Mathematical Physics

For a simple Lie algebra \mathfrak{g}, Kazhdan-Lusztig correspondence states that for certain values of the level k, there is an equivalence between two braided tensor categories: the category of modules of the affine Lie algebra of \mathfrak{g} at level k and the category of modules of the quantum group of \mathfrak{g} at q=e^{\pi i/k}. I will report on recent work to appear with T. Creutzig and T. Dimofte proving such a statement for a class of Lie superalgebras. These Lie superalgebras and their affine VOAs arise from the study of boundary conditions in 3d \mathcal{N}=4 abelian gauge theories. I will also explain how the corresponding supergroups act on the category of matrix factorizations.

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Zoom link https://pitp.zoom.us/j/92338197336?pwd=OG40V2p6V0FmalNCZnV2ZFF1NHAzZz09

I will introduce ($q$-)opers on a projective line in the presence of twists and singularities and will discuss the space of such opers. We will see how Bethe Ansatz equations for quantum spin chains and energy level equations of classical soluble models of Calogero-Ruijsenaars type naturally appear from the oper construction. Both can also be described in terms of so-called $QQ$-systems, which have their origins in algebra and representation theory. Our construction is universal and works for any simple, simply-connected complex Lie group $G$.

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Zoom link https://pitp.zoom.us/j/94393767928?pwd=Qkg2eG5CME5sQjNEV2lDVVNyMktDQT09

Abstract TBA

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Zoom link: https://pitp.zoom.us/j/91816652867?pwd=bnQzYzA0SXBuL3BTTmNUMFBMcEFWdz09

We study the quantum connection in positive characteristic for conical symplectic resolutions. We conjecture the equivalence of the p-curvature of such connections with (equivariant generalizations of) quantum Steenrod operations of Fukaya and Wilkins, which are endomorphisms of mod p quantum cohomology deforming the Steenrod operations. The conjecture is verified in a wide range of examples, including the Springer resolution, thereby providing a geometric interpretation of the p-curvature and a full computation of quantum Steenrod operations. The key ingredients are a new compatibility relation between the quantum Steenrod operations and the shift operators, and structural results for the mod p quantum connection recently obtained by Etingof--Varchenko.

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Zoom link: https://pitp.zoom.us/j/91010341249?pwd=QXJuMlJrWWd0dHpPdUpDUGVqVmYvZz09

In quantum field theories, locality and unitarity are essential properties. For functorial field theories, locality is manifested through compatibility with cutting and gluing of manifolds, which can be fully encoded in the definition of fully extended functorial field theories. However, unitarity or reflection positivity (its Euclidean version) has so far only been defined for non-extended or invertible field theories. In this talk, I will address the challenge of defining reflection positivity for extended topological field theories, proposing a definition based on a version of higher dagger categories.

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Zoom link: https://pitp.zoom.us/j/93806769415?pwd=TVJFbWlJK0JyZCtjbHMvOWYxVUlRUT09

In this talk I will explain my works with Y. Tachikawa to study anomaly in heterotic string theory via homotopy theory, especially the theory of Topological Modular Forms (TMF). TMF is an E-infinity ring spectrum which is conjectured by Stolz-Teichner to classify two-dimensional supersymmetric quantum field theories in physics. In the previous work (https://arxiv.org/abs/2108.13542), we proved the vanishing of anomalies in heterotic string theory mathematically by using TMF. Furthermore, we have a recent update (https://arxiv.org/abs/2305.06196) on the previous work. Because of the vanishing result, we can consider a secondary transformation of spectra, which is shown to coincide with the Anderson self-duality morphism of TMF. This allows us to detect subtle torsion phenomena in TMF by differential-geometric ways, and leads us to new conjectures on the relation between VOAs and TMF.

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Zoom link: https://pitp.zoom.us/j/95550331572?pwd=OW1oYlBvUWVxaGNJRWl5aHVrS0pJZz09

A consequence of the works of Costello and Lurie is that the Hochschild chain complex of a Calabi-Yau category admit the structure of a framed E_2 algebra (the genus zero operations). I will describe a new algebraic point of view on these operations which admits generalizations to the setting of relative Calabi-Yau structures, which do not seem to fit into the framework of TQFTs. In particular, we obtain a generalization of string topology to manifolds with boundary, as well as interesting operations on Hochschild homology of Fano varieties. Time permitting, I will explain some applications to quiver varieties. This is joint work with Christopher Brav.

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Zoom link: https://pitp.zoom.us/j/97363589637?pwd=T25YS0ZYQlBSWm5maXhQTklpSm50UT09

Unimodularity is a classical notion shows up in various fields like linear algebra, lattices, Poisson algebras, etc. In this talk, we focus on unimodular Hopf algebras and unimodular tensor categories. We will introduce unimodular module categories and use them to construct Frobenius algebras and unimodular tensor categories. These ideas will be illustrated with examples drawn from Hopf algebras.

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Zoom link https://pitp.zoom.us/j/98477599322?pwd=UDJmTklMTGxGODJZTm1Xc1VhL2tDdz09

A tower is an infinite sequence of deloopings of symmetric monoidal ever-higher categories. Towers are places where extended functorial field theories take values. Towers are a "deeper" version of commutative rings (as opposed to "higher rings" aka E∞-spectra). Notably, towers have their own opinions about Galois theory, and think that usual Galois groups are merely shallow approximations of deeper homotopical objects. In this talk, I will describe some steps in the construction and calculation of the deeper Galois group of a characteristic-zero field. In particular, I'll explain a homotopical version of the Kummer description of abelian extensions. This is joint work in progress with David Reutter.

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Zoom link: https://pitp.zoom.us/j/97950701035?pwd=Wk9FRSt2MkN3eWptTVltRVJncnFHdz09

In this talk, I will consider the centralizer of the quantum group U_q(sl_2) in the tensor product of three identical spin representations. The case of spin 1/2 (fundamental representation) is understood within the framework of the Schur-Weyl duality for U_q(sl_N), and the centralizer is known to be isomorphic to a Temperley-Lieb algebra. The case of spin 1 has also been studied and corresponds to the Birman-Murakami-Wenzl algebra. For a general spin, I will explain how to describe explicitly the centralizer (by generators and relations) using a combination of the braid group algebra and the Askey-Wilson algebra, which has been introduced in the context of orthogonal polynomials.

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Zoom link: https://pitp.zoom.us/j/98471794356?pwd=NjZFdjRFaDFON05HNkdTZS9hZTUvQT09

I will describe a correspondence between hyperkähler/quaternion-Kähler geometry and two dimensional chiral algebras that arises from twistor theory. As an application, I will explain how to use correlators of these chiral algebras to compute gravitational scattering amplitudes in four dimensional general relativity.

Zoom Link: https://pitp.zoom.us/j/98706048161?pwd=a3k1U3l2UXJlUjlqUElkOEZhZGwwUT09

In recent work, Aganagic proposed a categorification of quantum link invariants based on a category of A-branes, which is solvable explicitly. The theory is a generalization of Heegaard-Floer theory from gl(1|1) to arbitrary Lie algebras. I will describe in the detail the two simplest cases: the su(2) theory, categorifying the Jones polynomial, and the gl(1|1) theory, categorifying the Alexander polynomial. I will give an explicit algorithm for computing link homologies in these cases. I will also briefly describe the generalization to other simple Lie algebras and to Lie superalgebras of type gl(m|n). This talk is based on work to appear with Mina Aganagic and Miroslav Rapcak.

Zoom Link: TBD