Mathematical Physics

Given a semisimple group G and a smooth projective curve X over an algebraically closed field of arbitrary characteristic, let Bun_G(X) denote the moduli space of principal G-bundles over X. For a bundle P without infinitesimal symmetries, we describe the n^th order divided-power infinitesimal jet spaces of Bun_G(X) at P for each n. The description is in terms of differential forms on X^n with logarithmic singularities along the diagonals. Furthermore, we show the pullback of these differential forms to the Fulton-Macpherson compactification space is an isomorphism, thus illustrating a connection between infinitesimal jet spaces of Bun_G(X) and the Lie operad.

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A 2-group is a categorical generalization of a group: it's a category with a multiplication operation which satisfies the usual group axioms only up to coherent isomorphisms. The isomorphism classes of its objects form an ordinary group, G. Given a 2-group G with underlying group G, we can similarly define a categorical generalization of the notion of principal bundles over a manifold (or stack) X, and obtain a bicategory Bun_G(X), living over the category Bun_G(X) of ordinary G-bundles on X. For G finite and X a Riemann surface, we prove that this gives a categorification of the Freed--Quinn line bundle, a mapping-class group equivariant line bundle on Bun_G(X) which plays an important role in Dijkgraaf--Witten theory (i.e. Chern--Simons theory for the finite group G). This talk is based on joint work with Daniel Berwick-Evans, Laura Murray, Apurva Nakade, and Emma Phillips.

I will not assume previous knowledge of 2-groups: I will provide a quick overview in the main talk, as well as a more detailed discussion during a pre-talk on Tuesday. 

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We will discuss work, joint with Victor Ginzburg, on the quantization (non-commutative deformation) of the Ngô morphism, a morphism of group schemes which plays a key role in Ngô’s proof of the fundamental lemma in the Langlands program. We will also discuss how the tools used to construct this morphism can be used to prove conjectures of Ben-Zvi—Gunningham, which predict that this morphism gives “spectral decomposition” of DG categories with an action of a reductive group over the coarse quotient of a maximal Cartan subalgebra by the affine Weyl group.

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The quantum difference equation (qde) is the $q$-difference equation which is proposed by Okounkov and Smirnov to encode the $K$-theoretic twisted quasimap counting for the Nakajima quiver varieties. In this talk, we will give a direct quantum toroidal algebra $U_{q,t}(\hat{\hat{\mf{sl}}}_{n})$ construction for the qde of the affine type $A$ quiver varieties. We will show that there is a really explicit and concise formula for the quantum difference operators. Moreover we will show that the degeneration limit of the quantum difference equation is equivalent to the Dubrovin connection for the quantum cohomology of the affine type A quiver varieties, which will give the description of the monodromy representation of the Dubrovin connection via the monodromy operators in the quantum difference equation.

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Integrable field theories in two dimensions are known to originate as defect theories of 4d Chern-Simons theory and as symmetry reductions of the 4d anti-self-dual Yang-Mills equations. Based on ideas of Costello, it has been proposed in work of Bittleston and Skinner that these two approaches can be unified starting from holomorphic Chern-Simons theory in 6 dimensions. In this talk I will introduce the first complete description of this diamond of integrable theories for a family of deformed sigma models, going beyond the Dirichlet boundary conditions that have been considered thus far. The talk is based on the recent work https://arxiv.org/abs/2311.17551.

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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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