Intrinsic Approach to $1+1$D Carrollian Conformal Field Theory

TL;DR

This paper develops a purely Carrollian framework for 1+1D Carrollian conformal field theory, deriving Ward identities, operator product expansions, and the algebra of energy-momentum tensor modes using complex contour integrals and step-functions.

Contribution

It introduces a Carrollian perspective to 1+1D conformal field theories, deriving Ward identities and operator algebras from first principles with novel contour integral techniques.

Findings

Derived Ward identities using complex contour integrals.

Established the operator algebra via step-function based OPEs.

Showed energy-momentum tensor modes generate the extended Carrollian conformal algebra.

Abstract

The 3D Bondi-Metzner-Sachs (BMS$_3$) algebra that is the asymptotic symmetry algebra at null infinity of the $1+2$D asymptotically flat space-time is isomorphic to the $1+1$D Carrollian conformal algebra. Building on this connection, various preexisting results in the BMS$_3$-invariant field theories are reconsidered in light of a purely Carrollian perspective in this paper. In direct analogy to the covariant transformation laws of the Lorentzian tensors, the flat Carrollian multiplets are defined and their conformal transformation properties are established. A first-principle derivation of the Ward identities in a $1+1$D Carrollian conformal field theory (CCFT) is presented. This derivation introduces the use of the complex contour-integrals (over the space-variable) that provide a strong analytic handle to CCFT. The temporal step-function factors appearing in these Ward identities enable the translation of the operator product expansions (OPEs) into the language of the operator commutation relations and vice versa, via a contour-integral prescription. Motivated by the properties of these step-functions, the $i\epsilon$-forms of the Ward identities and OPEs are proposed that permit for the hassle-free use of the algebraic properties of the latter. Finally, utilizing the computational techniques developed, it is shown that the modes of the quantum energy-momentum tensor operator generate the centrally extended version of the infinite-dimensional $1+1$D Carrollian conformal algebra.