Asymmetric Galilean conformal algebras

The usual Galilean contraction procedure for generating new conformal symmetry algebras takes as input a number of symmetry algebras which are equivalent up to central charge. We demonstrate that the equivalence condition can be relaxed by inhomogeneously contracting the chiral algebras and present general results for the ensuing asymmetric Galilean algebras. Several examples relevant to conformal field theory are discussed in detail, including superconformal algebras and W-algebras. We also discuss how the Sugawara construction is modified in the asymmetric setting.

Galilean symmetry algebras can be constructed in several ways. In this paper, we focus on a parametric contraction procedure known as Galilean contraction, analogous to anÍnönü-Wigner contraction of Lie algebras [17,18]. Galilean algebras can also be constructed as so-called Takiff algebras, as in [19,20,21,22], or by using the so-called semigroup expansion method, as in [23,24,25]. Moreover, they appear naturally as the flat-space limit symmetries of models on AdS 3 spacetime [26,27,28,29,30].
The Galilean contraction procedure, as studied in [1,2,31,32], takes as input two symmetry algebras which are equivalent up to central parameters. That is, the chiral algebras are the same operator product algebra (OPA) [33,32], up to the value of their central charge. The procedure involves a parameter-dependent (ǫ) map on a particular set of fields such that in the limit ǫ → 0 the map becomes singular. If the limit is well-defined, the resulting algebra is referred to as the Galilean algebra corresponding to the input algebras. The most studied example of this construction involves contracting two copies of the Virasoro algebra, producing the so-called Galilean conformal algebra, also known as the Galilean Virasoro algebra [34,2,32].
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. In [21], the contraction procedure was further generalised to accommodate a tensor-product structure. This results in the so-called multi-graded Galilean contraction and produces Galilean algebras graded by integer sequences.
Here, we relax the condition on the equivalence of input algebras by inhomogeneously rescaling the fields generating the algebras. We refer to this as asymmetric Galilean contraction.
We require that each input algebra admits an embedding of a particular algebra, meaning that the input algebras have a pair of subalgebras which are equivalent up to the value of their central parameters. The fields of these subalgebras are contracted according to the usual Galilean procedure, yielding an order-two Galilean subalgebra. The fields not in the equivalent subalgebras are rescaled separately, and the choice of these rescalings determines the structure of the asymmetrically contracted algebra arising in the limit.
The paper is structured as follows. We begin by presenting a general description of asymmetric Galilean contractions. We then provide a number of detailed examples chosen to demonstrate the procedure and to provide interesting physical applications. Finally, we discuss a Sugawara construction for asymmetrically contracted affine Lie algebras.

Review of operator product algebras
The key structure in this paper is the symmetry algebra of a CFT. Such an algebra can be realised as an OPA, here denoted by A, which is particularly convenient for studying contractions. We note, however, that asymmetric contractions and our results apply more broadly, such as to finitedimensional Lie algebras and other algebraic structures encoding symmetries.
The fundamental objects of an OPA are the fields A(z) ∈ A, which have mode expansions where ∆ A ∈ R is known as the conformal weight of the field A. The basic product in an OPA is the operator product expansion (OPE). The OPE between two fields A(z), B(w) ∈ A is given by where [AB] n is a field of conformal weight ∆ A + ∆ B − n. The symbol ∼ indicates that all nonsingular terms are omitted. A set of fields is said to generate an OPA if all other fields in the OPA can be obtained from the generating set by taking OPEs, linear combinations, normally-ordered products, and derivatives.
An OPA is said to be conformal if it contains a distinguished field T (z) that generates a Virasoro subalgebra: A field A(z) ∈ A is called a scaling field if A scaling field A(z) is said to be quasi-primary if [T A] 3 = 0, and primary if [T A] n = 0 for all n > 2. Primary fields are thus quasi-primary, and we note that the Virasoro field T (z) is quasi-primary.
