Logarithmic Exotic Conformal Galilean Algebras
aa r X i v : . [ h e p - t h ] N ov Logarithmic Exotic Conformal Galilean Algebras
Malte Henkel a , Ali Hosseiny b,d , and Shahin Rouhani c,da Groupe de Physique Statistique, Institut Jean Lamour (CNRS UMR 7198),Universit´e de Lorraine Nancy,B.P. 70239, F–54506 Vandœuvre-l`es-Nancy Cedex, France. b Department of Physics, Shahid Beheshti UniversityG.C., Evin, Tehran 19839, Iran. c Department of Physics, Sharif University of TechnologyP.O. Box 11165-9161, Tehran, Iran. d School of Particles and Accelerators,Institute for Research in Fundamental Sciences (IPM)P.O. Box 19395-5531, Tehran, Iran. e-mail: [email protected] , al [email protected] and [email protected] Abstract
Logarithmic representations of the conformal Galilean algebra (CGA) and the Ex-otic Conformal Galilean algebra ( ecga ) are constructed. This can be achieved bynon-decomposable representations of the scaling dimensions or the rapidity indices,specific to conformal galilean algebras. Logarithmic representations of the non-exoticCGA lead to the expected constraints on scaling dimensions and rapidities and alsoon the logarithmic contributions in the co-variant two-point functions. On the otherhand, the ecga admits several distinct situations which are distinguished by differentsets of constraints and distinct scaling forms of the two-point functions. Two dis-tinct realisations for the spatial rotations are identified as well. The first example ofa reducible, but non-decomposable representation, without logarithmic terms in thetwo-point function is given.
Introduction
Logarithmic conformal field theories (LCFT) arose, by noticing that the independent solu-tions of the null vector equation governing the behaviour of the four point function, couldcoincide in certain cases; giving rise to new independent solutions involving logarithms [1, 2].Previously this possibility was ignored because unitarity ruled it out; however, applicationsfor such non-unitary theories could be found within condensed-matter or statistical physics(for reviews of LCFT and applications see [3–7]). On another front, recent developments hasattracted interest towards non-relativistic conformal field theories (NRCFT) [8–18]. Theseare theories based on attempted extensions of the Galilean symmetries, the motivation be-ing that they may apply to low-energy and/or time-dependent systems in condensed-matteror statistical physics. The best-known special cases of such symmetry algebras are theSchr¨odinger algebra and the conformal Galilei algebra (CGA), both to be defined below.The natural question arises as to whether logarithmic correlators may be found for suchNRCFTs [19–21], for a recent review see [22]. The answer is affirmative. Furthermore,applications including the one-dimensional contact process (Reggeon field-theory) and theone-dimensional Kardar-Parisi-Zhang equation have been suggested [23, 24]. In this paper,we present new logarithmic correlators for the Exotic Galilean Algebra ( ecga ) [25,26], whichis CGA in 2+1 dimensions, but with an ‘exotic’ central charge.Naturally, non-relativistic conformal symmetries are based on Galilean symmetry. AGalilean transformation ( x x ′ , t t ′ ) acts on a point x in d -dimensional Euclideanspace, at a given time t , according to: x ′ = R x + b t + a , t ′ = t + c (1.1)where R ∈ SO ( d ) is a d × d rotation matrix, b and a are d -dimensional vectors and c is aconstant. However, we shall look at larger symmetries. For instance the symmetry group(called the Schr¨odinger group ) of the free Schr¨odinger equation is larger: x ′ = R x + b t + a f t + g , t ′ = dt + cf t + g dg − f c = 1 (1.2)The Lie algebra ( Schr¨odinger algebra ) spanned by the infinitesimal generators of the trans-formations (1.2) is given below for 1 + 1 dimensions. Being non-semi-simple, this Lie algebraadmits a non-trivial central charge, related to projective transformations of the solutions ofthe Schr¨odinger equation. It is related to the (non-relativistic) ‘mass’ M of the system. Thiscan be generalised straightforwardly to what we shall call l -Galilei algebras by admitting amore complex transformation [28, 29]; x ′ = R x + b l t l + .... + b t + b ( f t + g ) l , t ′ = dt + cf t + g dg − f c = 1 (1.3)where the b i , i = 0 , , . . . , l are d -dimensional vectors. The transformations (1.3) form aclosed set and their infinitesimal generators span a closed Lie algebra only for l ∈ Z half-integer or integer. The Schr¨odinger group and its Lie algebra are recovered for l = ; the case In some papers these are referred to as spin − l Galilei algebras [27], however the index l has nothingto do with spin. = 1 gives the conformal Galilei group and its Lie algebra, the Conformal Galilean Algebra (CGA) [30]. These transformations, and more generally those of (1.3), have in common theexistence of a well-defined dynamical exponent z such that under a dilatation x → λ x , t → λ z t (1.4)such that z = 1 /l for the l -Galilei transformations (1.3). The two important special cases l = , t to arbitrary conformal transformations. Takingthe projective terms describing the transformation of the wave functions into account,the only cases with local generators which close as a Lie algebra are, besides evidentlyconformal transformations in space-time, the cases l = , z , restriction to time-like andlight-like geodesics reproduces exactly the Schr¨odinger algebra and the CGA, for z = 2and z = 1, respectively [16].Physical applications either refer to strongly anisotropic systems at equilibrium, where the‘time’ t is just a name for a peculiar spatial direction with strongly modified interactionssuch that z = θ is better referred to as an ‘anisotropy exponent’ (paradigmatic examples areuniaxial Lifshitz points in lattice spin models with competing interactions); or else to real dy-namics, at or far away from equilibrium. In the first case, co-variant n -point functions (suchas we shall calculate later on) will represent physical correlators; the second case, causalityconstraints imply that n -point functions are to be interpreted