]>NUPHB114680114680S0550-3213(19)30166-X10.1016/j.nuclphysb.2019.114680The AuthorsQuantum Field Theory and Statistical SystemsHigher-order Galilean contractionsJørgenRasmussenj.rasmussen@uq.edu.auChristopherRaymond⁎christopher.raymond@uqconnect.edu.auSchool of Mathematics and Physics, University of Queensland St Lucia, Brisbane, Queensland 4072, AustraliaSchool of Mathematics and PhysicsUniversity of Queensland St LuciaBrisbaneQueensland4072AustraliaSchool of Mathematics and Physics, University of Queensland St Lucia, Brisbane, Queensland 4072, Australia⁎Corresponding author.Editor: Hubert SaleurAbstractA 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 finite 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 W3 algebras.1IntroductionThe Galilean Virasoro algebra appears in studies of asymptotically flat three-dimensional spacetimes, see [1] and references therein. It can be constructed [2–6] as an Inönü-Wigner contraction [7–10] of a commuting pair of Virasoro algebras. The Galilean W3 algebra [11–14] likewise follows by contracting a pair of W3 algebras [15]. Many other Galilean conformal algebras with extended symmetries have been worked out [13,14,16], including contractions of higher-rank WN algebras [17–21]. Earlier works contributing to our understanding of non-relativistic systems with (typically non-affine) conformal symmetry can be found in [22–28].The constructions of the (affine) conformal algebras 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 finite number of symmetry algebras. 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.Higher-order Galilean Virasoro algebras have since appeared in work [29] on the so-called S-expansion method [30]. Moreover, in a recent study [31] of a non-abelian enlargement of the Poincaré symmetry algebra associated with a Chern-Simons theory on AdS3, the flat-space asymptotics of the ensuing AdS-Lorentz symmetry algebra [32] has been found to give rise to a third-order Galilean Virasoro algebra. As it has also been found [33–37] that the asymptotic algebra of higher-spin gravity on AdS3 exhibits a W-symmetry, it thus seems natural to expect that higher-order Galilean W-algebras will play a role in higher-spin Chern-Simons models on AdS spacetimes with enlarged Poincaré symmetry.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 [38] 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 W3 algebra and thereby obtain an infinite hierarchy of higher-order W3 algebras. Section 5 contains some concluding remarks.2Galilean contractions2.1Operator-product algebras and star relationsIt is often convenient to combine the generators of the symmetry algebra of a conformal field theory into generating fields of the form(2.1)A(z)=∑n∈−ΔA+ZAnz−n−ΔA, where ΔA 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(2.2)A(z)B(w)=∑n=−∞ΔA+ΔB[AB]n(w)(z−w)n. Here, if nonzero, [AB]n is a field of conformal weight ΔA+ΔB−n. As the nontrivial information of an OPE is stored in the singular terms, one often ignores the non-singular terms, writing(2.3)A(z)B(w)∼∑n=1ΔA+ΔB[AB]n(w)(z−w)n. 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(2.4)[TA]2=ΔAA,[TA]1=∂A. Such a field is quasi-primary if [TA]3=0, and primary if [TA]n=0 for all n≥3. Let BA denote a basis for the linear span of the quasi-primary fields in A. Relative to this, the OPE (2.3) reads(2.5)A(z)B(w)∼∑Q∈BACA,BQ(∑n=0ΔA+ΔB−ΔQβΔA,ΔBΔQ;n∂nQ(w)(z−w)ΔA+ΔB−ΔQ−n), with structure constants CA,BQ and(2.6)βΔA,ΔBΔQ;n=(ΔA−ΔB+ΔQ)nn!(2ΔQ)n,(x)n=∏j=0n−1(x+j). Compactly, we may represent the OPE (2.5) by the so-called star relation(2.7)A⁎B≃∑Q∈BACA,BQ{Q}, where {Q} represents the sum over n. We refer to [14,39] for more details on the algebraic structure of an OPA.2.2Contraction prescriptionFor N∈N, we consider the tensor-product algebra(2.8)A⊗N=⨂i=0N−1A(i), 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 ϵ∈C, let(2.9)Ai,ϵ=ϵi∑j=0N−1ωijA(j),ci,ϵ=ϵi∑j=0N−1ωijc(j),i=0,…,N−1, where A(j) (respectively c(j)) denotes the field A∈A(j) (respectively the central parameter c), and ω is the principal Nth root of unity,(2.10)ω=e2πi/N. For ϵ≠0, the map(2.11)A⊗N→A⊗N,(A(0),…,A(N−1))↦(A0,ϵ,…,AN−1,ϵ), (and similarly for the central parameters) is invertible, with(2.12)A(i)=1N∑j=0N−1ω−ijϵ−jAj,ϵ,i=0,…,N−1.In the special case N=2, we have ω=−1 and(2.13)A0,ϵ=A(0)+A(1),A1,ϵ=ϵ(A(0)−A(1)), with inverses(2.14)A(0)=12(A0,ϵ+1ϵA1,ϵ),A(1)=12(A0,ϵ−1ϵA1,ϵ). In [14], these fields are denoted by(2.15)A=A(0),A¯=A(1),Aϵ+=A0,ϵ,Aϵ−=A1,ϵ.For ϵ=0, the map (2.11) is singular (unless N=1), indicating that a new algebraic structure emerges in the limit ϵ→0, where(2.16)Ai,ϵ→Ai,ci,ϵ→ci. If the resulting algebra is a well-defined OPA, we refer to it as the Nth-order Galilean OPA AGN. 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 2.3.2.3Galilean Virasoro and affine Kac-Moody algebrasThe Virasoro OPA Vir of central charge c is of Lie type and generated by T, with star relation(2.17)T⁎T≃c2{I}+2{T}. The Galilean Virasoro algebra of order N, VirGN, is generated by the fields T0,…,TN−1, with central parameters c0,…,cN−1 and star relations(2.18)Ti⁎Tj≃{ci+j2{I}+2{Ti+j},i+j<N,0,i+j≥N. This yields an infinite family of extended Virasoro algebras, {VirGN|N∈N}, where VirG1≅Vir while VirG2 is the familiar Galilean Virasoro algebra [2–6,13,14]. For small N, the Galilean Virasoro algebras VirGN have recently appeared in [29].The OPE of two fields in an affine Kac-Moody (or current) algebra gˆ (where the central element K has been replaced by