# Workshop on Global Categorical Symmetries

## April 29, 2024 @ 9:00 am - May 3, 2024 @ 5:00 pm

The CMSA will be hosting a **Workshop on Global Categorical Symmetries** from April 29–May 3, 2024.

Participation in the workshop is by invitation.

There will be three **Symmetry Colloquia** open to the community on **Thursday, May 2, 2024.**

**Location: ** Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138

2:00 – 2:50 pm

**Speaker:** Clay Còrdova, University of Chicago

**Title:** Particle-Soliton Degeneracies from Spontaneously Broken Non-Invertible Symmetry

**Abstract:** We study non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken we show that the particle spectrum often has degeneracies dictated by the non-invertible symmetry and we deduce a procedure to determine the allowed multiplets. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. We exhibit these conclusions in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading Z2 odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum.

3:00 – 3:50 pm

**Speaker:** Thomas Dumitrescu, UCLA

**Title:** Symmetries, Invertible Field Theories, and Gauge Theory Phases

**Abstract:** I will start with a brief overview of gauge theory phases in 3+1 dimensions through the lens of higher symmetries — in particular the realization of 1-form symmetries acting on loop order parameters. I will then review recent progress in refining this characterization using invertible field theories, or equivalently symmetry protected topological phases (SPTs). This refinement leads to new results in gauge theories with fundamental matter, such as quantum chromodynamics (QCD), which do not possess 1-form symmetries. I will explain why these theories must sometimes undergo a phase transition between their confining and Higgs regimes, despite the fact that classic results and standard lore say they should be continuously connected.

4:30 – 5:20 pm

**Speaker:** Theo Johnson-Freyd, Dalhousie University and Perimeter Institute

**Title:** The universal target category

**Abstract:** Hilbert’s Nullstellensatz says that the complex numbers C satisfy a universal property among all R-algebras: every not-too-large nonzero commutative R-algebra maps to C. Deligne proved a similar statement in categorical dimension 1: every not-too-large symmetric monoidal category over R maps to the category sVec of super vector spaces. In other words, sVec (and not Vec!) is “algebraically closed”. These statements help explain why quantum field theory requires imaginary numbers and fermions. I will describe the universal symmetric monoidal higher category that extends the sequence C, sVec, …. This is joint work in progress with David Reutter, and builds on closely-related work by GCS collaborators Freed, Scheimbauer, and Teleman and Schlank et *al.*

Organizers:

**Dan Freed** (Harvard CMSA & Math)

**Constantin Teleman** (UC Berkeley)

*This event is co-sponsored by the Simons Foundation. *