Multi-graded Galilean conformal algebras

Galilean conformal algebras can be constructed by contracting a finite number of conformal algebras, and enjoy truncated $\mathbb{Z}$-graded structures. Here, we present a generalisation of the Galilean contraction procedure, giving rise to Galilean conformal algebras with truncated $\mathbb{Z}^{\otimes\sigma}$-gradings, $\sigma\in\mathbb{N}$. Detailed examples of these multi-graded Galilean algebras are provided, including extensions of the Galilean Virasoro and affine Kac-Moody algebras. We also derive the associated Sugawara constructions and discuss how these examples relate to multivariable extensions of Takiff algebras. We likewise apply our generalised contraction prescription to tensor products of $W_3$ algebras and obtain new families of higher-order Galilean $W_3$ algebras.

Following ideas put forward in [11], higher-order Galilean contractions were developed in [20], generalising the contraction procedure from pairs of symmetry algebras (or equivalently vertex algebras) to any finite number of symmetry algebras (or vertex algebras). In the case of Virasoro or affine Kac-Moody, the usual second-order Galilean algebras have been found [20] to be isomorphic to the Takiff algebras [21] considered in [22,23], while the higher-order counterparts provide N th-order generalisations (where N is the number of inputted symmetry algebras A). These higher-order Galilean algebras thus enjoy a truncated Z-grading whose truncation is determined by the order N of the contraction.
Here, we modify the higher-order contraction procedure to let it depend on a factorisation of N , where N = N 1 , . . . , N σ is a finite sequence of positive integers such that N = N 1 . . . N σ . We thus organise the N -fold tensor product A ⊗N in terms of N ℓ -fold tensor-product factors, and apply the higher-order contraction prescription of [20] to the factors 'simultaneously'. We find that the ensuing Galilean algebra, A N G , is Z ⊗σ -graded, truncated according to the sequence N 1 , . . . , N σ . Because of this graded structure, we refer to the generalised contraction as multi-graded contraction. We also observe that the contractions are independent of the ordering of the factors in the factorisation of N . In case the mode algebra underlying A is a Lie algebra, we find that A N G is isomorphic to a multivariable generalisation of the Takiff algebras discussed in [20], with the number of variables given by the length σ of the contraction sequence N.
In Section 2, we outline the multi-graded contraction procedure and illustrate it by working out the corresponding Galilean Virasoro and affine Kac-Moody algebras. We also discuss the ensuing grading structures and relate the corresponding Galilean algebras to a multivariable generalisation of the Takiff algebras. In Section 3, we construct a Sugawara operator for each of the multi-graded Galilean Kac-Moody algebras; its central charge is given by the product of the contraction order N and the dimension of the underlying Lie algebra. We also show that the Sugawara construction commutes with the contraction procedure. In Section 4, we apply multi-graded contractions to the W 3 algebra and thereby obtain a new class of Galilean W 3 algebras. Section 5 contains some concluding remarks.

Contraction procedure
We find it advantageous to describe the multi-graded contractions and ensuing algebras in the language of operator-product algebras (OPAs), and refer to [11,24] for details on the structure of an OPA. We say that an OPA is of Lie type if the corresponding mode algebra is a Lie algebra, as is the case for the Virasoro and Kac-Moody algebras. Throughout, I denotes the identity field and ∆ A the conformal weight of the scaling field A.

Star relations in OPAs
For the space of quasi-primary fields in the OPA A, we let B A denote a basis consisting of quasi-primary fields only. Only keeping the non-singular terms, the operator-product expansion of A, B ∈ B A can then be expressed as Convenient for our purposes, the essential part of the operator-product expansion (2.1) is synthesised in the so-called star relation where {Q} represents the sum over n displayed in (2.1). For example, the Virasoro algebra Vir of central charge c is of Lie type and generated by T , with star relation Likewise, the nontrivial star relations in an affine Kac-Moody algebra g are given by where f ab c ∈ C are structure constants, k ∈ C the level and κ the Killing form of the underlying finitedimensional complex Lie algebra g. As is customary, we do not display summations over repeated group indices, here the summation over c ∈ {1, . . . , dim g}. We note that g is of Lie type.