Let Q denote a basis for the space of quasi-primary fields of a conformal OPA A. The OPE between two fields in Q is given by where f AB C are structure constants, and The expression in parentheses in (5) is known as the conformal chain (in reference to similar quantities appearing in [48]), and is determined by ∆ A , ∆ B , and Q. Using this observation, the explicit dependence of the fields on the variables z, w will be dropped in the following, and we will write the OPE (5) as where we have used × and ≃ to distinguish this form of the product from that of (5).
An important family of OPAs are those whose underlying algebra of modes is an infinite-dimensional Lie algebra. Such OPAs are known as Lie-type OPAs. Usually, we consider symmetries described by infinite-dimensional Lie algebras with central extensions. As we are working with the field algebras associated with these infinite-dimensional Lie algebras, the value that the central extension takes on the vacuum representation becomes a parameter of the theory. We will refer to this parameter as a central parameter and denote it by c ∈ C generally, although it may be, for example, the level k of an affine current algebra.
By definition, only the structure constants accompanying the identity field depend on central parameters for a Lie-type algebra. Moreover, the structure constants accompanying the identity are linear in c. Hence, the OPE of a Lie-type algebra may be written as where I denotes the identity field of the OPA. The remaining terms are a sum overQ ≡ Q \ {I}.
Later in the paper, we will employ the following summation convention: When the identity field is exhibited explicitly on the right-hand side, accompanying summations are taken to be over all fields excluding the identity.

Asymmetric Galilean algebras
We begin by considering a Lie-type OPA H, and denote a corresponding set of elementary fields by H. We then consider embeddings of H into Lie-type OPAs A 1 and A 2 , where A 1 and A 2 are not necessarily equivalent up to central parameters. We denote the image of H under the embedding into A ℓ by H ℓ , ℓ = 1, 2. The central parameter of the algebra A ℓ is denoted by c (ℓ) . In the corresponding infinite-dimensional Lie algebras, where one has central elements rather than central parameters, the subalgebras H ℓ are indeed isomorphic.
For each ℓ, we can extend the set of elementary fields of H ℓ , denoted by H ℓ , to a set of elementary fields for A ℓ , denoted by G ℓ . We thus have the partitions G ℓ = H ℓ ⊔Ḡ ℓ . For simplicity, we will assume that the elementary fields of H 1 and H 2 have been selected such that (loosely speaking) H 1 = H 2 . We denote the vector subspace of A ℓ spanned by the fields in G ℓ , H ℓ , andḠ ℓ , along with their derivatives, by g ℓ , h ℓ , andḡ ℓ , respectively.
We will use notation A (ℓ) , B (ℓ) , C (ℓ) for fields in H ℓ , and X (ℓ) , Y (ℓ) , Z (ℓ) for fields inḠ ℓ . Moreover, we generally denote the structure constants of A 1 and A 2 by f (1) and f (2) , respectively, but will denote the structure constants of H simply by f AB C . We remark that, although we assume in the following that the input algebras are Lie-type OPAs, it is possible to perform asymmetric contractions on non-Lie-type algebras. Examples of asymmetric Galilean contractions of non-Lie-type W -algebras are thus explored in Section 5.
A Galilean-type contraction on the OPA begins by choosing a map on the set of elementary fields G 1 ∪ G 2 , which is parametrized by ǫ ∈ C. For each pair m 1 , m 2 ∈ R ≥0 , we thus introduce new fields and along with new central parameters The inverse map on fields is given by and correspondingly, we have an inverse map on central parameters, given by For simplicity, for each ℓ, all elements X (ℓ) ∈Ḡ ℓ are scaled by the same ǫ-monomial. We will comment on this assumption at the end of this Section.
The contraction is now performed by taking the limit ǫ → 0. If the limit exists, it defines a map A → A G to the corresponding asymmetric Galilean algebra, where The images of the elementary fields of the OPA A form a set of elementary fields of A G .
