitchin fibration are central to many thriving research areas, such as mirror symmetry, non-abelian Hodge theory and the geometric Langlands program.  A group-theoretic or multiplicative version was introduced by Frenkel and Ngo in 2011 to give a geometric interpretation of orbital integrals and trace formulas from automorphic representation theory. Since then, there is an ongoing program to replicate the theory of Higgs bundles for the multiplicative case. One notable development is the study of multiplicative affine Springer fibers. Like the usual ones, they are local analogues of multiplicative Hitchin fibers. In this talk, I discuss my work in continuing this program and providing a multiplicative version of Z. Yun's global Springer theory. This involves the study of parabolic multiplicative Higgs bundles and affine Springer fibers. 

Counting partitions in diverse dimensions is a long-standing problem in enumerative combinatorics. It also plays a prominent role in the physics of instanton counting and in algebraic geometry through the computation of Donaldson-Thomas invariants. In this talk I will give an overview of these counting problems, and discuss how recent developments in the computation of instanton/Donaldson-Thomas partition functions clarify some open problems in the enumeration of higher-dimensional partitions.

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Zoom link https://pitp.zoom.us/j/92547375606?pwd=VDBiTTV6QjBtWThnSjJPc0phVEI1dz09

I will discuss an explicit way to construct Hopf algebras and quasi-triangular Hopf algebras (their Drinfeld doubles) within 3d TQFT, using extended operators on boundary conditions -- dubbed `spark' algebras. The representation categories of these algebras capture boundary and bulk line operators. Overall, the construction realizes Tannakian duality geometrically; in perturbative TQFT's, it is closely connected to the holographic Koszul duality of Costello and Paquette. I'll illustrate the construction for Dijkgraaf-Witten theory (a.k.a. gauge theory with finite gauge group), and then sketch an application to the B-type topological twist of 3d N=4 gauge theories, which initially motivated these investigations. (Work in progress with T. Creutzig and W. Niu.)

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Zoom link https://pitp.zoom.us/j/93746215441?pwd=YjdhaDNFeko3VDVKQW5ZV1MzL1cvUT09