as response functions withrespect to some external perturbation. See [18] for an introduction and overview on recentresults. For brevity, we shall refer throughout to the two-point functions to be computed as‘correlators’.Finally returning to the Lie algebra of the symmetry transformations (1.3), in 1 + 1dimensions it spanned by the generators: H = − ∂ t , P n = − t n ∂ x ; n = 0 , ..., l,D = − ( t∂ t + lx∂ x ) , C = − (2 ltx∂ x + t ∂ t ) . (1.5)with the following non-vanishing commutators[ D, H ] = H, [ D, C ] = − C, [ C, H ] = 2 D, [ D, P n ] = ( l − n ) P n , [ H, P n ] = − nP n − , [ C, P n ] = (2 l − n ) P n +1 . (1.6)Known physical realisations of these algebras are known for l = 1 / , for l = 1 as the CGA and for l = 2 and l = 3 in the Lifshitz points of first and An algebraic method of derivation uses an embedding into a parabolic sub-algebra of the conformalalgebra in d + 2 dimensions, see [31] for details. Especially in the phase-ordering kinetics far from equilibrium for spin systems quenched to temperatures
T < T c below the critical temperature T c >
0, when for a non-conserved order parameter one has naturally z = 2 [18, 32, 33]. It remains an open problem to find physical realisations for genericvalues of l .CGA is special because it can be obtained from the relativistic conformal algebra throughcontraction. When contracting, in some sense we are investigating the symmetry for lowvelocities. In other words we allow: x → x c , t → t, c → ∞ . (1.7)In 1 + 1 dimensions, CGA is even more special since it has an infinite-dimensional extension(which is called ‘full CGA/altern-Virasoro algebra’ in the literature, contains a Virasorosub-algebra and admits two independent central charges [38]) which in turn can be obtainedfully from contraction [39,40]. This infinite-dimensional extension of the CGA is almost solv-able [41], a property which helps to investigate logarithmic representations and holographicrealisation easily [19, 20].Here, we study some properties of the finite-dimensional CGA (and leave aside its infinite-dimensional extensions). In 2 + 1 dimensions, CGA admits a non-trivial central extension(the so-called “exotic” central charge [25, 26]) which forbids Galilean boosts to commute,reminiscent of non-commutative theories. Its physical significance has been of interest [42,43].The central charge can also be obtained by contraction and two-point function is realisedusing auxiliary coordinates [27]. In this paper we consider this exotic algebra ecga andshow that logarithmic representations exist. A new feature arises in the CGA and the ecga in that the rapidities allow for extra types of logarithmic representations, which we shallconstruct. We work out two-point functions for realisations in which the rapidity index isincluded. The ‘exotic’ extension of the CGA in 2 + 1 dimensions leads to several unexpectedresults on the form of the two-point functions; notably, we discuss the consequences of twodistinct realisations of rotation-invariance (which from a purely algebraic point of view areindistinguishable). We hope these results to be useful in future attempts in identifyingspecific models with conformal galilean symmetries.This paper is organised as follows: In section 2 we give a very brief presentation ofLCFT, and recall the derivation of the two-point functions in logarithmic representationsof the LCFT, the Schr¨odinger algebra and the CGA, using the elegant formalism of nil-potent variables. In section 3 we give a short introduction to the exotic CGA, and derivethe two-point functions, both for scalar and logarithmic representations. Some conclusionsare presented in section 4, with a table summarising our findings in a compact manner.Several appendices treat technical aspects of the calculations, either in the ecga or onrotation-invariance. See [12, 28]. When considering the uniaxial Lifshitz points in systems with competing interactions suchas the ANNNO( n ) model, field-theoretic two-loop calculations have shown that the anisotropy exponent θ − = O( ε ) in d = 4 . − ε dimensions or θ − = O(1 /n ), which known, non-vanishing coefficients whichare of the order ≈ − − − [34–37]. The ANNNS model corresponds to n → ∞ . Logarithmic CFT: background
Logarithmic conformal field theories (LCFTs) arise when indecomposable but reducible rep-resentations of the Virasoro algebra are taken [1, 2] (for reviews see [3–6]). When the actionof the scaling operator on the Verma module is not diagonal it gives rise to staggered mod-ules [44, 45]. In the simplest case, the highest weight primary operator and its logarithmicpartner form a rank-2 Jordan cell: L φ h ( Z ) | i = hφ h ( Z ) | i , L ψ h ( Z ) | i = hψ h ( Z ) | i + φ h ( Z ) | i . (2.1)There is a simple method for dealing with case by introducing nilpotent variables θ i whichsatisfy the following relations: θ i = 0 , θ i θ j = θ j θ i . (2.2)These nilpotent variables also admit complex conjugation which go into the anti-holomorphicpart of the primary operators:¯ θ i = 0 , ¯ θ i θ j = θ j ¯ θ i . (2.3)Now we can define our super-fields asΦ( z, θ ) = φ ( z ) + θψ ( z ) , (2.4)and thereby equation (2 .
1) is written compactly as [46]: L | h + θ i = ( h + θ ) | h + θ i , (2.5)where the state | h + θ i is: | h + θ i = | h, i + θ | h, i . (2.6)This method allows a quick calculation of the two-point function. Concentrating on theholomorphic part of quasi-primary operators we obtain [46]: G ( z , θ ; z , θ ) = h Φ ( z , θ )Φ ( z , θ ) i = g ( θ , θ )( z − z ) − ( h + θ + h + θ ) δ h ,h . (2.7)where g ( θ , θ ) is given by g ( θ , θ ) = a ( θ + θ ) + bθ θ . (2.8)and a, b are normalisation constants. Now, expanding both sides of (2 .
7) in powers of θ , ,one obtains the well-known logarithmic CFT two-point functions (with z := z − z )) h φ ( z ) φ ( z ) i = 0 , h φ ( z ) ψ ( z ) i = a z − h δ h ,h , h ψ ( z ) ψ ( z ) i = z − h ( b − a ln z ) δ h ,h . (2.9)This offers a simple and fast way of obtaining logarithmic correlators in other algebras aswell, as we shall demonstrate below. Of course, we merely discussed here the most simplescenario for the appearance of logarithmic representations and shall leave to future work thedescription of more complex situations. 4 .2 Logarithmic representations of the Schr¨odinger algebra The Schr¨odinger algebra is the symmetry of the free Schr¨odinger equation. It is naturallytied in with Galilean symmetry. It is the smallest ( l = 1 /