We denote the asymmetric Galilean contraction of A 1 and A 2 with respect to particular embeddings ρ ℓ of the subalgebra H by , or simply by (A m1 1 ←֓ H ֒→ A m2 2 ) G . By construction, there is an exchange symmetry in the asymmetric Galilean contracted algebras given by A m1 The contracted algebra is graded according to the ǫ-monomials used in the procedure. Concretely, we have that the set of elementary fields of A G splits as H 0 ⊔H 1 ⊔Ḡ m1 1 ⊔Ḡ m2 2 , where the superscript denotes the Galilean grading of the fields. We denote the vector subspaces of A G formed by the fields in H 0 , H 1 ,Ḡ m1 1 , andḠ m2 2 , along with their derivatives, by h 0 , h 1 ,ḡ m1 1 , andḡ m2 2 , respectively. The resulting algebra A G has an order-two Galilean subalgebra, denoted by H G , with H G = H 0 ⊔ H 1 as a set of elementary fields.
In general, for Lie-type OPAs, products between elementary fields are sufficient to determine the algebraic structure. As such, we use notation h i × h j to denote the space of OPEs between fields in h i and h j , i, j ∈ {0, 1}. For example, we write to describe that OPEs between elementary fields in H 0 and H 1 , and their derivatives, only produce fields in h 1 .
Naturally, the structure of A G is sensitive to the choice of the contraction parameters m ℓ ; indeed, A G may only exist for certain choices. A simplified situation occurs if A 2 ∼ = H 2 . In that case, there is only one contraction parameter asḠ 2 is empty, and we denote the resulting Galilean algebra simply by (H ֒→ A n ) G .
The algebraic structure for the subalgebra H G is given by The remaining algebraic structure depends on the choice of m 1 and m 2 . The OPEs involving fields inḡ m ℓ ℓ , ℓ = 1, 2, are determined by the value of m ℓ . The OPEs for ℓ = 1, 2, are independent of each other and fall into the following cases: 3. If m ℓ = 1, which is always possible, then 1 2 , 1}, and the contraction limit is well-defined, then Proof: To determine the structure of the contracted algebra, we begin by calculating OPEs in the algebra A before taking the contraction limit. As the algebras A 1 and A 2 are Lie-type, their structure constants accompanying the identity are linear in c. Recalling that 1}, the OPEs between elementary fields in the subalgebras H ℓ are given by where i, j ∈ {0, 1}, and we have that The remaining products between transformed elementary fields of A are given by and Finally, X 1,ǫ × Y 2,ǫ ≃ {0}.
In the contraction limit ǫ → 0, the OPEs in the subalgebra H G are given by For m 1 ∈ 0, 1 2 , one must have for the limit to exist. This condition is equivalent to g 1 ×ḡ 1 ⊆ḡ 1 . For The conditions for m 2 ∈ 0, 1 2 and m 2 ∈ 1 2 , 1 are the same as those for m 1 , but with f (1) replaced by f (2) .
In summary, the possible products on the contracted algebra fall into the following cases: 2. For m ℓ = 0, we have 3. For m ℓ = 1 2 , we have 4. For m ℓ = 1, we have Following these, the chosen values of the contraction parameters then determine the structure of the resulting Galilean algebra.
The case m 1 = m 2 = 1 2 is, in a sense, the natural one. Examples of Lie algebra contractions with these parameter values have been seen before [37,38,49] in the study of WZW models on noncompact Lie groups. It is the only rescaling such that the products between elements inḠ m1 1 and G m2 2 only produce fields in H G . The resulting product structure is equivalent to that of a Z 2 -graded Lie algebra, where the even space is given by h G , and the odd space is the spaceḡ We now return to the earlier remark about uniform ǫ-scaling of the fields inḠ ℓ . For our general description, we have only considered rescaling all fields by the same ǫ-monomial. However, in some cases it may be interesting to rescale generating fields individually. For example, suppose that each field inḠ ℓ is rescaled separately by some s r ∈ R ≥0 , where r = 1, . . . , |Ḡ ℓ |. Then, , so to produce a field A 1 ∈ H G , we must have s 1 + s 2 = 1. Thus, there is substantial freedom in choosing s 1 and s 2 . An example of this is considered in Section 4.
