Mathematical Physics

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

We will begin by reviewing the work of Kazhdan and Lusztig, who established an equivalence between certain affine Lie algebra representations and quantum group representations. It can be thought of as a logarithmic version of the CS-WZW correspondence. In a joint work with Lin Chen, we used factorization algebras to establish an extended version of this equivalence. We will explain the main structure of the proof, and draw some connections with Kazhdan and Lusztig's original proof. The key idea is that of Kac-Moody localization, a parametrized version of factorization homology. Time permitting, we will also explain how this idea plays a role in the recently announced proof (by Gaitsgory et al.) of the de Rham geometric Langlands conjecture.

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Zoom link https://pitp.zoom.us/j/98932564942?pwd=WWdMLzI5WktkZnMrQmplU0J3Mk43dz09