kI, with k the level) is given by(2.19)Ja(z)Jb(w)∼κabk(z−w)2+fabcJc(w)z−w, where fabc 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 gˆGN is generated by {Jia|a=1,…,dimg;i=0,…,N−1}, with nontrivial star relations(2.20)Jia⁎Jjb≃κabki+j{I}+fabc{Ji+jc},i+j∈{0,…,N−1}.In the limit N→∞, we obtain the algebra gˆG∞; it is generated by {Jia|a=1,…,dimg;i∈N0}, with nontrivial star relations(2.21)Jia⁎Jjb≃κabki+j{I}+fabc{Ji+jc}. It follows that(2.22)gˆG∞≅gˆ⊗C[t] and(2.23)gˆGN≅gˆ⊗C[t]/〈tN〉, extending to general N the construction of the Takiff algebras considered in [40,41]. We similarly have(2.24)VirG∞≅Vir⊗C[t],VirGN≅Vir⊗C[t]/〈tN〉.3Generalised Sugawara constructionsIn [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 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 3.2.3.1Galilean Sugawara constructionFor the generators of VirGN, we make the ansatz(3.1)Ti=∑r,s=0N−1λir,sκab(JraJsb),i=0,…,N−1, where κab are elements of the inverse Killing form on g. The task is now to determine the coefficients λir,s such that(3.2)Ti⁎Jja≃{{Ji+ja},i+j∈{0,…,N−1},0,i+j≥N. We show below that this is indeed possible. It subsequently follows that VirGN=〈T0,…,TN−1〉, with central charges(3.3)c0=Ndimg,c1,…,cN−1=0.First, we compute the OPE(3.4)Jja(z)Ti(w)=∑r,s=0N−1λir,s(z−w)2{kj+rJsa(w)+kj+sJra(w)+2h∨Jj+r+sa(w)}+∑r,s=0N−1λir,sκbcz−w{fabd(Jj+rdJsc)(w)+facd(JrbJj+sd)(w)}, where h∨ is the dual Coxeter number of g, arising through the relation κbcfabdfdce=2h∨δea. To satisfy (3.2), the first sum must equal Ji+ja(w)/(z−w)2 while the second sum must vanish. The second-sum constraint implies that(3.5)λir,s={λiℓ,N−1,r+s=N−1+ℓ(ℓ=0,…,N−1),0,r+s<N−1. This leaves N coefficients, λi0,N−1,…,λiN−1,N−1, for each i∈{0,…,N−1}. The first-sum constraint then requires that(3.6)2∑n=jN−1∑ℓ=0n−jλiℓ,N−1kN−1−n+j+ℓJna+2Nh∨λi0,N−1δj,0JN−1a={Ji+ja,i+j≤N−1,0,i+j≥N. For each i, this translates into a lower-triangular system of linear equations:(3.7)2(kN−1kN−2kN−1⋮⋱⋱k1⋱kN−1k0′k1⋯kN−2kN−1)(λi0,N−1⋮⋮λiN−1,N−1)=(0⋮1⋮0), where k0′=k0+Nh∨, 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 kN−1≠0, in which case the problem reduces to inverting the lower-triangular Toeplitz matrix(3.8)A=(1a11a2⋱⋱⋮⋱⋱1aN−1⋯a2a11), where(3.9)am=kN−1−m+Nh∨δm,N−1kN−1,m=1,…,N−1. The inverse is itself a lower-triangular Toeplitz matrix with 1's on the diagonal,(3.10)A−1=(b0b1b0b2⋱⋱⋮⋱⋱b0bN−1⋯b2b1b0),b0=1, and we find that the nontrivial matrix elements are given by(3.11)bn=∑p∈(N0)n(−1)|p|δ||p||,n|p|!p1!