2) element of the l − Galilei algebras, plus a central extension: [ P i , P j ] = M δ ij (2.10)where the scalar M is the non-relativistic mass and i, j = 1 , . . . , d . The Schr¨odinger algebra sch ( d ) has a well-known infinite-dimensional extension (with a Virasoro sub-algebra) whichis now usually called the ‘Schr¨odinger-Virasoro algebra’ ( sv ) [10]. In 1 + 1 dimensions, thealgebra sv is represented by differential operators as (with n ∈ Z and m ∈ Z + ): X n = − t n +1 ∂ t −
12 ( n + 1) t n x∂ x − n ( n + 1) M t n − x − ( n + 1) ht n ,Y m = − t m +1 / ∂ x − ( m + 12 ) t m − / M x,M n = −M t n . (2.11)These generators make up the dynamical symmetry algebra of the free Schr¨odinger equation S φ = 0, with the Schr¨odinger operator S := 2 M ∂ t − ∂ x = 2 M X − − ( Y − ) . All generators(2.11) of sch (1) := h X , ± , Y ± , M i commute with S , with the two exceptions (cid:2) S , X (cid:3) = −S , (cid:2) S , X (cid:3) = − tS − M (cid:18) h − (cid:19) (2.12)such that solutions of S φ = 0 which have h = will be mapped onto another solution (andan obvious generalisation to d ≥ Representations of this algebra sch (1) maybe constructed by invoking scaling states: X | h i = 0 . (2.13)Now, similar to CFT, a rank 2 logarithmic representation may be found where two statesexist, | h, i and | h, i such that the action of X on them is non-diagonizable X | h, i = 0 , X | h, i = | h, i . (2.14)We follow the formalism of the previous sub-section. The two-point function is well known [19]: h Φ ( x , t , θ ) , Φ ∗ ( x , t , θ ) i = δ h ,h δ M , M t − h exp (cid:20) − M x t (cid:21) × (cid:0) b ( θ + ¯ θ ) + θ ¯ θ ( c − b ln t ) (cid:1) (2.15) There is an unitary bound h ≥ d/ sch ( d ) in d dimensions [47]. At first sight, one might believe that because of the commutator (2.10), with
M 6 = 0, invariance underspace-translations and Galilei-transformations could not be required simultaneously. However, invarianceunder M gives the Bargman super-selection rule M [2] = M + M = 0. Hence the action of the commutator(2.10) vanishes on any n -point function. Here, the ‘complex conjugate’ Φ ∗ is obtained from Φ by changing the sign of the mass: M 7→ −M [18]. t := t − t , x := x − x and b, c are normalisation constants. Expanding, one has h φ ( t , x ) φ ∗ ( t , x ) i = 0 , h φ ( t , x ) ψ ∗ ( t , x ) i = b t − h exp (cid:20) − M x t (cid:21) δ h ,h δ M , M , (2.16) h ψ ( t , x ) ψ ∗ ( t , x ) i = t − h ( c − b ln t ) exp (cid:20) − M x t (cid:21) δ h ,h δ M , M . dimensions Galilean conformal algebra in 1 + 1 and 2 + 1 dimensions is special. In 1 + 1 dimensions, it isunique because it can be obtained directly from contracting 2-dimensional CFTs. Followingthis contraction many aspects of the fields can be extracted from CFT . In 2 + 1 dimensions,it admits an ‘exotic’ central charge [25, 26].For the moment, and to remain faithful to the method of previous section , consider 1+1dimensions: P = − ∂ x , K = − t∂ x − γ, F = − t ∂ x − tγ,H = − ∂ t , D = − ( t∂ t + x∂ x ) − ∆ , C = − (2 tx∂ x + t ∂ t ) − t ∆ , (2.17)in which ∆ is eigenvalue of dilation D and γ is eigenvalue of K which is called rapidity . Thesecan be further embedded into an infinite-dimensional set of generators (where X − , , = H, D, C and Y − , , = P, K, F ) which generate the infinite-dimensional Lie algebra called‘Full CGA/altern-Virasoro algebra’ in the literature ( n ∈ Z ): X n = − (cid:2) t n +1 ∂ t + ( n + 1) t n x∂ x + ( n + 1)( t n ∆ + nt n − γx ) (cid:3) Y n = − (cid:2) t n +1 ∂ x + ( n + 1) t n γ (cid:3) (2.18)with the commutators[ X m , X n ] = ( m − n ) X m + n , [ X m , Y n ] = ( m − n ) Y m + n , [ Y m , Y n ] = 0 . (2.19)Co-variant two-point functions are [12, 39, 48] (with x := x − x and t := t − t ) h φ ( t , x ) φ ( t , x ) i CGA = a δ ∆ , ∆ δ γ ,γ t − exp (cid:20) − γ xt (cid:21) (2.20)As mentioned above, the interesting point regarding CGA in d = 1 + 1 dimensions is that wecan obtain them from two-dimensional conformal symmetry by contraction. Briefly, d = 2conformal symmetry consists of two commuting Virasoro algebras, with generators: L n = − z n +1 ∂ z , L n = − z n +1 ∂ z . (2.21)Under the contraction limit (1 . X n = L n + L n Y n = 1 c ( L n − L n ) (2.22)6enerate the algebra given by (2.19). The central charges of the two chiral copies of Virasoroalgebra, namely C and ¯ C combine to give the two independent central charges in the CGA,making it non-unitary [41]. are a contracted limit of CFT , it might be possible that itsrepresentations are contracted limit of CFT representations [41]. To observe this considerprimary states in CFT : X | h, h i = ( L + L ) | h, h i = ( h + h ) | h, h i ,Y | h, h i = L − L c | h, h i = h − hc | h, h i . (2.23)We observe that the scaling states of CFT are scaling states of CGA, too. Now, they areidentified by their scaling weight and rapidity. In other words | h, h i → | ∆ , γ i , (2.24)in which ∆ := h + hγ := h − hc (2.25)Now, to build a logarithmic representation of the full CGA, we expect logarithmic part-ners to appear in (2 . γ by 2 × d = 2 will be needed in section 3 below for the ecga )∆ b ∆ := (cid:18) ∆ ∆ ′ (cid:19) , γ b γ := (cid:18) γ γ ′ γ ′′ γ (cid:19) (2.26)where we already used that one of the two matrices can without restriction of the generalitybe assumed to have a (non-diagonalisable) Jordan form. In order to find the most generaladmissible form, we write the above representations (2.18) of the CGA as follows (and includeall terms which describe the transformation of the scaling operators), with n ∈ Z X n = − t n +1 ∂ t − ( n + 1) t n x · ∂ x − ( n + 1) t n (cid:18) ∆ ∆ ′ (cid:19) − n ( n + 1) t n − (cid:18) γ γ ′ γ ′′ γ (cid:19) · xY n = − t n +1 ∂ x − ( n + 1) t n (cid:18) γ γ ′ γ ′′ γ (cid:19) (2.27) R = − ǫ ij x i ∂ j − ǫ kℓ γ k ∂∂γ ℓ − ǫ kℓ γ ′ k ∂∂γ ′ ℓ − ǫ kℓ γ ′′ k ∂∂γ ′′ ℓ (with ∂ j := ∂/∂x j ) such that the non-vanishing commutators become[ X n , X m ] = ( n − m ) X n + m + ( n + 1)( m + 1)( m − n ) t n + m − ∆ ′ γ ′′ · x (cid:18) − (cid:19) [ X n , Y m ] = ( n − m ) Y n + m + ( n + 1)( m + 1) t n + m ∆ ′ γ ′′ (cid:18) − (cid:19) (2.28) This discussion is quite analogous to the one which applies to the logarithmic representations of the‘ageing’ sub-algebra of the Schr¨odinger algebra (without time-translations) [23]; the two scaling dimensions x, ξ used therein and their matrix generalisations play the same rˆoles as ∆ , γ in the CGA studied here. Y ni , R ] = − ǫ iℓ Y nℓ . In order to recover the commutators (2.19) of the CGA, we musthave ∆ ′ γ ′′ = (2.29)Hence, either ∆ ′ = 0 such that the matrix b ∆ = ∆ (cid:18) (cid:19) and b γ is either diagonalisable(which would give a pair of non-logarithmic representations) or else it has a Jordan formwhere one can always arrange for γ ′′ = . Alternatively, we have directly γ ′′ = . Therefore, one can always set γ ′′ = in (2.26). In summary without loss of generality, we can repeat eq. (2.14) by admitting as the mostgeneral case states : X | ∆ , γ ; 1 i = 0 , X | ∆ , γ ; 2 i = ∆ ′ | ∆ , γ ; 1 i Y | ∆ , γ ; 1 i = 0 , Y | ∆ , γ ; 2 i = γ ′ | ∆ , γ ; 1 i (2.30)In the language of nilpotent variables, we define an eigenstate | e ∆ , e γ i where e ∆ = ∆ + ∆ ′ θ, e γ = γ + γ ′ θ. (2.31)Equation (2.30) then reads as: X | e ∆ , e γ ; 2 i = ∆ ′ θ | e ∆ , e γ ; 1 i , Y | e ∆ , e γ ; 2 i = γ ′ θ | e ∆ , e γ ; 1 i . (2.32)Now we follow on to calculate two-point functions: F ( x , t , θ ; x , t , θ ) = h e ∆ , e γ | φ ( x , t ) φ ( x , t ) | e ∆ , e γ i (2.33)Before going further let’s redefine our parameters such that for the general variable w weset: w = w − w w + = w + w . (2.34)For instance: t = t ! − t t + = t + t θ = θ − θ θ + = θ + θ ∆ = ∆ − ∆ ∆ + = ∆ + ∆ etc. (2.35)Two-point functions should be invariant under the Ward identities arising out of CGA el-ements X − , X , X , Y − , Y , Y . First, since F must be invariant under space- and time-translation, it must be a function merely of t and x an not of t + and x + . Invariance under Y is expressed as ( t ∂ x + γ + γ ′ θ + t ∂ x + γ + γ ′ θ ) F = 0 , (2.36) Implicitly, ∆ ′ and γ ′ are assumed to have the same value for both scaling operators. t∂ x + γ + + γ ′ θ + γ ′ θ ) F = 0 (2.37)restricting F to: F = e − ( γ + + γ ′ θ + γ ′ θ ) xt [ g ( t ) + g ( t ) θ + g ( t ) θ + g ( t ) θ θ ] . (2.38)Now, invariance under Y gives: t + ( t∂ x + γ + + γ ′ θ + γ ′ θ ) F + t ( γ + γ ′ θ − γ ′ θ ) F = 0 . (2.39)So, we find that γ = 0 , g ( t ) = 0 , γ ′ g = γ ′ g , (2.40)reducing F to: F = e − ( γ + + γ ′ θ + γ ′ θ ) xt [ w ( t ) γ ′ θ + w ( t ) γ ′ θ + g ( t ) θ θ ] δ γ ,γ . (2.41)in which w ( t ) = g ( t ) /γ ′ . Now let’s look at X which appears as( t∂ t + x∂ x + ∆ + + ∆ ′ θ + ∆ ′ θ ) F = 0 , (2.42)Inserting F from (2.41) in the above equation leads to:( t∂ t + ∆ + ) g ( t ) = 0 , ( t∂ t + ∆ + ) g ( t ) + ∆ ′ γ ′ w ( t ) + ∆ ′ γ ′ w ( t ) = 0 , (2.43)which results in w ( t ) = at − ∆ + ,g ( t ) = t − ∆ + ( b − a (∆ ′ γ ′ + ∆ ′ γ ′ ) ln | t | ) . (2.44)Action of X gives nothing new but super-selection rules:∆ = 0 , ∆ ′ γ ′ = ∆ ′ γ ′ . (2.45)This constraint will appear in G and G . Since under exchange 1 ↔ G ↔ G we can renormalize φ in a way to arrange for a perfect symmetry under exchange of thescaling operators, such that h φ ψ i = G
12 ! = G = h ψ φ i . This leads to the strongerconstraints: ∆ ′ = ∆ ′ , γ ′ = γ ′ . (2.46)So, the final result is F = e − (2 γ + γ ′ θ + ) xt (cid:2) at − θ + + t − ( b − a ∆ ′ ln | t | ) θ + θ + (cid:3) δ γ ,γ δ ∆ , ∆ δ ∆ ′ , ∆ ′ δ γ ′ ,γ ′ . (2.47)9xpanding both sides of (2.33) in terms of nilpotent variables, we find the two-point functionsfor logarithmic primaries of CGA h φ φ i = 0 , h φ ψ i = ae − γ xt t − δ ∆ , ∆ δ ∆ ′ , ∆ ′ δ γ ,γ δ γ ′ ,γ ′ , h ψ ψ i = e − γ xt t − h − a ∆ ′ ln | t | − a γ ′ xt + b i δ ∆ , ∆ δ ∆ ′ , ∆ ′ δ γ ,γ δ γ ′ ,γ ′ . (2.48)One needs to notice that since φφ term is equal to zero, then we can re-scale φ so that∆ ′ = ∆ ′ and thereby γ ′ = γ ′ . These results can be obtained as well by contraction from aLCFT where both chiral components have logarithmic partners.Since we wrote the generators in (2.27) in a arbitrary number of space dimensions d , itis now straightforward to write down the extension of (2.48) to d + 1 dimensions h φ φ i ( t, x ) = 0 h φ ψ i ( t, x ) = a | t | − e − γ · x /t δ ∆ , ∆ δ γ , γ δ ∆ ′ , ∆ ′ δ γ ′ , γ ′ (2.49) h ψ ψ i ( t, x ) = | t | − e − γ · x /t h b − a x t · γ ′ − a ∆ ′ ln | t | i δ ∆ , ∆ δ γ , γ δ ∆ ′ , ∆ ′ δ γ ′ , γ ′ with a manifest invariance under the spatial rotations (2.27). We also list explicitly theseveral super-selection rules, as they apply to the non-vanishing elements of the matrices b ∆and b γ . The exotic
CGA ( ecga ) is a centrally extended CGA in 2 + 1 dimensions. The generators
P, K, F now become 2-dimensional vectors P , K , F (or equivalently Y n is replaced by Y n )such that the immediate extension of the commutators (2.19) is centrally extended by thenontrivial commutators [25, 26]:[ K i , K j ] = Ξ ǫ ij , [ P i , F j ] = − ǫ ij ; i, j = 1 , , (3.1) ǫ ij are the elements of the totally antisymmetry matrix b ǫ = (cid:18) − (cid:19) and and ǫ = 1.For realising the central charge, following [27] one may invoke two operators χ i such that[ χ i , χ j ] = Ξ ǫ ij [ χ i , Ξ] = 0 . (3.2)Since Ξ is central, one may represent it by its eigenvalue Ξ = ξ . The ecga generators read: H = − ∂ t , D = − x i ∂ i − t∂ t , C = − tx i ∂ i − t ∂ t − x i χ i ,P i = − ∂ i , K i = − t∂ i − χ i , F i = − t ∂ i − tχ i − x j ǫ ij ξ,J = − ǫ ij x i ∂ j − ξ χ i χ i , (3.3) In principle, the constants a, b can depend on the vectors γ and γ ′ . Rotation-invariance then implies a = a ( γ , γ ′ , γ · γ ′ ) and analogously for b . J of rotations was explicitly included as well. Its commutators with theother generators of the ecga read[ J, H ] = [
J, D ] = [
