Mathematical Physics

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.

Identifying which boundary conditions preserve integrability in two-dimensional sigma-models is a basic structural question, relevant from open string dynamics to impurity problems.  I will discuss a simple analytic approach that determines integrable boundary conditions through the divisor structure of the Lax connection, giving a generalisation of previous constructions.  As an application, I will treat open strings on AdS3 x S3 with mixed NSNS-RR flux, where the standard approaches cease to work at generic flux, and show how the background supports integrable D-branes that wrap twisted conjugacy classes in SL(2,R)xSU(2).  If time permits I will discuss this example and construction from the point of view of 4d Chern-Simons theory.

I will explain a recent joint work with A. Shapiro in which we prove that the quantized K-theoretic BFN Coulomb branch ring associated to a quiver gauge theory is isomorphic to a quantum upper cluster algebra, when the gauge theory quiver is without 1-cycles. The new ingredient needed to upgrade our earlier partial results in this direction is a precise description of the image of the Coulomb ring for gauge group G inside the localized equivariant K-theory of the affine Grassmannian Gr_T, where T \subset G is the maximal torus. This description combines conditions on residues at possible simple poles at shifted root hyperplanes with divisibility conditions controlled by the representation N determining the matter content of the theory.

Given a vertex operator algebra V, one can define sheaves of conformal blocks on moduli spaces of curves following constructions of Ben-Zvi--Frenkel and Damiolini--Gibney--Tarasca. When V is strongly rational, these sheaves are vector bundles equipped with a projectively flat connection. In this talk, I will explain how these bundles satisfy the compatibility conditions required to form a modular functor. A key consequence of this result is that the category C_V of admissible V-modules is a modular fusion category. This provides a purely algebro-geometric construction of the tensor product on C_V, which is expected to agree with the tensor product defined by Huang and Lepowsky using analytic methods. Time permitting, I will discuss open questions concerning extensions beyond the rational setting. This is joint work with Lukas Woike.

In recent years, non-onsite symmetry representations on the
lattice have enabled new insights in both anomalous and non-invertible
symmetries.  In (1+1)D, these can be represented as matrix product
operators (MPO), a type of tensor network that captures the correlated
action on neighbouring sites.  In this talk, I will present the
underlying mathematical structures that govern these symmetries, and
show that these MPOs provide the lattice representation theory of
generalised symmetries in (1+1)D.  Time permitting, I will discuss some
applications of these ideas, in particular how they can be leveraged to
improve computational performance in state of the art in tensor network
algorithms.

Based on arXiv:2509.03600, arXiv:2408.06334