⋯pn!a1p1⋯anpn, where(3.12)|p|=∑i=1npi,||p||=∑i=1nipi,p=(p1,…,pn). It follows that(3.13)λiℓ,N−1={0,ℓ=0,…,i−1,bℓ−i2kN−1,ℓ=i,…,N−1, so the unique expression for Ti of the form (3.1) is given by(3.14)Ti=∑n=0N−1−ibn2kN−1∑t=0N−1−i−nκab(Ji+n+taJN−1−tb). For N=2, we thus recover the Galilean Sugawara construction obtained in [14],(3.15)T0=κab2k1{(J0aJ1b)+(J1aJ0b)}−k0+2h∨2(k1)2κab(J1aJ1b),T1=κab2k1(J1aJ1b), whereas for N=3, we find the new expressionsT0=κab2k2{(J0aJ2b)+(J1aJ1b)+(J2aJ0b)}−k1κab2(k2)2{(J1aJ2b)+(J2aJ1b)}+(k1)2−(k0+3h∨)k22(k2)3κab(J2aJ2b),T1=κab2k2{(J1aJ2b)+(J2aJ1b)}−k1κab2(k2)2(J2aJ2b),T2=κab2k2(J2aJ2b).For each i=0,…,N−1, the value of the central parameter ci follows from the leading pole in the OPE T0(z)Ti(w). Using (3.14), we compute(3.16)T0(z)Ti(w)∼∑n=0N−1−ibn2kN−1∑t=0N−1−i−nκabκabkN−1+i+n(z−w)4+⋯, suppressing all subleading poles. Since ka=0 for a≥N, this term is zero unless n+i=0, that is, unless n=i=0. From κabκab=dimg, we then obtain the announced result (3.3).3.2Sugawara before Galilean contractionOn the individual factors of gˆ⊗N, the Sugawara construction is given by(3.17)T(i)=κab2(k(i)+h∨)(J(i)aJ(i)b),c(i)=k(i)dimgk(i)+h∨,i=0,…,N−1. Changing basis as in (2.9) introduces(3.18)Ti,ϵ=ϵi∑j=0N−1ωijT(j)=ϵi∑j=0N−1ωij∑ℓ,ℓ′=0N−1ω−j(ℓ+ℓ′)ϵ−ℓ−ℓ′κab(Jℓ,ϵaJℓ′,ϵb)2N(∑m=0N−1ω−jmϵ−mkm,ϵ+Nh∨)=12NkN−1,ϵ∑j,ℓ,ℓ′=0N−1(ωjϵ)N−1+i−ℓ−ℓ′κab(Jℓ,ϵaJℓ′,ϵb)1+∑m=1N−1am,ϵ(ωjϵ)m, where(3.19)am,ϵ=kN−1−m,ϵ+Nh∨δm,N−1kN−1,ϵ,m=1,…,N−1. Now, using that a lower-triangular N×N Toeplitz matrix of the form (3.8) decomposes as(3.20)A=I+a1η+⋯+aN−1ηN−1, where I is the identity matrix and η the N×N matrix(3.21)η=(0100⋱⋱⋮⋱⋱00⋯010), we can use the result for A−1 in (3.10)-(3.11) to expand the expression for Ti,ϵ in powers of ϵ. We thus find that(3.22)Ti,ϵ=12NkN−1,ϵ∑j,ℓ,ℓ′=0N−1(ωjϵ)N−1+i−ℓ−ℓ′κab(Jℓ,ϵaJℓ′,ϵb)(∑n=0N−1bn,ϵ(ωjϵ)b+O(ϵN))=12NkN−1,ϵ∑ℓ,ℓ′,n=0N−1bn,ϵκab(Jℓ,ϵaJℓ′,ϵb)∑j=0N−1(ωjϵ)N−1+i−ℓ−ℓ′+n+O(ϵi+1), where(3.23)b0,ϵ=1,bn,ϵ=∑p∈(N0)n(−1)|p|δ||p||,n|p|!p1!⋯pn!a1,ϵp1⋯an,ϵpn,n=1,…,N−1. The summation over j yields a factor of the form(3.24)∑j=0N−1ωj(N−1+i−ℓ−ℓ′+n)={N,N−1+i−ℓ−ℓ′+n≡0(modN),0,N−1+i−ℓ−ℓ′+n≢0(modN), and since N−1+i−ℓ−ℓ′+n>−N, it follows that the Ti,ϵ-coefficients to ϵm for m negative are 0. The limit ϵ→0 is therefore well-defined, resulting in(3.25)Ti=12kN−1∑ℓ,ℓ′,n=0N−1bnκab(JℓaJℓ′b)δN−1+i−ℓ−ℓ′+n,0, whose nonzero terms are seen to match the expression in (3.14).For the central parameters, we evaluate(3.26)ci,ϵ=ϵi∑j=0N−1ωijc(j)=ϵi∑j=0N−1ωij∑ℓ=0N−1ω−jℓϵ−ℓkℓ,ϵdimg∑ℓ′=0N−1ω−jℓ′ϵ−ℓ′kℓ′,ϵ+Nh∨=dimgkN−1,ϵ∑ℓ,n=0N−1bn,ϵkℓ,ϵ∑j=0N−1(ωjϵ)N−1+i−ℓ+n+O(ϵi+1), from which it follows that(3.27)ci=dimgkN−1∑ℓ,n=0N−1bnkℓδN−1+i−ℓ+n,0=Ndimgδi,0, again confirming (3.3).4Galilean W3 algebrasHigher-order Galilean contractions can also be applied to W-algebras. Below, we present the results for the W3 algebra.4.1W3 algebraThe W3 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(4.1)T⁎T≃c2{I}+2{T},T⁎W≃3{W},W⁎W≃c3{I}+2{T}+3222+5c{Λ2,2}, where(4.2)Λ2,2=(TT)−310∂2T, is quasi-primary.4.2Galilean W3 algebra of order 