J, C ] = 0[ J, P ] = b ǫ P , [ J, K ] = b ǫ K , [ J, F ] = b ǫ F (3.4)Martelli and Tachikawa [27] obtain the above algebra by making a contraction of the2 + 1 dimensional conformal algebra where spin has been taken into account. In other wordsthey start by:(with µ and ν = 0 , , f M µν = − x µ ∂ν + x ν ∂ µ − Σ µν , e P µ = − ∂ µ , e K µ = − x ν x ν ∂ µ + 2 x µ x ν ∂ ν + 2 x ν Σ µν e D = − x µ ∂ µ , (3.5)where Σ µν is the spin. Now under the contraction limit (1.7) and redefining operators as: P i = e P i c , H = P K i = f M i cD = e D F i = e K i c C = − e K ,χ i = Σ i c ξ = Σ c ; J = f M + 12 ξc (Σ i Σ i ) + Σ (3.6)they end up with (3.3). Clearly, the operators χ i and the central charge ξ are remnants ofthe spin components.The operators χ i and the central charge ξ can be realised explicitly via an auxiliary spacewith coordinates ν , ν : χ i = ∂ ν i − ǫ ij ν j ξ (3.7)Alternatively, one may use instead of J a more natural-looking generator of infinitesimalrotations, including its action on the auxiliary space R := − ǫ ij x i ∂ x j − − ǫ ij γ i ∂ γ j − ǫ ij ν i ∂ ν j (3.8)which obeys the same algebraic properties as the generator J .In the above realisation, one expects the operators D and Ξ to have simultaneous eigen-vectors since they commute, which is the primary state we use to construct the correlators.In order to include the rapidities as well, and to simplify the computation of two-point func-tions, we include all those terms which describe the transformation of the scaling operatorsinto the generators. Then the action of all generators on two-point functions simply vanishes.The important Bargman super-selection rule of the central charge ξ [2] = ξ + ξ = 0 follows.This is completely analogous to the treatment of the central charge in the Schr¨odinger alge-bra in section 2.2. In this new representation, the generators of the ecga read (those given11n [27, 43] are included as special cases) H = X − = − ∂ t , D = X = − x i ∂ i − t∂ t − ∆ ,C = X = − tx i ∂ i − t ∂ t − t − x i ∂ ν i + ǫ ij ν j x i ξ − γ i x i ,P i = Y − i = − ∂ i , K i = Y i = − t∂ i − ∂ ν i + 12 ǫ ij ν j ξ − γ i ,F i = Y i = − t ∂ i − t∂ ν i + tǫ ij ν j ξ − x j ǫ ij ξ − tγ i (3.9)This set is to be completed by a rotation generator for which we shall choose either J or R . Two-point functions may now be derived explicitly from (3.9). We observe that rotationinvariance under the action of the generator R leads to a different result than requiringco-variance under the rotation generator J from (3.3), as used in [27]. In what follows, wedistinguish these two cases and speak of ‘ J -invariance ’ if J is used along with the generatorsof (3.9) and of ‘ R -invariance ’ when R is used.Quite analogously to Schr¨odinger-invariance treated above, the generators (3.9) are dy-namical symmetries of the wave equation S φ = 0, where the Schr¨odinger operator is S := − ξ∂ t + ǫ ij ( χ i + γ i ) ∂ j = − ξ∂ t + ( χ + γ ) b ǫ ∂ x (3.10)The only generators of the ecga (3.9) which do not commute with S are D and C :[ S , D ] = −S , [ S , C ] = − t S − ξ (∆ −
1) (3.11)Rotation-invariance holds as well: [ S , J ] = [ S , R ] = 0. Hence one has a dynamical symmetryon the space of solutions of the equation S φ = 0 where ∆ = ∆ φ = 1, consistent with theknown unitary bound ∆ ≥ As a first step towards the logarithmic two-point functions from the ecga , we begin withthe non-logarithmic case. This was done first by Martelli and Tachikawa [27], but only forvanishing rapidities γ i = . It is one of the aims of this section to allow for γ i = and toanalyse systematically the possible constraints. The two-point function is defined as F := F ( t , t ; x , x ; ν , ν ) = h φ ( t , x , ν ) φ ( t , x , ν ) i . (3.12)We shall use throughout the variables as defined in (2.35) and apply the ecga -Ward identitiesderived from (3.9) to F . Space- and time-translation-invariance restrict F to be a function of t = t − t and x = x − x . The invariance under the central charge Ξ gives the importantBargman super-selection rule ξ + ξ = 0 (3.13)Invariance under the dilatations D and the two generalised Galilei-transformations K gives( − t∂ t − x · ∂ − ∆ − ∆ ) F = 0 (3.14)( − t ∂ − γ − γ − χ − χ ) F = 0 (3.15)12ather than using these to parametrise immediately an explicit scaling form, we preferto use these identities first to simplify the remaining conditions and especially to derivethe constraints the two-point function F must obey. Therefore, we next rewrite the Wardidentity of the two generators F , which gives. ( − t ∂ − t ( χ + γ ) − ξ b ǫ x ) F = 0 andwhere the eqs. (3.13,3.15) have been used. Using again (3.15), this is further simplified to( − t ( χ − χ + γ − γ ) − ξ b ǫ x ) F = 0 (3.16)Similarly, invariance under C gives ( − t ∂ t − t x · ∂ − t − χ + γ ) · x ) F = 0, whereeqs. (3.13,3.14,3.15) have been used. Applying (3.14), again, leads to( − t x · ∂ − (∆ − ∆ ) t − χ + γ ) · x ) F = 0 (3.17)Eqs. (3.14,3.15,3.16,3.17) contain the complete available information on the shape of thetwo-point function F , up to rotation-invariance, to be discussed below.Multiplying (3.16) with x , one has x · ( χ + γ ) F = x · ( χ + γ ) F (3.18)such that comparison of eqs. (3.15,3.17) leads to (∆ − ∆ ) tF = 0. Hence, the constraint∆ = ∆ follows. The remaining three independent equations can be further simplified via the ansatz F = t − e − ( γ + γ ) · u f ( u , ν , ν ) , u := x /t (3.19)which leads to the following two conditions( ∂ u + χ + χ ) f = 0 , ( χ − χ + γ − γ + 2 ξ b ǫ u ) f = 0 (3.20)Only now, we use the explicit form (3.7). Further, we introduce the new variables ν ± := ( ν ± ν ) and write f = f ( u , ν + , ν − ) such that (cid:0) ∂ u + ∂ ν + − ξ b ǫ ν − (cid:1) f = 0 , (cid:0) ∂ ν − − ξ b ǫ ν + + 2 ξ b ǫ u + γ − γ (cid:1) f = 0 (3.21)The first of those is solved by the ansatz f = exp [ ξ u b ǫ ν − ] φ ( w , ν − ), with w := u − ν + .The last function φ can be found from the equation ( ∂ ν − + ξ b ǫ w + γ − γ ) φ = 0. Hence φ (cid:0) w , ν − (cid:1) = φ ( w ) exp (cid:2) − ξ ν − · b ǫ w − ν − · ( γ − γ ) (cid:3) (3.22)where φ is an arbitrary differentiable function, which besides on w , can in principle alsodepend on the parameter vectors γ , .Summarising our results, we can write