Galilean contractions of W -algebras

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.

Higher-order Galilean contractions

“…In the general construction, these algebras are all assumed identical, up to their central charges, although asymmetric contractions are possible, as we discuss briefly. These results solidify ideas put forward in [14] and give rise to new infinite hierarchies of higher-order Galilean conformal algebras.…”

“…where {Q} represents the sum over n. We refer to [14,39] for more details on the algebraic structure of an OPA.…”

“…Vir 2 G is the familiar Galilean Virasoro algebra [2,3,4,5,6,13,14]. For small N , the Galilean Virasoro algebras Vir N G have recently appeared in [29].…”

“…It can be constructed [2,3,4,5,6] as an Inönü-Wigner contraction [7,8,9,10] of a commuting pair of Virasoro algebras. The Galilean W 3 algebra [11,12,13,14] likewise follows by contracting a pair of W 3 algebras [15]. Many other Galilean conformal algebras with extended symmetries have been worked out [16,13,14], including contractions of higher-rank W N algebras [17,18,19,20,21].…”

“…The Galilean W 3 algebra [11,12,13,14] likewise follows by contracting a pair of W 3 algebras [15]. Many other Galilean conformal algebras with extended symmetries have been worked out [16,13,14], including contractions of higher-rank W N algebras [17,18,19,20,21]. Earlier works contributing to our understanding of non-relativistic systems with (typically non-affine) conformal symmetry can be found in [22,23,24,25,26,27,28].…”