Higher-order contractions
Higher-order Galilean contractions were developed in [11]. Here, we recast them in a notation suitable for their multi-graded generalisation introduced in Section 2.3. Thus, for N ∈ N, let where A (0) , . . . , A (N −1) are copies of the same OPA A, up to the values of their central parameters (such as central charges or levels). In effect, we are viewing the central parameters of A as independent indeterminants. We then write where A (i) (respectively c (i) ) denotes the field A ∈ A (i) (respectively a central parameter of A (i) ). For ǫ ∈ C, we also let and where ω is the principal N th root of unity, It follows that (for ǫ = 0) Thus, with is invertible for ǫ = 0. For ǫ = 0, on the other hand, the map is singular (unless N = 1). If a well-defined OPA arises in the limit ǫ → 0, where the ensuing algebra is known [11] as the N th-order Galilean OPA A N G . Note that A 1 G ∼ = A. For small N , the Galilean Virasoro algebras Vir N G also appeared in [25].

Generalised higher-order contractions
Fix σ ∈ N. As in Section 1, we denote complex-number sequences of length σ by S = S 1 , . . . , S σ etc, with 0 = 0, . . . , 0 the zero sequence. Linear combinations are readily formed α i + β j = αi 1 + βj 1 , . . . , αi σ + βj σ , α, β ∈ C, (2.15) and two sequences can be compared as If every element of S is nonzero, we let S −1 denote the sequence S −1 1 , . . . , S −1 σ . The set of integer sequences bounded strictly by N admits the canonical order where i appears before j if and only if for each m such that i m > j m there exists ℓ < m such that i ℓ < j ℓ . This corresponds to the usual ordering of basis vectors for the tensor product space Using the same ordering prescription, we now form the N -dimensional vectors where A (i) (respectively c (i) ) denotes the field A ∈ A (i) (respectively a central parameter of A (i) ). With and for The map If a well-defined (N th-order Galilean) OPA arises in the limit ǫ → 0, where In Section 2.4, we use this result to determine the structure of A N G for A of Lie type.

Multi-grading
Still treating central parameters as indeterminants, we assign the following grades to the generators and parameters of the Galilean algebras: The action of gr is then extended linearly and to rational functions of the central parameters and normal-ordered products and derivatives of the fields, with gr(∂) = 0, so that, for instance, We say the algebra is multi-graded if the grading is compatible with the product structure of operatorproduct expansions, in the sense that A priori, it is not guaranteed that all terms appearing in the decomposition of A i * B j have a well-defined grade, let alone the same grade. However, as we will argue, all Galilean algebras of the type A N G are, in fact, multi-graded. Moreover, the grading is finitely truncated by N in the sense that A i * B j = 0 unless i + j < N.
Let A, B ∈ B A and consider the star relation (2.3). If A is of Lie type, then the only structure constants C Q AB that can depend on central parameters have Q = I, as in (2.4) and (2.5). To indicate this, we write The nonlinearity of an OPA that is not of Lie type obscures the question of its grading structure, as witnessed in sections 3 and 4. However, as already indicated, all the Galilean algebras we have analysed are nevertheless multi-graded in the sense outlined above.
Virasoro algebras: The multi-graded Galilean Virasoro algebra Vir N G is generated by the fields {T i | 0 ≤ i < N} and has central parameters {c i | 0 ≤ i < N}, with star relations given by Note that T 0 generates a subalgebra isomorphic to Vir with central charge c 0 , and that, for every i, T i is quasi-primary with respect to T 0 .
Affine Kac-Moody algebras: The multi-graded Galilean Kac-Moody algebra g N G is generated by Note that {J a 0 | a = 1, . . . , dim g} generates a subalgebra isomorphic to g at level k 0 .

