Higher-order Galilean contractions

A Galilean contraction is a way to construct Galilean conformal algebras from a pair of infinite-dimensional 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 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.


Introduction
The Galilean Virasoro algebra appears in studies of asymptotically flat three-dimensional spacetimes, see [1] and references therein. 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]. These constructions are all based on contractions of pairs of symmetry algebras, or equivalently, contractions of tensor products of two vertex algebras. In this note, we present a generalisation to allow for inputs of any number of symmetry algebras. This solidifies ideas put forward in [14] and gives rise to new infinite hierarchies of higher-order Galilean conformal algebras.
In Section 2, we outline the generalised contraction prescription and illustrate it by working out the higher-order Galilean Virasoro and affine Kac-Moody algebras. In Section 3, we construct a Sugawara operator [22] for each Galilean Kac-Moody algebra; its central charge is given by the product of the contraction order and the dimension of the underlying Lie algebra. We also show that the Sugawara construction commutes with the Galilean contraction procedure. In Section 4, we apply the Galilean contractions to the W 3 algebra and thereby obtain an infinite hierarchy of higher-order W 3 algebras. Section 5 contains some concluding remarks.

Galilean contractions 2.1 Operator-product algebras and star relations
It is often convenient to combine the generators of the symmetry algebra of a conformal field theory into generating fields of the form where ∆ A is the conformal weight of A. We are interested in the corresponding operator-product algebra (OPA) A, where the operator-product expansion (OPE) of the two fields A, B ∈ A is given by Here, if nonzero, [AB] n is a field of conformal weight ∆ A + ∆ B − n. As the nontrivial information of an OPE is stored in the singular terms, one often ignores the non-singular terms, writing The normal ordering of A, B ∈ A is given by (AB) = [AB] 0 . We use I to denote the identity field. An OPA A is said to be conformal if it contains a distinct field T generating a Virasoro subalgebra. In that case, a field A ∈ A is called a scaling field if with structure constants C Q A,B and Compactly, we may represent the OPE (2.5) by the so-called star relation where {Q} represents the sum over n. We refer to [14,23] for more details on the algebraic structure of an OPA.

Contraction prescription
For N ∈ N, we consider the tensor-product algebra where, for simplicity, A (0) , . . . , A (N −1) are copies of the same OPA A, up to the value of their central parameters (such as central charges). For ǫ ∈ C, let where A (j) (respectively c (j) ) denotes the field A ∈ A (j) (respectively the central parameter c), and ω is the principal N th root of unity: ω = e 2πi/N . For ǫ = 0, the map (and similarly for the central parameters) is invertible, with In the special case N = 2, we have ω = −1 and with inverses In [14], these fields are denoted by (2.14) For ǫ = 0, the map (2.10) is singular (unless N = 1), indicating that a new algebraic structure emerges in the limit ǫ → 0, where If the resulting algebra is a well-defined OPA, we refer to it as the N th-order Galilean OPA A N G . In particular, if A is an OPA of Lie type (that is, the underlying algebra of modes is a Lie algebra), then all the corresponding higher-order Galilean contractions are indeed well-defined and readily obtained. This is illustrated by the Virasoro and affine Kac-Moody algebras in Section 2.3.

Galilean Virasoro and affine Kac-Moody algebras
The Virasoro OPA Vir of central charge c is of Lie type and generated by T , with star relation The Galilean Virasoro algebra of order N , Vir N G , is generated by the fields T 0 , . . . , T N −1 , with central parameters c 0 , . . . , c N −1 and star relations This yields an infinite family of extended Virasoro algebras, 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 [24]. The OPE of two fields in an affine Kac-Moody (or current) algebra g (where the central element K has been replaced by k I, with k the level) is given by where f ab c are structure constants and κ the Killing form of the underlying finite-dimensional Lie algebra g. (As is customary, the summation over the basis label c is not displayed.) The corresponding OPA is of Lie type, and we find that extending to general N the construction of the Takiff algebras considered in [25,26]. We similarly have

Generalised Sugawara constructions
In [14], we constructed a Sugawara operator for Galilean affine Kac-Moody algebras (of order 2), and showed that this process commutes with the Galilean contraction procedure. We find that a similar result holds for the higher-order Galilean affine Kac-Moody algebras, manifested by the commutativity of the diagram

Gal Gal
Gal Sug To verify this, separate analyses of the two branches are presented in the following two subsections: The lower branch is considered in Section 3.1; the upper one in Section 3.2.

Galilean Sugawara construction
For the generators of Vir N G , we make the ansatz where κ ab are elements of the inverse Killing form on g. The task is now to determine the coefficients λ r,s i such that We show below that this is indeed possible. It subsequently follows that where h ∨ is the dual Coxeter number of g, arising through the relation κ bc f ab d f dc e = 2h ∨ δ a e . To satisfy (3.2), the first sum must equal J a i+j (w)/(z − w) 2 while the second sum must vanish. The second-sum constraint implies that The first-sum constraint then requires that For each i, this translates into a lower-triangular system of linear equations: where k ′ 0 = k 0 + N h ∨ , and where the only nonzero component on the right-hand side is a 1 in position i + 1. To solve these systems, we must assume that k N −1 = 0, in which case the problem reduces to inverting the lower-triangular Toeplitz matrix The inverse is itself a lower-triangular Toeplitz matrix with 1's on the diagonal, and we find that the nontrivial matrix elements are given by ip i , p = (p 1 , . . . , p n ). (3.12) It follows that so the unique expression for T i of the form (3.1) is given by (3.14) For N = 2, we thus recover the Galilean Sugawara construction obtained in [14], 15) whereas for N = 3, we find the new expressions For each i = 0, . . . , N − 1, the value of the central parameter c i follows from the leading pole in the OPE T 0 (z)T i (w). Using (3.14), we compute suppressing all subleading poles. Since k a = 0 for a ≥ N , this term is zero unless n + i = 0, that is, unless n = i = 0. From κ ab κ ab = dim g, we then obtain the announced result (3.3).

