# Galilean contractions of $W$-algebras

## TL;DR

This paper introduces a new contraction method for $W$-algebras and related structures, leading to the discovery of Galilean $W$-algebras and their properties, including their construction and potential applications.

## Contribution

It develops a contraction prescription for operator-product algebras, enabling the construction of new Galilean $W$-algebras from known superconformal and $W$-algebras.

## Key findings

- Galilean $W$-algebras are constructed from contractions of known algebras.
- Galilean affine algebras admit a Sugawara construction with level-independent central charge.
- The contraction method applies to various superconformal and $W$-algebras, revealing new algebraic structures.

## Abstract

Infinite-dimensional Galilean conformal algebras can be constructed by contracting pairs of symmetry algebras in conformal field theory, such as $W$-algebras. Known examples include contractions of pairs of the Virasoro algebra, its $N=1$ superconformal extension, or the $W_3$ algebra. Here, we introduce a contraction prescription of the corresponding operator-product algebras, or equivalently, a prescription for contracting tensor products of vertex algebras. With this, we work out the Galilean conformal algebras arising from contractions of $N=2$ and $N=4$ superconformal algebras as well as of the $W$-algebras $W(2,4)$, $W(2,6)$, $W_4$, and $W_5$. The latter results provide evidence for the existence of a whole new class of $W$-algebras which we call Galilean $W$-algebras. We also apply the contraction prescription to affine Lie algebras and find that the ensuing Galilean affine algebras admit a Sugawara construction. The corresponding central charge is level-independent and given by twice the dimension of the underlying finite-dimensional Lie algebra. Finally, applications of our results to the characterisation of structure constants in $W$-algebras are proposed.