Higher-order Galilean contractions

Abstract

A Galilean contraction is a way to construct Galilean conformal algebras from a pair of infinitedimensional conformal algebras, or equivalently, a method for contracting tensor products of vertex algebras. Here, we present a generalisation of the Galilean contraction prescription to allow for inputs of any finite number of conformal algebras, resulting in new classes of higher-order Galilean conformal algebras. We provide several detailed examples, including infinite hierarchies of higher-order Galilean Virasoro algebras, affine Kac-Moody algebras and the associated Sugawara constructions, and W 3 algebras.

Multi-graded Galilean conformal algebras

“…Although our presentation focuses on the case of two input algebras, the generalisation to any number of input algebras is straightforward. Such algebras have higher-order Galilean subalgebras [35] with OPEs between the fields in G n ℓ ℓ determined by Proposition 3.1. Our asymmetric Galilean contraction procedure provides a unifying framework for several interesting examples in the CFT literature, from early work on non-compact WZW models to current work on tensionless strings and super-BMS symmetries.…”

“…For n = 0, the contraction results in a well-defined algebra with two conformal-weight 2 quasiprimary fields. This is a novel Galilean structure that cannot be realised using higher-order, or multi-graded, Galilean contractions [35,21].…”

“…The Galilean contraction procedure has been generalised to allow for the input of any number of symmetry algebras, leading to so-called higher-order Galilean contractions [24,35]. The ensuing Galilean algebras exhibit graded structures of any integer order N 2.…”

“…A method to perform Galilean contractions of W -algebras was developed in [31,32,35], and many concrete examples of Galilean W -algebras have been constructed.…”

“…The Galilean Virasoro subalgebra is generated by T 0 , T 1 , and the fields W ℓ , U 1 are primary with respect to the Virasoro field T 0 . The remaining OPEs are determined by applying the techniques detailed in [32,35].…”