Mathematical Physics

Poisson sigma models sit at the intersection of deformation theory, geometry, and quantum field theory; specifically, the perturbative expansion of the two-dimensional Poisson sigma model with boundary is known to recover Kontsevich’s deformation quantization formula. In this talk, I introduce a higher-dimensional holomorphic–topological generalization of Poisson sigma models and explain their connections to the deformation quantization of holomorphic–topological factorization algebras. We also give a systematic construction of extended objects, including interfaces, enriched boundaries and defects in these theories. We explain how these models relate to the construction of quantum group-like structures via Koszul duality, and explore examples related to the quantization of Lie bialgebras, W-algebras, and Yangians.

The assignment of scattering data to monopoles is a classical construction in gauge theory useful to understand the geometry of monopole moduli spaces; from the modern perspective, it is of interest because monopole scattering matrices are quasiclassical limits of R-matrices of shifted Yangians. I will describe an analogous construction of quasiclassical R-matrices of affine Yangians using moduli spaces of noncommutative instantons; this can be interpreted via defect intersections in a semiclassical limit of the 5d Chern-Simons theory. In the simplest possible case, corresponding R-matrices reproduce Miura operators, but otherwise they seem to be new. If time permits I will describe a (dual) construction of the quantization of these R-matrices using geometry of quasimap moduli spaces.

Various operations on two-dimensional quantum field theories such as the GSO projection and various dualities involve a sum over spin structures. This talk will discuss a subtle mod-16 anomaly in the spin structure, extending the usual mod-8 anomaly of fermion parity, which can obstruct this sum. As such one must account for this anomaly to define these operations properly or face surprising consequences. We will see how this phenomenon is captured cleanly by homotopy theory, specifically the non-vanishing cohomology of a sequence of cobordism groups. From there, we will discuss applications in diverse dimensions and with additional symmetries, and the corresponding SymTFT.

Semisimple categorical symmetries, and their physical implications have been studied in great detail in the continuum and on the lattice. In particular, the nice properties of semisimple categories allows one to construct hamiltonians which enjoy this symmetry, and study its symmetric gapped phases. Using the anyonic chain framework in (1+1)d, I will present a model that has a finite but non-semisimple categorical symmetry. Sacrificing Hermiticity, I will present several symmetric, frustration-free, gapped Hamiltonians with real spectra and analyze their ground state subspaces. I then discuss two intriguing phenomena. First, I will identify a smooth path of gapped symmetric Hamiltonians whose ground states transform inequivalently under the symmetry. Second, I will discuss a model where a product state and the so-called W state spontaneously break the symmetry, and propose an explanation for the indistinguishability of these two states in the infinite-volume limit in terms of the symmetry category.

The amplituhedron was introduced by Arkani-Hamed and Trnka as a geometric object which "encodes" scattering amplitudes in N=4 SYM theory. One of their motivations was the Britto-Cachazo-Feng-Witten (BCFW) recursion for amplitudes, which in particular produced many different formulas for the same amplitude. Their expectation was that the amplitude is the "volume" of the amplituhedron, and the different BCFW formulas correspond to different ways of decomposing the amplituhedron into small pieces. This expectation is in fact correct. I'll discuss some of the mathematics going into the proof, some unexpected phenomena arising along the way, and some lingering questions. This is joint work with (various subsets of) Chaim Even-Zohar, Tsviqa Lakrec, Matteo Parisi, Ran Tessler, and Lauren Williams.

The theory of quiver varieties provides a fundamental bridge between representation theory, enumerative geometry, and physics. From 3d mirror symmetry, any quiver variety comes with a dual variety known as the Coulomb branch. A conjecture proposed by Bullimore-Dimofte-Gaiotto-Hilburn-Kim and, independently, Okounkov, asserts that the cohomology of the moduli space of quasimaps to a quiver variety admits a canonical action by the quantized coordinate ring of the dual BFN Coulomb branch. In this talk, I will report on progress on refining this conjecture and proving it. The construction relies on a -1 shifted symplectic structure on the moduli space of quasimaps and the theory of cohomological Hall algebras. Based on work in preparation with Spencer Tamagni.