CMSA

Residues and homotopy Lie algebras
2025-05-08 10:00 - 11:00 (America/New_York) | Virtual
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Zhenping Gui, Shanghai Institute for Mathematics and Interdisciplinary Sciences
Title: Residues and homotopy Lie algebras
Abstract: I will introduce the notion of a chiral operad for any compact Riemann surface. This operad consists of compositions of residue operations, which give rise to the Chevalley-Cousin complex and lead to the definition of chiral homology (derived conformal blocks). I will explain how to use this machinery to rigorously define certain Feynman integrals in 2D chiral CFTs. Subsequently, I will present a polysimplicial construction of a series of chain models for the configuration space of points in an affine space and study residue operations. These residue operations can be described by a homotopy Lie algebra structure, and the latter defines a higher-dimensional analog of the Chevalley-Cousin complex. This is based on joint work in progress with Charles Young and Laura Felder.

Resurgence, number theory, and quantum mirror curves
2025-05-15 10:00 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Claudia Rella (IHES)
Title: Resurgence, number theory, and quantum mirror curves
Abstract: Resurgence provides a powerful toolbox to access the non-perturbative sectors hidden within the divergent asymptotic series of quantum theories. Under some special assumptions, the non-perturbative data extracted via resurgent methods possess intrinsic number-theoretic properties that are deeply rooted in the symmetries and arithmetic of the geometry underlying the quantum theory. The framework of modular resurgence aims to formalise this observation. In this talk, after introducing the basics of modular resurgence, I will consider the TS/ST correspondence for toric Calabi-Yau threefolds and focus on the fermionic spectral traces of quantum mirror curves. Here, a complete realisation of the modular resurgence paradigm is found in the spectral theory of local P^2—where the bridge between non-perturbative physics and the arithmetic properties of the geometry takes the form of an exact strong-weak symmetry—and is now being generalised to all local weighted projective spaces. This talk is based on arXiv:2212.10606, 2404.10695, 2404.11550, and work in progress.

Higher Gauge Theory and Integrability
2025-05-22 10:00 - 11:00 | Virtual
Mathematical Physics and Algebraic Geometry Seminar
Speaker: Joaquin Liniado, Instituto de Física La Plata
Title: Higher Gauge Theory and Integrability
Abstract: Integrable field theories are remarkable for possessing an infinite number of conserved quantities, which often allow for their exact solvability. In two dimensions, this structure is elegantly captured by the existence of a Lax connection, whose path ordered exponential allows for the systematic construction of an infinite number of conserved quantities. In 2019, Costello, Witten and Yamazaki introduced a four-dimensional holomorphic extension of Chern-Simons theory that provides the first attempt at explaining the appearance of the Lax connection, whose origin had remained somewhat mysterious until then.
In this talk, we present a generalization of these ideas to three-dimensional field theories, guided by the so-called “categorical ladder = dimensional ladder” principle. The central idea is that conserved quantities arise from surface-ordered exponentials of higher-rank tensors, defining a higher categorical notion of the Lax connection. We show that such a structure naturally emerges from a five-dimensional holomorphic extension of higher Chern-Simons theory. This work, carried out in collaboration with Hank Chen, provides a framework that enables the systematic construction of integrable field theories in three dimensions.