the explicit two-point function, with u = x /t h φ φ i = f (cid:18) u − ν + ν (cid:19) | t | − e − ( γ + γ ) · u − ( γ − γ ) · ( ν − ν ) e ξ u ∧ ( ν − ν )+ ξ ν ∧ ν δ ∆ , ∆ δ ξ + ξ , (3.23) For the non-exotic CGA, (3.16) or (3.18) would further imply γ = γ . ↔ One alsouses the notation of the skew product a ∧ b := a b ǫ b = ǫ ij a i b j . If the rotation-invariance istaken into account as well, the undetermined function written above as f = f ( w ) becomes J -invariance : f = f (cid:0) γ , γ , γ · γ , w + b ǫ ( γ − γ ) (2 ξ ) − (cid:1) R -invariance : f = f (cid:0) w , γ , γ , w · γ , w · γ (cid:1) (3.24)as is shown in appendix B.Remarkably, the ecga -covariant two-point function no longer needs to obey the con-straint γ = γ of the rapidities which we had obtained in (2.49) for the non-exotic case. We are finally prepared for the computation of the co-variant two-point functions in thelogarithmic representation of the ecga , which in the most simple case can be obtainedformally from the representation (3.9) by making the replacements∆ b ∆ := (cid:18) ∆ ∆ ′ (cid:19) , γ b γ := (cid:18) γ γ ′ γ (cid:19) (3.25)see section 2.3. We seek the two-point functions F = h φ φ i , G = h φ ψ i , G = h ψ φ i , H = h ψ ψ i (3.26)where the arguments are implicit. Surprisingly, it turns out that the non-modified contri-bution F = h φ φ i does not necessarily vanish, in contrast with the non-exotic CGA (2.49).Throughout, temporal and spatial translations-invariance and invariance under the centralcharge Ξ shall be used, such that all two-point functions depend on t and x and the Bargmansuper-selection rule (3.13) is valid.We now state the explicit result and refer to appendix A for the details of the calculation.Two distinct cases must be recognised, namely1. Case 1 : ∆ ′ = 0 or ∆ ′ = 0 and F = 0. This is the most direct extension of thelogarithmic representations of the non-exotic CGA.2. Case 2 : ∆ ′ = ∆ ′ = 0 and F = 0. Here, only the rapidity matrices b γ i will take aJordan form, while b ∆ = ∆ b is diagonal.In what follows, we shall use the notations from section 3.1.In Case 1 , we have F = 0 and G = G ( t, x ) = G ( − t, − x ) = G =: G such that G = | t | − e − ( γ + γ ) · u − ( γ − γ ) · ( ν − ν ) e ξ u ∧ ( ν − ν )+ ξ ν ∧ ν g ( w ) H = | t | − e − ( γ + γ ) · u − ( γ − γ ) · ( ν − ν ) e ξ u ∧ ( ν − ν )+ ξ ν ∧ ν h ( u , ν , ν ) (3.27) h = h ( w ) − g ( w ) (cid:18) ′ ln | t | + u · ( γ ′ + γ ′ ) + 12 ( ν − ν ) · ( γ ′ − γ ′ ) (cid:19) The correlator obtained from the free-field solution of S φ = 0 is of this form, with f = cste. [27]. u = x /t and w := u − ( ν + ν ) and the constraints∆ = ∆ , ∆ ′ = ∆ ′ and ξ + ξ = 0. The undetermined functions g ( w ) and h ( w ) still aresubject to rotation-invariance, see below.In Case 2 , we find the constraints ∆ = ∆ and ξ + ξ = 0 and F = | t | − e − ( γ + γ ) · u − ( γ − γ ) · ( ν − ν ) e ξ u ∧ ( ν − ν )+ ξ ν ∧ ν f ( w ) G = | t | − e − ( γ + γ ) · u − ( γ − γ ) · ( ν − ν ) e ξ u ∧ ( ν − ν )+ ξ ν ∧ ν g ( u , ν , ν ) G = | t | − e − ( γ + γ ) · u − ( γ − γ ) · ( ν − ν ) e ξ u ∧ ( ν − ν )+ ξ ν ∧ ν g ( u , ν , ν ) (3.28) H = | t | − e − ( γ + γ ) · u − ( γ − γ ) · ( ν − ν ) e ξ u ∧ ( ν − ν )+ ξ ν ∧ ν h ( u , ν , ν )where g = g ( w ) − f ( w ) (cid:18) u −
12 ( ν − ν ) (cid:19) · γ ′ g = g ( w ) − f ( w ) (cid:18) u + 12 ( ν − ν ) (cid:19) · γ ′ (3.29) h = h ( w ) − g ( w ) (cid:18) u · ( γ ′ + γ ′ ) + 12 ( ν − ν ) · ( γ ′ − γ ′ ) (cid:19) + 12 f ( w ) (cid:18) u + 12 ( ν − ν ) (cid:19) · γ ′ (cid:18) u −
12 ( ν − ν ) (cid:19) · γ ′ Finally, in both cases, rotation-invariance must be taken into account, see appendix Bfor the details. If we use R -invariance, in both cases the functions f ( w ), g ( w ) and h ( w )are short-hand notations for undetermined functions of 9 rotation-invariant combinations of w , γ , and γ ′ , , for example f = f (cid:16) w , γ , γ , γ ′ , γ ′ , w · γ , w · γ , w · γ ′ , w · γ ′ (cid:17) (3.30)and analogously for g and h . We point out that in both cases, there is no constraint neitheron the γ i , nor the γ ′ i . Alternatively, if we use J -invariance we find that the γ -matrices arediagonal, viz. γ ′ = γ ′ = . Then only case 1 retains a logarithmic structure and we have g = g (cid:0) γ , γ , γ · γ , w + b ǫ ( γ − γ ) (2 ξ ) − (cid:1) h = h (cid:0) γ , γ , γ · γ , w + b ǫ ( γ − γ ) (2 ξ ) − (cid:1) (3.31)So, the task is done and two-point functions of logarithmic representations of the ecga havebeen worked out. The exotic Galilean algebra corresponds to d = 2 and l = 1 case of l -Galilei algebras. Thisalgebra arises as the singular limit of the conformal algebra when the speed of light tendsto infinity. In other words it should describe the low velocity limit of conformal systems.15lgebra eq. constraintsCGA (2.20) ∆ = ∆ γ = γ L-CGA (2.49) ∆ = ∆ ∆ ′ = ∆ ′ γ = γ γ ′ = γ ′ ecga (3.23) ∆ = ∆ ξ + ξ = 0(3.27) ∆ = ∆ ∆ ′ = ∆ ′ ξ + ξ = 0 R1L- ecga (3.28) ∆ = ∆ ∆ ′ = ∆ ′ = 0 ξ + ξ = 0 R2(3.27) ∆ = ∆ ∆ ′ = ∆ ′ γ ′ = γ ′ = ξ + ξ = 0 JTable 1: Summary on the constraints obeyed by co-variant two-point functions in severalvariants of conformal galilean algebras. The first column indicates the non-exotic algebraCGA or the exotic ecga , where a prefix ‘L-’ indicates a logarithmic representation. Theequation labels refer to the explicit form of the two-point function, as derived in the text.The various constraints on either scaling dimensions ∆, rapidities γ or the Bargman super-selection rule on the ‘masses’ ξ are listed. The last three lines refer to the logarithmicrepresentations of the exotic ecga . Therein, the extra labels refer to the two distinct choicesof the rotation generator: either R -invariance with the two distinct case 1 (R1) and case 2(R2), or else J -invariance (J).However the low energy limit is often described by the Schr¨odinger algebra which is the l = case of l -Galilei algebras. This is rather paradoxical and the physical candidates forthe realisation of CGA have proved hard to find.In this work, we analysed the generic form of co-variant two-point functions, for scalarand logarithmic representations of conformal galilean algebra. The