Permutation invariance
In all the Galilean algebras we have analysed, we observe that Together with the grading property, this implies that all inequivalent decompositions of star relations arise as A 0 * B j for some A, B ∈ A and 0 ≤ j < N. It also implies that the multi-graded contraction procedure is independent of the ordering of the elements in the contraction sequence. That is, where π = π 1 , . . . , π σ is a permutation of the integers 1, . . . , σ. Moreover, as the tensorial structure of the contraction process ensures that we see that

Multivariable Takiff algebras
In this limit, the algebra This Z ⊗σ -graded algebra is seen to be isomorphic to a multivariable polynomial ring, and we likewise recognise the isomorphism This extends to multiple variables the Takiff algebras considered in [20], themselves extensions to general order N of the second-order (one-variable) Takiff algebras considered in [22,23]. We similarly have (2.48) In this limit, the Galilean algebra A N G becomes A ∞s G , where, for example,

Generalised Sugawara construction
The objective here is to construct a Sugawara operator for each Galilean affine Kac-Moody algebra g N G and to show that this process commutes with the Galilean contraction procedure, thereby establishing the commutativity of the diagram The lower branch is analysed in Section 3.1; the upper one in Section 3.2.

Galilean Sugawara construction
For the Sugawara generators of Vir N G , we make the ansatz where κ ab are elements of the inverse Killing form on g, and our goal is to determine the coefficients λ r,s i such that To this end, we compute the operator-product expansion where the dual Coxeter number h ∨ of g has arisen through To satisfy (3.2), the sum multiplying the single pole in (3.3) must be zero while the sum multiplying the double pole must equal J a i+j (w). The single-pole constraint implies that  As the generators {J a m | 0 ≤ m < N} are linearly independent, the constraint (3.7) translates into a lower-triangular system of linear equations in the variables {λ n;N−1 i | 0 ≤ n < N}. Indeed, considering λ * ;N−1 i as the N -vector with components λ n;N−1 i ordered canonically according to n ∈ I N , such that All the diagonal entries are thus given by 2k N−1 . The only nonzero component on the righthand side of (3.8) is a 1 in the position corresponding to i ∈ I N . The structure of M resembles a lower-triangular Toeplitz matrix, but with some entries set to 0. Indeed, where each M i 1 ,i 2 ∈ {M i 1 ,1 , . . . , M i 1 ,N 2 } is an N N 1 N 2 × N N 1 N 2 lower-triangular matrix of similar form, and so on. The innermost lower-triangular matrices appearing in this nested description of M are N σ × N σ Toeplitz matrices of the form For the sequence N = 3, 2, 3, for example, we thus have (3.14) written using the simplified notation The inverse of M , has the same nested Toeplitz-like structure, with all diagonal entries given by 1/(2k N−1 ). With this, we solve (3.8) and find λ n;N−1 The Galilean Sugawara construction (3.1) is thus given by For each i, the value of the central parameter c i follows from the leading pole in the OPE Since k h = 0 unless h < N, the only contribution to the leading-pole term appears for n = 0, hence for i = 0. The term thus reduces to from which it follows that This result for the central charge c 0 resembles similar results [27] for Sugawara constructions associated with so-called double extensions [26].

Sugawara before Galilean contraction
As above, let N = N 1 , . . . , N σ and N = N 1 . . . N σ . Accordingly, on the individual factors of g ⊗N , we denote the Sugawara construction by let T * and c * denote the corresponding N -vectors formed as in (2.20), and change basis as in (2.25): It follows that where We can thus write For each ℓ ∈ {1, . . . , σ}, the summation over j ℓ yields a factor of the form (3.36) it follows that the T i,ǫ -coefficients to negative powers of any of the ǫ ℓ 's are all zero. The limit ǫ → 0 is therefore well-defined, and we find This is seen to agree with (3.17) In the affirmative, the relations (3.39) follow from the similarity in structures of M m,n in (3.8) and a m in (3.32). We have thus established the commutativity of Galilean contractions and Sugawara constructions, without explicit knowledge of the coefficients b n,i andâ n appearing in the inversion of the coefficient matrix M and the series expansion of the denominator of T i,ǫ , respectively. The coefficients are readily obtained case by case, but cumbersome to express for general parameters. For σ = 1, the coefficients are given in [20]. For the central parameters, we have (3.40) so in the limit ǫ → 0, 41) in accordance with (3.20).