We need not restrict ourselves to producing a field A 1 ∈ h G in the OPE X 1,ǫ × Y 1,ǫ . One can also use individual rescalings to produce interesting graded structures on the fields coming fromḠ ℓ . Comparing to (21), we see that uniform scaling with m ℓ > 0 cannot produce a field Z ℓ . However, if the field Z (ℓ) is rescaled by ǫ s1+s2 , the term will no longer vanish in the contraction limit. An example where one can introduce such a grading is discussed in Section 5.

Examples with H ∼ = A 2
We begin our discussion of examples of asymmetric contractions with the case where one has H ∼ = A 2 . We use this simplified setting to explore values of m = 1 2 . We remark that since H ∼ = A 2 , the parameter m 2 is not required.

4.1
The affine Nappi-Witten algebra gl(1) ֒→ sl (2) 1 2 G Here, we consider the contraction of an affine sl(2) algebra at level k (1) , with an affine gl(1) algebra at level k (2) , where the asymmetric contraction parameter is m = 1 2 . This example was first discussed in the papers [37,38], where the authors considered WZW models on non-compact Lie groups. In particular, they remark that a contraction procedure leads to the affine Nappi-Witten algebra [50], which we denote by H 4 . It has also been realised by alternative contraction procedures, namely in the context of pp-waves and Penrose limits [51,52].
We present it again here as a demonstration of a known case of our general construction, and for its physical relevance. In the asymmetric Galilean framework, this contraction (formally) leads to a 2-parameter generalisation of H 4 . However, we remark that one parameter may freely be set to zero using automorphisms of the algebra (see, e.g. [53,54]).
The OPA A 1 , corresponding to the Lie algebra sl(2) at level k (1) , is generated by fields The OPA H, corresponding to gl(1) at level k (2) , is generated by the field a, with OPE relation We see that the field h ∈ sl(2) generates such a gl(1) subalgebra at level k (1) .

4.2
The asymmetric Galilean Virasoro algebra Vir ֒→ (Vir 2 G ) n G Our next example is the contraction of the Galilean Virasoro algebra Vir 2 G with a Virasoro algebra Vir. This example is well-defined for m = 0, as the underlying mode algebra of Vir 2 G has an abelian ideal. The Galilean Virasoro OPA Vir 2 G is generated by the fields {T 0 , T 1 } and has OPEs given by The OPA Vir 2 G ⊗ Vir is generated by the fields {(T 0 ) (1) , (T 1 ) (1) , T (2) }. To perform the contraction (leaving the parameter m free), we form the fields and central parameters The resulting Galilean algebra Vir ֒→ (Vir 2 G ) m G is generated by the fields T 0 , T 1 ,T 1 , with nontrivial OPEs For m = 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 asymmetric N
In this example, we consider the N = 2 superconformal OPA. The algebra is generated by the fields {T, J, G + , G − }, where T, J are bosonic, and G ± are fermionic. The defining nontrivial OPE relations are The fields T, J generate a bosonic subalgebra which we denote by W (1, 2).

Examples with H A 1 and H A 2
In this section, we present examples where H ℓ is a proper subalgebra of A ℓ , for each ℓ = 1, 2. We take m 1 = m 2 = 1 2 to exhibit the behaviour seen in (30), of interest in the literature [37]. The first example involves OPAs associated with affine Lie algebras, which are algebras of Lie-type. The second example involves algebras which are not Lie-type: the asymmetric contraction of a W 4 = W (2, 3, 4) algebra with a W 3 = W (2, 3) algebra.
The nontrivial OPE relations in the ensuing Galilean algebra are given by In the Galilean algebra, there are no nontrivial relations of the form e r 1 × e s 1 or f r 1 × f s 1 for r, s ∈ {1, 2, 3}.
Here, Λ r,s,...,t denotes a quasi-primary field associated with the normally-ordered product of generating fields with conformal weights r, s, . . . , t, respectively. Our convention is that the normallyordered product of more than two fields is right-nested. Primes accompanying an index denote the number of derivatives acting on that component field. For example, We choose to normalise so that the term corresponding to the superscript has coefficient 1. Given a normally-ordered product of generating fields and their derivatives, the corresponding normallyordered field is determined, up to normalisation, by its conformal transformation properties.