2Following [13,14], we now recall the structure of the second-order Galilean W3 algebra [11,12,44]. It is generated by the four fields T0,T1,W0,W1, with central parameters c0 and c1, and nontrivial star relations(4.3)Ti⁎Tj≃ci+j2{I}+2{Ti+j},Ti⁎Wj≃3{Wi+j},i+j∈{0,1}, and(4.4)W0⁎W0≃c03{I}+2{T0}+645c1{Λ0,12,2}−32(44+5c0)25c12{Λ1,12,2},W0⁎W1≃c13{I}+2{T1}+325c1{Λ1,12,2}, where(4.5)Λ0,12,2=(T0T1)−310∂2T1,Λ1,12,2=(T1T1) are quasi-primary. We note that a nonzero c1 can be scaled away by renormalising as T1,W1→Tˆ1=T1c1,Wˆ1=W1c1.4.3Infinite hierarchyFor any N∈N, the algebra W3⊗N is generated by the 2N fields {T(i),W(i)|i=0,…,N−1}, and has central charges {c(i)|i=0,…,N−1}. As outlined in the following, the corresponding Galilean algebra is well-defined. In tune with the general prescription in Section 2.2, we thus confirm that the Nth-order Galilean W3 algebra (W3)GN is generated by the fields {Ti,Wi|i=0,…,N−1} and has central parameters {ci|i=0,…,N−1}.First, it straightforwardly follows that(4.6)Ti⁎Tj≃ci+j2{I}+2{Ti+j},Ti⁎Wj≃3{Wi+j},i+j∈{0,…,N−1}, while(4.7)Ti⁎Tj≃Ti⁎Wj≃Wi⁎Wj≃0,i+j≥N. To determine Wi⁎Wj in (W3)GN for i+j=0,…,N−1, we compute the corresponding star relation Wi,ϵ⁎Wj,ϵ in W3⊗N,(4.8)Wi,ϵ⁎Wj,ϵ=ϵi+j∑r,s=0N−1ωir+jsW(r)⁎W(s)≃ϵi+j∑r=0N−1ω(i+j)r[c(r)3{I}+2{T(r)}+3222+5c(r){Λ(r)2,2}]=ci+j,ϵ3{I}+2{Ti+j,ϵ}+ϵi+j∑r=0N−13222+5c(r)ω(i+j)r{Λ(r)2,2}. Recycling the expansion techniques of Section 3, we find that(4.9)∑r=0N−13222+5c(r)(ωrϵ)i+jΛ(r)2,2=325NcN−1,ϵ∑n,ℓ,ℓ′=0N−1bn,ϵ∑r=0N−1(ωrϵ)N−1+i+j−ℓ−ℓ′+n(Tℓ,ϵTℓ′,ϵ)−4825cN−1,ϵ∑n,ℓ=0N−1bn,ϵ∑r=0N−1(ωrϵ)N−1+i+j−ℓ+n∂2Tℓ,ϵ+O(ϵi+j+1),where bn,ϵ (and bn appearing in (4.11) below) are given as in (3.23) (respectively (3.11)), but now based on(4.10)am,ϵ=cN−1−m,ϵ+22N5δm,N−1cN−1,ϵ,am=cN−1−m+22N5δm,N−1cN−1,m=1,…,N−1. In the limit ϵ→0, this yields(4.11)∑r=0N−13222+5c(r)(ωrϵ)i+jΛ(r)2,2→∑n=0N−1−i−j32bn5cN−1∑t=0N−1−i−j−n(Ti+j+n+tTN−1−t)−48N25cN−1∂2TN−1δi,0δj,0. Observing that, for every pair r,s∈{0,…,N−1} such that r+s∈{N−1,…,2N−2},(4.12)Λr,s2,2=(TrTs)−310∂2TN−1δr+s,N−1 is a quasi-primary field with respect to T0, we then conclude that, for i+j∈{0,…,N−1},(4.13)Wi⁎Wj≃ci+j3{I}+2{Ti+j}+∑n=0N−1−i−j32bn5cN−1∑t=0N−1−i−j−n{Λi+j+n+t,N−1−t2,2}. Using that Λr,s2,2=Λs,r2,2, this can be written as(4.14)Wi⁎Wj≃ci+j3{I}+2{Ti+j}+∑n=0N−1−i−j32bn5cN−1(∑t=0⌊N−2−i−j−n2⌋2{Λi+j+n+t,N−1−t2,2}+{ΛN−1+i+j+n2,N−1+i+j+n22,2}), where the last term is present only if N−1+i+j+n2 is integer.Let us illustrate our findings by summarising the nontrivial star relations for the third-order Galilean algebra (W3)G3: The six generating fields T0,T1,T2,W0,W1,W2 satisfy (4.6)-(4.7) with N=3 as well