Free-field construction of Carrollian $W_N$-algebras

TL;DR

This work constructs Carrollian $W_N$-algebras by ultra-relativistic contraction of two copies of relativistic $W_N$-algebras using a free-field Miura realization. At the classical level, the Carrollian algebras coincide with Galilean (or BMS$_3$) structures, while in the quantum regime two distinct contractions emerge: a flipped (time-reversed) Carrollian contraction that yields a quantum Galilean $W_N$-algebra, and a symmetric (averaged-ordering) Carrollian contraction that produces a proper Carrollian $W_N$-algebra with central charges matching the classical expressions. The construction provides explicit free-field realizations, including the Carrollian Miura transformation, and explicit $N=2$ (and higher) examples, enabling systematic study of representations and correlation functions within Carrollian (and Galilean) higher-spin symmetry. This framework supports applications to flat space holography and higher-spin gravity by furnishing an operator-based toolset to explore extended Carrollian symmetries and their representation theory.

Abstract

We study Carrollian contractions of $W_N$-algebras from a free-field perspective. Using a contraction of the Miura transformation, we obtain explicit free-field realizations of the resulting Carrollian $W_N$-algebras. At the classical level, they are isomorphic to the Galilean $W_N$-algebras. In the quantum case, we distinguish between two Carrollian constructions: a flipped Carrollian contraction, where the time direction is reversed in one sector, and a symmetric contraction. The flipped construction yields a quantum algebra isomorphic to the Galilean one, whereas the symmetric construction produces a distinct quantum Carrollian $W_N$-algebra whose basic structure constants are identical to those of the classical Carrollian $W_N$-algebra. These algebras provide a natural framework for studying extended symmetries in Carrollian conformal field theories, motivated by recent developments in flat space holography. Our construction provides tools for developing the representation theory of Carrollian (and Galilean) $W_N$-algebras using free-field techniques.