Sugawara before Galilean contraction
On the individual factors of g ⊗N , the Sugawara construction is given by Changing basis as in (2.9) introduces Now, using that a lower-triangular N × N Toeplitz matrix of the form (3.8) decomposes as where I is the identity matrix and η the N × N matrix we can use the result for A −1 in (3.10)-(3.11) to expand the expression for T i,ǫ in powers of ǫ. We thus find that where b 0,ǫ = 1, b n,ǫ = p∈(N 0 ) n (−1) |p| δ ||p||,n |p|! p 1 ! · · · p n ! a p 1 1,ǫ · · · a pn n,ǫ , n = 1, . . . , N − 1. (3.23) The summation over j yields a factor of the form (3.24) and since N − 1 + i − ℓ − ℓ ′ + n > −N , it follows that the T i,ǫ -coefficients to ǫ m for m negative are 0. The limit ǫ → 0 is therefore well-defined, resulting in whose nonzero terms are seen to match the expression in (3.14).
For the central parameters, we evaluate from which it follows that

Galilean W 3 algebras
Higher-order Galilean contractions can also be applied to W-algebras. Below, we present the results for the W 3 algebra.

W 3 algebra
The W 3 algebra [15] of central charge c is generated by a Virasoro field T and a primary field W of conformal weight 3, with star relations is quasi-primary.

Galilean W 3 algebra of order 2
Following [13,14], we now recall the structure of the second-order Galilean W 3 algebra [11,12,29]. It is generated by the four fields T 0 , T 1 , W 0 , W 1 , with central parameters c 0 and c 1 , and nontrivial star relations and are quasi-primary. We note that a nonzero c 1 can be scaled away by renormalising as T 1 ,

Infinite hierarchy
For any N ∈ N, the algebra W First, it straightforwardly follows that To determine W i * W j in (W 3 ) N G for i + j = 0, . . . , N − 1, we compute the corresponding star relation Recycling the expansion techniques of Section 3, we find that where b n,ǫ (and b n appearing in (4.11) below) are given as in (3.23) (respectively (3.11)), but now based on (4.10) In the limit ǫ → 0, this yields (4.11) Observing that, for every pair r, s ∈ {0, . . . , N − 1} such that r + s ∈ {N − 1, . . . , 2N − 2}, is a quasi-primary field with respect to T 0 , we then conclude that, for i + j ∈ {0, . . . , N − 1}, Using that Λ 2,2 r,s = Λ 2,2 s,r , this can be written as where the last term is present only if N −1+i+j+n 2 is integer. Let us illustrate our findings by summarising the nontrivial star relations for the third-order Galilean algebra (W 3 ) 3 G : The six generating fields T 0 , T 1 , T 2 , W 0 , W 1 , W 2 satisfy (4.6)-(4.7) with N = 3 as well as where are quasi-primary.

Renormalisation
We now consider (W 3 ) N G in the special case where Correspondingly, the inverse of the matrix A in (3.20) is given by so (for N > 2) Let us also introduce the renormalised generators In terms of these, the nontrivial star relations are given by (i + j ∈ {0, . . . , N − 1}) and The central parameter c has thus been absorbed by a renormalisation of the algebra generators. A similar absorption is also possible in the Galilean Sugawara construction of Section 3, with where k i = k i , i = 1, . . . , N − 1, for some k ∈ C × . The renormalised Galilean Virasoro generators are then given by while the nontrivial star relations read (i + j ∈ {0, . . . , N − 1})

Discussion
In our continued exploration [13,14] of Galilean contractions, we have presented a generalisation of the contraction prescription to allow for inputs of any number of OPAs or vertex algebras. This has resulted in hierarchies of higher-order Galilean conformal algebras, including Virasoro, affine Kac-Moody and W 3 algebras. Asymmetric Galilean N = 1 superconformal algebras, corresponding to an N = (1, 0) supersymmetry, can be obtained [27,28,29,30] from a Galilean contraction of the tensor product, SVir ⊗ Vir, of an N = 1 superconformal algebra, SVir, and the Virasoro algebra. As we hope to discuss in detail elsewhere, this extends to contractions of a conformal symmetry algebra with any subalgebra thereof. For example, one readily generalises our contraction prescription to the asymmetric tensor product W 3 ⊗ Vir, where one contracts the Virasoro subalgebra of W 3 with a separate Virasoro algebra. This yields an OPA generated by fields T 0 , T 1 , W , with nonzero star relations (i + j ∈ {0, 1}) There is significant freedom in such contractions, leading to a variety of inequivalent Galilean algebras.
Other avenues for future research include representation theory and free-field realisations. The representation theory of the Galilean Virasoro algebra, also known as the W (2, 2) algebra, has already been studied in some detail [31,32,33,34,35,36]. In general, though, the representation theory of Galilean algebras remains largely undeveloped and is entirely unexplored in the case of the higher-order algebras introduced in the present note.
Free-field realisations [37,38,39,17,40,41,42,43,44,45] have been central to many developments in and applications of conformal field theory, and it seems natural to expect that free fields will play a similar role when Galilean conformal symmetries are present. This includes the representation theory of the Galilean algebras alluded to above. Although realisations of the Galilean Virasoro algebra and some of its superconformal extensions have been considered [46,36,30], a systematic approach and general results are still lacking.