as(4.15)W0⁎W0≃c03{I}+2{T0}+645c2{Λ0,22,2}+325c2{Λ1,12,2}−64c15(c2)2{Λ1,22,2}−32[(66+5c0)c2−5(c1)2]25(c2)3{Λ2,22,2},(4.16)W0⁎W1≃c13{I}+2{T1}+645c2{Λ1,22,2}−32c15(c2)2{Λ2,22,2},(4.17)W0⁎W2≃W1⁎W1≃c23{I}+2{T2}+325c2{Λ2,22,2}, where(4.18)Λ0,22,2=(T0T2)−310∂2T2,Λ1,12,2=(T1T1)−310∂2T2,Λ1,22,2=(T1T2),Λ2,22,2=(T2T2) are quasi-primary.4.4RenormalisationWe now consider (W3)GN in the special case where(4.19)ci=ci,i=1,…,N−1, for some c∈C×, leaving only two independent central parameters: the central charge c0 and c. The am coefficients in (4.10) then simplify to(4.20)am=c−m(1+[c0+22N5−1]δm,N−1),m=1,…,N−1. Correspondingly, the inverse of the matrix A in (3.20) is given by(4.21)A−1=I−c−1η+[1−c0−22N5](c−1η)N−1, so (for N>2)(4.22)b0=1,b1=−c−1,bn=0(1<n<N−1),bN−1=[1−c0−22N5]c−(N−1). Let us also introduce the renormalised generators(4.23)Tˆi=c−iTi,Wˆi=c−iWi,i=0,…,N−1, and ditto quasi-primary fields(4.24)Λˆr,s2,2=c−r−sΛr,s2,2. In terms of these, the nontrivial star relations are given by (i+j∈{0,…,N−1})(4.25)Tˆi⁎Tˆj≃c0δi+j,02{I}+2{Tˆi+j},Tˆi⁎Wˆj≃3{Wˆi+j}, and(4.26)Wˆi⁎Wˆj≃c0δi+j,03{I}+2{Tˆi+j}+325[1−c0−22N5]{ΛˆN−1,N−12,2}δi+j,0+325∑n=0,1∑t=0N−1−i−j−n(−1)n{Λˆi+j+n+t,N−1−t2,2}. 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(4.27)Jˆia=k−iJia,Tˆi=k−iTi,i=0,…,N−1, where ki=ki, i=1,…,N−1, for some k∈C×. The renormalised Galilean Virasoro generators are then given by(4.28)Tˆi=12∑n=0,1∑t=0N−1−i−nκab(Jˆi+n+taJˆN−1−tb)+12[1−k0−Nh∨]κab(JˆN−1aJˆN−1b)δi,0, while the nontrivial star relations read (i+j∈{0,…,N−1})(4.29)Jˆia⁎Jˆjb≃κabk0δi+j,0{I}+fabc{Jˆi+jc},Tˆi⁎Jˆja≃{Jˆi+ja},Tˆi⁎Tˆj≃Ndimg2{I}δi+j,0+2{Tˆi+j}.5DiscussionIn our continued exploration [13,14] of Galilean contractions, we have presented a generalisation of the contraction prescription to allow for inputs of any finite number of OPAs or vertex algebras. This has resulted in hierarchies of higher-order Galilean conformal algebras, including Virasoro, affine Kac-Moody and W3 algebras.Asymmetric Galilean N=1 superconformal algebras, corresponding to an N=(1,0) supersymmetry, can be obtained [42–45] 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 W3⊗Vir, where one contracts the Virasoro subalgebra of W3 with a separate Virasoro algebra. This yields an OPA generated by fields T0,T1,W, with nonzero star relations (i+j∈{0,1})(5.1)Ti⁎Tj≃ci+j2{I}+2{Ti+j},T0⁎W≃3{W},W⁎W≃c13{I}+2{T1}+325c1{Λ1,12,2}. 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 [46–51]. 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 [17,52–60] 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 [45,51,61], a systematic approach and general results are still lacking.AcknowledgementsJR was supported by the Australian Research Council under the Discovery Project scheme, project number DP160101376. CR was funded by a University of Queensland Research Scholarship. 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