Non-Invertible Theta Symmetries

Abstract: It is by now well-known that the symmetries of a d-dimensional quantum field theory form a fusion (d-1)-category. This contains a fusion 1-category piece formed by genuine topological line operators and non-genuine topological local operators which is extremely well understood both physically and mathematically. This talk will discuss recent physical and mathematical understandings of a fusion 2-category piece formed by genuine topological surface operators and non-genuine topological line & local operators, which contains within it the above fusion 1-category piece. For simplest physical constructions, the fusion 2-category can be identified with the 2-category formed by 2-representations of a finite group G. We will discuss the mathematical definition of 2-representations and provide a simple universal physical reasoning for their appearance: namely that any QFT that can be obtained by gauging a 0-form symmetry group G contains dual non-invertible symmetries described by surface operators labeled by 2-representations of G. We refer to such universal symmetries as theta symmetries because they generalize the notion of theta angle. We will then proceed to discuss the inclusion of non-universal symmetries that depend on the specifics of the QFT under consideration. We will see that the physical reasoning involved in the discussion of universal theta symmetries will continue to play a major role also for non-universal symmetries, leading to the notion of "twisted theta symmetries". Many interesting physical phenomena like symmetry fractionalization on condensation defects will be encountered. Towards the end of the talk, we will briefly discuss the generalization of these studies to fusion 3-categories and higher. This involves various physical and mathematical subtleties having to do with the fact that TQFTs in spacetime dimension bigger than two form an extremely rich mathematical structure.

HEP - phenomenologyHEP - theorymathematical physicscategory theory

Audience: researchers in the topic

Symmetry Seminar

Series comments: Weekly seminar on generalized symmetries in QFT, string theory and related topics.