Similarly, the W -algebra W 3 is generated by fields {T, W }, where T generates a Virasoro subalgebra, and W is a primary field of conformal weight 3. The OPE relations for W 3 are given by We remark that the W 3 algebra is not a subalgebra of the W 4 algebra.
The change of basis for the generating fields in this example is given by The asymmetrically contracted algebra is then generated by the fields {T 0 , T 1 , W 1 , W 2 , U 1 }, and the Lie-type OPE relations are given by 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].
To illustrate, consider the OPE Applying (13) to c (1) and expanding as a power series about ǫ small, we have and where λ (1) is the expression (45) as a function of c (1) . The fields appearing on the right-hand side of (50) are expanded using the inverse maps (13).
The resulting expressions are combined, and the limit ǫ → 0 is taken, resulting in where the subscripts on the field Λ 2,2 1,1 denote the corresponding subscripts on the leading normallyordered term, that is, Λ 2,2 1,1 = (T 1 T 1 ). The remaining nontrivial products on the contracted algebra are given by In this case, OPEs between primary fields produce only Galilean Virasoro fields and normallyordered products thereof. We also could have chosen to rescale W and U separately in the W 4 algebra. For example, we could scale W (1) by ǫ 1 2 and U (1) by ǫ 1 , thereby introducing a Galileantype structure amongst fields coming fromḠ 1 . With these changes in rescaling, the non-Lie-type OPEs become 6 The asymmetric Sugawara construction Here, we want to understand when it is possible to construct a Virasoro field from an asymmetric Galilean affine Lie algebra in a process analogous with the Sugawara construction. As Galilean contracted algebras contain an abelian ideal, they are not semisimple. There is a substantial amount of literature devoted to developing a Sugawara construction for non-semisimple Lie algebras [60,36,38,39,49].
The goal of the construction is to build a field from bilinears and derivatives of the currents which satisfies the Virasoro algebra OPE relations, and with respect to which the currents are primary fields of conformal weight 1. Such a field naturally satisfies all the constraints placed by conformal symmetry on the stress-energy tensor of a particular model. However, we cannot say exactly for what model the field would be a stress-energy tensor, as our interest is purely algebraic. It would be of interest in the physics literature to identify and understand the models for which the constructed field is in fact the stress-energy tensor.
It was shown in [36,49] that a necessary and sufficient condition for such a field to exist for an infinite-dimensional Lie algebra is that the algebra possesses a non-degenerate invariant symmetric bilinear form, here denoted by Ω = (Ω ab ).
From [49], we have that the general form of Ω is given by where κ is the Killing form (κ ab = f ac d f bd c ), and g = (g ab ) is an invariant bilinear form arising in the OPEs of the current algebra: The invariance of g is necessary for the associativity of the OPE. For simple algebras, Ω is a scalar multiple of the Killing form: and the corresponding Sugawara operator T (z) is then given by the well-known expression where (Ω ab ) and (κ ab ) are the inverses of (Ω ab ) and (κ ab ), respectively.
We begin with affine Lie OPAs g 1 , g 2 , each with an embedded affine Lie subalgebra h ℓ . Our notation for the currents of the algebra g ℓ is as follows: Currents in h ℓ are labelled with Roman letter group indices and are denoted by J a (ℓ) , a ∈ 1, . . . , dim h; the remaining currents are labelled with Greek letter group indices and are denoted by J α (ℓ) , α = 1, . . . , dim g ℓ − dim h; where g ℓ and h are the Lie algebras underlying g ℓ and h, respectively.
The algebra g ℓ then has OPEs given in their most general form by As before, we have dropped the subscript label for the structure constants of the subalgebras h ℓ , namely M ab and f ab c . We assume that each of the input algebras admits a Sugawara construction. We want to determine conditions under which the asymmetric Galilean algebra g m1 admits a non-degenerate symmetric invariant bilinear form Ω.