Galilean W 3 algebras
Our generalised Galilean contractions can also be applied to W-algebras. Below, we present the results for the factorisation sequence N = 2, 3 applied to the W 3 algebra, giving rise to the sixth-order Galilean algebra (W 3 ) 2,3 G . In preparation, we first recall the structure of the W 3 algebra and its second-and third-order Galilean counterparts.

W 3 algebra
The W 3 algebra [28] 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
and As will become clear below, it is convenient to introduce (4.5) (W 3 ) 2 G algebra: For N = 2, the star relations between the W -fields are given by where are quasi-primary with respect to T 0 .
(W 3 ) 3 G algebra: For N = 3, the star relations between the W -fields are given by 10) and where are quasi-primary with respect to T 0 . Both (W 3 ) 2 G and (W 3 ) 3 G are readily seen to be (multi-)graded, truncated according to their order (2 and 3, respectively).

Galilean algebra
The N th-order Galilean algebra (W 3 ) N G is generated by the fields {T i , W i | 0 ≤ i < N} and has central parameters {c i | 0 ≤ i < N}, with the star relations involving the T fields given by (2.37) and  where, for i + j ≥ N − 1, Λ i; j = (T i T j ) − 3 10 ∂ 2 T N−1 δ i+j, N−1 (4.23) is quasi-primary with respect to the Virasoro generator T 0 . It follows that the sixth-order Galilean algebra (W 3 ) 2,3 G is multi-graded, with truncation dictated by the sequence 2, 3.

Discussion
In our continued exploration [9,11,20] of Galilean contractions, we have presented a generalisation of the contraction procedure to multi-graded Galilean algebras. Our construction uses factorisations of the order parameter N , and has resulted in whole new families of higher-order Galilean conformal algebras, including Virasoro, affine Kac-Moody and W 3 algebras. We have also discussed how some of these algebras are related to a multivariable extension of Takiff algebras. W -algebras related to Takiff algebras were introduced in [29,30] and constructed as principal Walgebras built on the centralizer of a nilpotent element in gl(n). The construction is carried out in the context of (Poisson) vertex algebras, and it appears natural that it is linked to the one presented here. In particular, the nilpotent element being characterised by a partition λ = λ 1 , λ 2 , . . ., the algebras have an indexation comparable to N = N 1 , N 2 , . . . used in our multi-graded contraction procedure.
Other avenues for future work include asymmetric contractions and free-field realisations. Asymmetric Galilean N = 1 superconformal algebras were constructed in [10,[31][32][33] from a Galilean contraction of the tensor product SVir ⊗ Vir, where one contracts the Virasoro subalgebra of an N = 1 superconformal algebra, SVir, with a separate Virasoro algebra. The ensuing Galilean superconformal algebra can be viewed as encoding a (1, 0) supersymmetry. This was extended in [20] to a contraction of the asymmetric tensor product W 3 ⊗ Vir, giving rise to a Galilean W 3 algebra generated by fields T 0 , T 1 , W . There is significant freedom in such contractions, and we hope to return elsewhere with a partial classification of the inequivalent Galilean algebras that can arise this way.
Free-field realisations [34][35][36][37][38][39][40][41][42][43] are ubiquitous in conformal field theory, and we find it natural to expect that they will continue to play a central role when Galilean conformal symmetries are present. Some work on this has been done [33,44,45], but a systematic approach and general results remain outstanding. We hope to report such advances in the near future.