Quantum Groups as Global Symmetries

TL;DR

This paper explores quantum field theories with quantum group symmetries, revealing their non-local operator structure, deriving symmetry constraints, and providing an example with $U_q(sl_2)$ that connects to the Ising model.

Contribution

It introduces the concept of non-local operators associated with quantum group symmetries and demonstrates their properties through a solvable $U_q(sl_2)$ symmetric CFT example.

Findings

Operators are generally non-local and live at ends of topological lines.

Correlation functions satisfy Ward identities from quantum group symmetry.

A special case reproduces the fermionic Ising model formulation.

Abstract

We study quantum field theories which have quantum groups as global internal symmetries. We show that in such theories operators are generically non-local, and should be thought as living at the ends of topological lines. We describe the general constraints of the quantum group symmetry, given by Ward identities, that correlation functions of the theory should satisfy. We also show that generators of the symmetry can be represented by topological lines with some novel properties. We then discuss a particular example of $U_q(sl_2)$ symmetric CFT, which we solve using the bootstrap techniques and relying on the symmetry. We finally show strong evidence that for a special value of $q$ a subsector of this theory reproduces the fermionic formulation of the Ising model. This suggests that a quantum group can act on local operators as well, however, it generically transforms them into non-local ones.