Before taking the limit ǫ → 0, we have the following ǫ-dependent OPEs: and The form g can be read off these relations. Relative to the ordered basis {J a 0,ǫ , J a 1,ǫ , J α 1,ǫ , J α 2,ǫ }, (g •• ) is a block matrix given by If the contraction parameters m 1 and m 2 are such that the limit ǫ → 0 does not exist for general M •• (ℓ) in the expressions above, then we are forced only to consider algebras for which some blocks in M •• (ℓ) (labelled by the upper indices) are zero, allowing the limit to be taken: • For m 1 = 0, we require M aβ (1) = M αβ (1) = 0, in which case 1 = 2 = 5 = 0.
• For m 2 = 1, we have 4 = 6 = 0 in the contraction limit. The Killing form on the pre-contraction algebra with respect to the same ordered basis is given by the block matrix where we have introduced the shorthand a i for currents J a i , and α 1ℓ for J α ℓ . The entries are given by with the remaining ones following from the symmetry of (κ •• ).
The entries κ a0a1 , κ a1a1 , κ a1α1 , κ a1α2 , all vanish in the contraction limit. We remark that it may occur that, for the contraction limit to exist for particular values of the contraction parameters, some structure constants in the above expression must be zero.
There are only four possible structures for the Killing form in the contraction limit. If m 1 = m 2 = 0, we have If m 1 = 0 and m 2 = 0, we have Similarly, if m 2 = 0 and m 1 = 0, we have Finally, if m 1 = 0 and m 2 = 0, the Killing form becomes From this, we see that the form Ω = 2g + κ can only be invertible if m 1 , m 2 ∈ {0, 1 2 , 1} and hence, only in these cases is a Sugawara construction possible on the contracted algebra, subject to the exact values of the structure constants M and f . By the results of [49], the central charge of the resulting Virasoro algebra generated by the Sugawara field is given by c = dim g 1 + dim g 2 − Ω rs κ rs . (73) In the situation that a Sugawara operator cannot be constructed (that is, one cannot find a nondegenerate invariant bilinear form), one could still consider an extension of the field algebra whereby one introduces an additional Virasoro field. This would result in the extended algebra being a conformal algebra; however, the resulting Virasoro field cannot be realised in terms of bilinears and derivatives of the currents which make up the original affine Galilean algebra.
We remark that in the papers [50,37,38,49], the authors consider algebras which can be realised using the general description above. In particular, examples given in [50,37,38] correspond to cases with g 2 ∼ = h 2 and m = 1 2 .

Discussion
In this paper, we have introduced a framework for performing Galilean-type contractions where the input algebras are not alike, relaxing a prior restriction on such contractions. 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 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. In a sense, Galilean contraction procedures provide an opposite construction to semigroup expansions which also give rise to Galilean subalgebras [23,61,24]. That is, the asymmetric contraction constrains a larger symmetry rather than enlarging a smaller symmetry. However, the question of whether every semigroup expanded algebra can be obtained by a suitable choice of contraction remains open.
In Section 6, we have demonstrated that for particular values of the contraction parameters, namely m 1 , m 2 ∈ {0, 1 2 , 1}, asymmetric Galilean algebras arising from affine Lie algebras admit a Sugawara construction. It remains to be determined whether one can also construct an accompanying field of conformal weight 2 which, along with the Sugawara field, generates a Galilean Virasoro algebra Vir 2 G (see Section 4). Such a field arises in the Sugawara construction for other Galilean algebras (see [32,35,21]); however, those results rely upon concrete expressions for the Sugawara fields of the input algebras.
We have not considered the representation theory of asymmetric Galilean algebras in this paper. However, representations of algebras that arise in the asymmetric Galilean framework have been studied elsewhere. For example, the representation theory of the affine Nappi-Witten algebra H 4 is considered in [62,63,64,54]. In particular, the authors of [54] demonstrate the existence of indecomposable yet reducible representations, indicating that H 4 may encode the symmetries of a logarithmic CFT.
It would be interesting to see if logarithmic behaviour can arise from the action of the Galilean Virasoro algebra, as the grading on Galilean algebras produces a non-diagonalisable action of the generators. For example, it is well known that the algebra H 4 admits a Sugawara construction. In addition, one can also construct a Galilean partner field such that these fields generate an action of the Galilean Virasoro algebra Vir 2 G .