Mathematical Physics

Recent work by Davide Gaiotto and collaborators introduced a new type of parametric Feynman integrals to compute BRST anomalies in topological and holomorphic quantum field theories. The integrand of these integrals is a certain differential form in Schwinger parameters. In a new article together with Simone Hu, we showed that this "topological" differential form coincides with a "Pfaffian" differential form that had been used by Brown, Panzer, and Hu, to compute cohomology of the odd graph complex and of the linear group. In my talk, I will review some aspects of the graph complex and the role played by the Pfaffian form there, sketch the proof of equivalence, and comment on various observations on either side of the equivalence and their natural counterparts on the other side.

In this talk, I will discuss a birational realization of King's conjecture which is indeed true, and its connections with noncommutative algebraic geometry and mirror symmetry. In particular, I will also establish the A-side analog of this result using constructible sheaves and promote the celebrated Coherent-Constructible Correspondence to a global family. The talk is based on recent joint work https://arxiv.org/abs/2501.00130 and work in preparation with David Favero.

In 2010, Kontsevich and Soibelman defined Cohomological Hall Algebras for quivers and potential as a mathematical construction of the algebra of BPS states. These algebras are modeled on the cohomology of vanishing cycles, which makes these algebras particularly hard to study but often result in interesting algebraic structures. A deformation of a particular case of them gives rise to a positive half of Maulik-Okounkov Yangians. The goal of my talk is to give an introduction to these ideas and explain how for the case of tripled cyclic quiver with canonical cubic potential, this algebra turns out to be one-half of the universal enveloping algebra of the Lie algebra of matrix differential operators on the torus, while its deformation turn out be one half of an explicit integral form of the Affine Yangian of gl(n).

Vertex operator algebras (VOAs) arise in many corners of supersymmetric quantum field theory. One particularly influential instance is in 4d N=2 superconformal field theories, whereby the VOA is realized as the cohomology of a suitable supercharge. Unitarity of the underlying SCFT imposes strong constraints on the structure of the resulting VOA. In this talk, I will describe one aspect of how the unitary of the underlying SCFT constrains this VOA: in the context of superconformal gauge theories, the resulting BRST complex shares a striking resemblance to the de Rham complex of a compact Kähler manifolds. I will finish with several consequences of this observation, e.g. the formality of these BRST complexes as in the work of Deligne-Griffiths-Morgan-Sullivan on compact Kähler manifold. This is based on work in progress with C. Beem.

In this talk, I will present a sewing-factorization theorem for conformal blocks in arbitrary genus associated to a (possibly nonrational) $C_2$-cofinite VOA $V$. This result gives a higher genus analog of Huang-Lepowsky-Zhang's tensor product theory. Moreover, I will explain the relation between our result and pseudotraces, and confirm some of the conjectures by Gainuditnov-Runkel. The relationship between our result and coends will also be discussed. The talk is based on an ongoing project (arXiv: 2305.10180, 2411.07707) joint with Bin Gui.

After an introduction to the notion of Leonard pairs, I explain their different uses. Then, I provide the definition of the associated Heun operator and how it allows us to simplify the computation of the quantum entanglement entropy. Finally, I show, in the simplest example, how the Bethe ansatz can be used to diagonalize the Heun operator.

In this talk, I will present the construction of the Yangian of a cotangent Lie algebra from the geometry of the equivariant affine Grassmannian. Furthermore, I will discuss how this quantum group can be dynamically twisted to a quantum groupoid over a neighborhood in the moduli space of G-bundles over a compact Riemann surface. These constructions are motivated by relations between a certain holomorphic-topological 4d gauge field theory and the geometric Langlands correspondence. Representations of the Yangian are pertubative line operators of said theory, while the dynamical twist of the Yangian controls the action of these operators via Hecke modifications in this setting. This talk is based on the two joint works arXiv:2405.19906 and arXiv:2411.05068 with Wenjun Niu.