Conference

We identify infinitely many non-invertible generalized global symmetries in QED and QCD for the real world in the massless limit. In QED, while there is no conserved Noether current for the axial symmetry because of the ABJ anomaly, for every rational angle, we construct a conserved and gauge-invariant topological symmetry operator. Intuitively, it is a composition of the axial rotation and a fractional quantum Hall state coupled to the electromagnetic U(1) gauge field. These conserved symmetry operators do not obey a group multiplication law, but a non-invertible fusion algebra over TQFT coefficients. These non-invertible symmetries lead to selection rules, which are consistent with the scattering amplitudes in QED. We further generalize our construction to QCD, and show that the neutral pion decay can be understood from a matching condition of the non-invertible global symmetry.

I'll discuss a vertex algebra whose correlators are scattering amplitudes (and form factors) of self-dual Yang-Mills theory, for certain gauge groups and matter. The vertex algebra is a kind of vertex quantum group, and is a cousin of the affine Yangian. This is joint work with Natalie Paquette.

"Motivated by applications to soft supersymmetry breaking, we revisit the Seiberg-Witten solution for N=2 super Yang-Mills theory in four dimensions with gauge group SU(N). We present a simple exact Taylor series expansion for the periods obtained at the origin of moduli space, thereby generalizing earlier results for SU(2) and SU(3). With the help of these analytic results and others, we analyze the global structure of the Kahler potential, presenting evidence for a conjecture that the unique global minimum is the curve at the origin of moduli space. Two applications of these results are considered. Firstly, we analyze candidate walls of marginal stability of BPS states on special slices for which the expansions of the periods simplify. Secondly, we consider soft supersymmetry breaking of the N=2 theory to non-supersymmetric four-dimensional SU(N) gauge theory with two massless adjoint Weyl fermions (""adjoint QCD""). The Seiberg-Witten Kahler potential and strong coupling spectrum play a crucial role in this analysis, which ultimately leads to an exploration of the adjoint QCD phase diagram."

This will be an introductory discussion of our joint work with Roman Bezrukavnikov. Given a symplectic resolution X, one may study its Gromov-Witten theory and the monodromy group of the curve-counting functions in the K\"ahler variables. There is also a large group of derived autoequivalences of X coming from its quantization in large prime characteristic, as studied by Bezrukavnikov and collaborators. Conjecturally, the action of the latter group on K(X) is identified with the former group, and we prove this for many $X$.

In this talk I will review some recent progress in the study of non-invertible symmetries in dimensions d>2. After introducing known constructions and describing how they lead to constraints on RG flows, I will discuss how non-invertible symmetries can also be used to obtain new RG flows. This involves the notion of ``non-invertible twisted compactification,” which can be used to construct e.g. novel 3d N=6 theories from 4d N=4 SYM. Finally, I will describe upcoming work in which we give a partial criterion for determining whether a given non-invertible symmetry is ``intrinsically non-invertible”, or whether it can be recast (in an appropriate sense) as a standard invertible symmetry with an anomaly.

The Thirring Model is a covariant quantum field theory of interacting fermions, sharing many features in common with effective theories of two-dimensional electronic systems with linear dispersion such as graphene. For a small number of flavors and sufficiently strong interactions the ground state may be disrupted by condensation of particle- hole pairs leading to a quantum critical point. With no small dimensionless parameters in play in this regime the Thirring model is plausibly the simplest theory of fermions requiring a numerical solution. I will review what is currently known focussing on recent results and challenges from simulations employing Domain Wall Fermions, a formulation drawn from state-of-the-art lattice QCD, to faithfully capture the underlying symmetries at the critical point.

In the last few years the concept of symmetry has been significantly expanded. One exotic example of the generalized symmetries, is the “type-II subsystem symmetry”, where the conserved charge is defined on a fractal sublattice of an ordinary lattice. In this talk we will discuss examples of models with the fractal symmetries. In particular, we will introduce a quantum many-body model with a “Pascal’s triangle symmetry”, which is an infinite series of fractal symmetries, including the better known Sierpinski-triangle fractal symmetry. We will also construct a gapless multicritical point with the Pascal’s triangle symmetry, where the generator of all the fractal symmetries decay with a power-law. If time permits, we will also mention a few potential experimental realizations for models with fractal symmetries.

In frustrated magnets, novel phases characterized by fractionalized excitations and emergent gauge fields can occur. A paradigmatic example is given by the Kitaev model of localized spins 1/2 on the honeycomb lattice, which realizes an exactly solvable quantum spin liquid ground state with Majorana fermions as low-energy excitations. I will demonstrate that the Kitaev solution can be generalized to systems with spin and orbital degrees of freedom. The phase diagrams of these Kitaev-Kugel-Khomskii spin-orbital magnets feature a variety of novel phases, including different types of quantum liquids, as well as conventional and unconventional long-range-ordered phases, and interesting phase transitions in between. In particular, I will discuss the example of a continuous quantum phase transition between a Kitaev spin-orbital liquid and a symmetry-broken phase. This transition can be understood as a realization of a fractionalized fermionic quantum critical point.