Mathematical Physics

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

The Monster Lie Algebra $\mathfrak m$ has two well-known avatars: It is a Borcherds' algebra that is also a quotient of the physical space of a specific tensor product of vertex algebras. In this talk, I will discuss a construction of vertex algebra elements that project to bases for subalgebras of $\mathfrak m$ isomorphic to $\mathfrak{gl}_2$, corresponding to each of the imaginary simple roots of the Monster Lie algebra.

Furthermore, for a fixed imaginary simple root, I will illustrate how the action of the Monster simple group on the Moonshine module induces an action of the Monster group on the set of the $\mathfrak{gl}_2$ subalgebras constructed this way. I will discuss this action and related open questions.

This talk is based on joint work with Darlayne Addabbo, Lisa Carbone, Elizabeth Jurisich, and Scott H. Murray.

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Zoom link TBA

W-algebras are a family of vertex algebras obtained as Hamiltonian reductions of affine vertex algebras parametrized by nilpotent orbits. The W-algebras associated with regular nilpotent orbits enjoy the Feigin-Frenkel duality. More recently, Gaiotto and Rap\v{c}\'ak generalize this result to hook-type W-algebras with the triality for vertex algebras at the corner. In this talk, I will present the correspondence of representation categories for the hook-type W-superalgebras and how to gain general W-algebras in type A from hook-type W-algebras. The talk is based on joint works with my collaborators.

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Zoom link https://pitp.zoom.us/j/92163414611?pwd=a1A5NHUrbEpxUUVuS3pEd1VYQk5kdz09