Remarks on BMS${}_3$ invariant field theories: correlation functions and nonunitary CFTs

TL;DR

This paper explores the structure of BMS${}_3$ invariant field theories, analyzing correlation functions and nonunitary conformal field theories using algebraic isomorphisms and modified Ward identities.

Contribution

It introduces a novel operator for nonunitary CFTs based on nilpotent variables and clarifies the role of extended symmetries in BMS${}_3$ invariant theories.

Findings

Extended symmetry acts trivially in unitary theories.

A new operator organizes Verma modules in nonunitary CFTs.

Modified Ward identities impose specific conditions on correlation functions.

Abstract

We use the isomorphism between the BMS${}_3$ and the $W(2,2)$ algebras to reconsider some generic aspects of CFTs with the BMS${}_3$ algebra defined as a chiral symmetry. For unitarity theories, it is known that the extended symmetry generator acts trivially, and the resulting theory is equivalent to a CFT with a Virasoro symmetry only. For nonunitary CFTs, we define an operator depending on a nilpotent variable, and we organize the Verma module through the action of this new operator. Finally, we find the conditions imposed by the modified Ward identity.