transformation of quasi-primary scaling operators under these algebras can be characterised in terms of a scalingdimension ∆, a rapidity vector γ and in the case of the exotic ecga also in terms of a ‘mass’ ξ . If one considers logarithmic representations, the scaling dimension and the rapidities canacquire a matrix form. Restricting to the most simple case of two-component logarithmicrepresentations, these matrices have been shown to be simultaneously of a Jordan form∆ b ∆ = (cid:18) ∆ ∆ ′ (cid:19) , γ b γ = (cid:18) γ γ ′ γ (cid:19) (4.1)Qualitatively very different results were obtained for the non-exotic CGA and the exotic ecga , as summarised in table 1.1. When considering the CGA, the extension to logarithmic representation essentiallyproduced the expected generalisations of the constraints on both the scaling dimension∆ and the rapidity γ also to the non-diagonal elements of the corresponding matrices,according to (4.1). In addition, the various two-point functions simply take up thesame kind of logarithmic prefactors, see eq. (2.49), as one would have expected fromlogarithmic conformal or even logarithmic Schr¨odinger-invariance, see eqs. (2.9,2.16).2. Therefore, by comparing the results (2.9) of logarithmic conformal invariance and(2.16) of logarithmic Schr¨odinger-invariance, one might have believed that going over16o the ecga would merely lead to the naturally expected Bargman super-selectionrule ξ + ξ = 0, which would be the analogue of non-relativistic mass conservationin Schr¨odinger-invariant systems. Remarkably, it turned out that the form of theco-variant two-point functions in the exotic ecga is considerably richer.3. For scalar representations, the presence of extra internal dimensions needed to realisethe extra exotic structure releases the constraint on the rapidities γ i of the two scalingoperators. A finer difference arise through the possibility of two distinct choices forthe generator of rotation, labelled J and R , and referred to as ‘ J -invariance ’ and ‘ R -invariance ’. The precise shape of a last undetermined scaling function f depends onwhether J -invariance or R -invariance is assumed, see eq. (3.24).4. Stronger qualitative differences appear in the logarithmic representations of the ecga .For R -invariance, two distinct cases emerge. The first one, labelled R1 in table 1,keeps the conventional result that the two-point function F = h φφ i = 0 of the partnervanishes. But if the matrices b ∆ are diagonal, a new case arises, labelled R2 in table 1,where F = 0 and new additional terms in the remaining two-point functions h φψ i and h ψψ i arise. In both cases, the remaining scaling functions are of the generic form(3.30). On the other hand, for J -invariance, labelled J in table 1, the rapidity matrices b γ are forced to be diagonal, such that the logarithmic terms reduce to those knownfrom the non-exotic CGA. Here, only case 1 retains a logarithmic structure and theform of the remaining scaling functions is given in (3.31).5. What has been referred to in the literature as “logarithmic” conformal field theory,uses in fact reducible but non-decomposable representations of the conformal algebra.In all cases known so far, the correlators also acquired a logarithmic term as well aspower-law-dependence on distance, which motivated the name ‘ logarithmic ’. Here, afirst example has been found (case R2 of the L- ecga in table 1) where a reducible butnon-decomposable representation does not lead to explicit logarithms in the two-pointfunctions.The present study looked at co-variant two-point functions of conformal galilean algebrasfrom an abstract point of view. We hope that the results presented here will turn out tobe helpful in identifying specific physical model with one of them as a dynamical symmetry.We hope to come back to this in future work. Acknowledgments
We are grateful to S. Moghimi-Araghi and A. Naseh for discussions. Two of us (MH andSR) would like to thank the organisers of the ADCFTA at the IHP Paris 2011, for warmhospitality, where this work was initiated. MH thanks the organisers of the Symposium‘Models from statistical mechanics in applied sciences’ at Warwick University 2013 for warmhospitality and acknowledges partial support from the Coll`ege Doctoral franco-allemandNancy-Leipzig-Coventry (Syst`emes complexes `a l’´equilibre et hors ´equilibre) of UFA-DFH.SR would like to thank warm hospitality at Van der Waals-Zeeman Instituut, University of17msterdam,Nederlands and Groupe de Physique Statistique, Institut Jean Lamour, Univer-sity of Lorraine, France where part of this work were carried out.
Appendix A. Calculational details
The results (3.27) and (3.28,3.29) of the main text, respectively, are derived.Starting from the definitions (3.26), the function F was already found in section 3.1.As we shall see that F = 0 may occur, we revert to the standard formulation of LCFT.Temporal and spatial translation-invariance and the Bargman super-selection rule ξ + ξ = 0are obvious.We begin with the two-point function G = h φ ψ i . Co-variance under the generators X , Y , Y and X , via calculations totally analogous to the ones presented in section 3.1,lead to the conditions ( − t∂ t − x · ∂ − ∆ − ∆ ) G − ∆ ′ F = 0( − t ∂ − γ − γ − χ − χ ) G − γ ′ F = 0( − t ( γ − γ + χ − χ ) − ξ b ǫ r ) G + t γ ′ F = 0 (A.1)( − t r · ∂ − t (∆ − ∆ ) − r · ( χ + γ )) G + t ∆ ′ F = 0If F = 0, then from section 3.1 we have ∆ = ∆ . Otherwise, if F = 0, the above conditionsare then identical to those treated in section 3.1, so that again ∆ = ∆ follows. Hence, we always have the constraint ∆ = ∆ . Next, multiply the 3 rd eq. (A.1) with x . On theother hand, simplify the 4 th eq. (A.1) by using again Y -covariance. This gives the twosimultaneous conditions t x · ( χ − χ + γ − γ ) G − t x · γ ′ F = 0 (A.2) x · ( χ − χ + γ − γ ) G − x · γ ′ F − t ∆ ′ F = 0which requires that t ∆ ′ F = 0 (A.3)Similarly, if we consider the other mixed two-point function G = h ψ φ i , we find t ∆ ′ F = 0.Therefore, the following cases must be distinguished: Case 1 : Let ∆ ′ = 0 or ∆ ′ = 0 . Then F = 0 .2. Case 2 : Let ∆ ′ = ∆ ′ = 0 . Then F = 0 is possible. Before we enter into this distinction, we write down the conditions for the last two-pointfunction H = h ψ ψ i . Proceeding as in section 3.1, we find( − t∂ t − x · ∂ − ∆ − ∆ ) H − ∆ ′ G − ∆ ′ G = 0( − t ∂ − γ − γ − χ − χ ) H − γ ′ G − γ ′ G = 0( − t ( γ − γ + χ − χ ) − ξ b ǫ r ) H − t ( γ ′ G − γ ′ G ) = 0 (A.4)( − t r · ∂ − t (∆ − ∆ ) − r · ( χ + γ )) H − t (∆ ′ G − ∆ ′ G ) − γ ′ · x G = 0 We leave out here distributional contributions F ∼ δ ( t ) , δ ′ ( t ). rd of these by x and re-use Y -covariance on the 4 th , along with theknown constraint ∆ = ∆ . This gives simultaneously x · ( χ − χ + γ − γ ) H + x · ( γ ′ G − γ ′ G ) = 0 (A.5) x · ( χ − χ + γ − γ ) H + x · ( γ ′ G − γ ′ G ) + t (∆ ′ G − ∆ ′ G ) = 0which implies the constraint t (∆ ′ G − ∆ ′ G ) = 0 (A.6)In case 2, we have ∆ ′ = ∆ ′ = 0 and this constraint is already satisfied. In case 1, F = 0and the form of G and G can be read off from the non-logarithmic representation ofsection 3.1. Since under the exchange 1 ↔ G ↔ G , one can always arrangetheir amplitudes such that ∆ ′ = ∆ ′ (A.7) G ( t, x ) = G = G = G ( − t, − x )Now, we can consider the two cases separately and work out the two-point functions explic-itly. Case 1 . With the two constraints ∆ = ∆ and ∆ ′ = ∆ ′ , H is to be found from the threeindependent equations ( t∂ t + x · ∂ + 2∆ ) H + 2∆ ′ G = 0( t ∂ + γ + γ + χ + χ ) H + ( γ ′ + γ ′ ) G = 0 (A.8)(( γ − γ + χ − χ ) + 2 ξ b ǫ u ) H + ( γ ′ − γ ′ ) G = 0We also require the explicit form of the operators χ i ( χ + χ ) f = (cid:0) ∂ ν + − ξ b ǫ ν − (cid:1) f , ( χ − χ ) f = (cid:0) ∂ ν − − ξ b ǫ ν + (cid:1) f (A.9)with the variables ν ± := ( ν ± ν ). Since F = 0, the mixed correlator G can be read offfrom (A.1). The result has already been obtained in section 3.1 and reads G = | t | − e − ( γ + γ ) · u g ( u , ν + , ν − )= g ( w ) | t | − e − ( γ + γ ) · u − ( γ − γ ) · ν − e ξ ( u + w ) b ǫ ν − (A.10)where w := u − ν + and g ( w ) is an undetermined function. Analogously, we write H = | t | − e − ( γ + γ ) · u h ( t, u , ν + , ν − ) (A.11)and proceed to extract h from the three conditions (A.8). The first one reduces to t∂ t h +2∆ ′ g = 0 and has the solution h ( t, u , ν + , ν − ) = − ′ ln | t | g ( u , ν + , ν − ) + h ( u , ν + , ν − ) (A.12)Next, the second condition (A.8) becomes (cid:0) ∂ u + ∂ ν + − ξ b ǫ ν − (cid:1) h + ( γ ′ + γ ′ ) g = 0 (A.13)19his is solved by the transformation h = e ξ u b ǫ ν − h such that the resulting equation for h is readily integrated, with the result h = − u · ( γ ′ + γ ′ ) g ( w ) e − ( γ − γ ) · ν − e ξ u b ǫ ν − + e ξ u b ǫ ν − h ( w , ν − ) (A.14)The last condition (A.8) has the form( ∂ ν − + ξ b ǫ ( u + w ) + γ − γ ) h + ( γ ′ − γ ′ ) g = 0 (A.15)Inserting h from (A.14), and with the transformation h = e − ν − b ǫ w e − ν − · ( γ ′ − γ ′ ) h ( w , ν − ),this reduces to ∂ ν − h + ( γ ′ − γ ′ ) g ( w ). Hence h = − ν − · ( γ ′ − γ ′ ) g ( w ) + h ( w ) such thatfinally h = − u · ( γ ′ + γ ′ ) g ( w ) e − ( γ − γ ) · ν − e ξ ( u + w ) b ǫ ν − + h ( w ) e − ( γ − γ ) · ν − e ξ ( u + w ) b ǫ ν − (A.16)and where h ( w ) remains undetermined. Summarising, we have found that F = 0 and G = | t | − e − ( γ + γ ) · u − ( γ − γ ) · ν − e ξ ( u + w ) ∧ ν − g ( w ) H = | t | − e − ( γ + γ ) · u − ( γ − γ ) · ν − e ξ ( u + w ) ∧ ν − h ( u , w , ν − ) (A.17) h = h ( w ) − g ( w ) (cid:0) ′ ln | t | + u · ( γ ′ + γ ′ ) + ν − · ( γ ′ − γ ′ ) (cid:1) We have the constraints ∆ = ∆ , ∆ ′ = ∆ ′ and ξ + ξ = 0. At this stage, the functions g ( w ) and h ( w ) remain undetermined.The further consequences of rotation-invariance are discussed in appendix B. Case 2 . Since now ∆ ′ = ∆ ′ = 0, the first mixed correlator G is obtained from the firstthree equations (A.1). Similarly, the other mixed correlator G is found from the equations( t∂ t + x · ∂ + 2∆ ) G = 0( t ∂ + γ + γ + χ + χ ) G + γ ′ F = 0(( γ − γ + χ − χ ) + 2 ξ b ǫ u ) G + γ ′ F = 0and the last one is determined from ( t∂ t + x · ∂ + 2∆ ) H = 0( t ∂ + γ + γ + χ + χ ) H + γ ′ G + γ ′ G = 0 (A.18)(( γ − γ + χ − χ ) + 2 ξ b ǫ u ) H + γ ′ G − γ ′ G = 0and all subject to the constraints ∆ = ∆ and ξ + ξ = 0. Since F was already found insection 3.1, we can also adapt eq. (A.8) from the Case 1 treated above and write directlydown the mixed correlators, with the result F = | t | − e − ( γ + γ ) · u − ( γ − γ ) · ν − e ξ ( u + w ) ∧ ν − f ( w ) G = | t | − e − ( γ + γ ) · u − ( γ − γ ) · ν − e ξ ( u + w ) ∧ ν − (cid:2) g ( w ) − f ( w ) (cid:0) u − ν − (cid:1) · γ ′ (cid:3) (A.19) G = | t | − e − ( γ + γ ) · u − ( γ − γ ) · ν − e ξ ( u + w ) ∧ ν − (cid:2) g ( w ) − f ( w ) (cid:0) u + ν − (cid:1) · γ ′ (cid:3) ↔ G ↔ G . Then f and g remain undetermined. The correlator H is writtenin the form H = | t | − e − ( γ + γ ) · u − ( γ − γ ) · ν − e ξ ( u + w ) ∧ ν − h ( u , w , ν − ) (A.20)Then the first of eqs. (A.18) is automatically satisfied, whereas the second and third lead tothe system ∂ u h + ( γ ′ + γ ′ ) g − (cid:0) γ ′ (cid:0) u − ν − (cid:1) · γ ′ + γ ′ (cid:0) u + ν − (cid:1) · γ ′ (cid:1) f = 0 ∂ ν − h + ( γ ′ − γ ′ ) g − (cid:0) γ ′ (cid:0) u − ν − (cid:1) · γ ′ − γ ′ (cid:0) u + ν − (cid:1) · γ ′ (cid:1) f = 0 (A.21)These are decoupled by defining y ± := ( u ± ν − ). Considering h = h ( y + , y − , w ), one has ∂ y + h = − γ ′ g + 2 γ ′ y − · γ ′ f , ∂ y − h = − γ ′ g + 2 γ ′ y + · γ ′ f (A.22)such that finally, with h ( w ) an undetermined function h = h ( w ) − g ( w ) y + · γ ′ − g ( w ) y − · γ ′ + 2 f ( w ) y + · γ ′ y − · γ ′ = h ( w ) − g ( w ) (cid:0) u + ν − (cid:1) · γ ′ − g ( w ) (cid:0) u − ν − (cid:1) · γ ′ + 12 f ( w ) (cid:0) u + ν − (cid:1) · γ ′ (cid:0) u − ν − (cid:1) · γ ′ (A.23)The discussion of rotation-invariance is analogous to Case 1 and carried out in appendix B.Combining the results eqs. (A.19,A.20,A.23) and reverting to the original coordinates, onearrives at the final form (3.28,3.29) stated in the main text. Appendix B. On rotation-invariance in the ecga
Having discussed in the main text the shape of two-point functions co-variant under the ecga , we now consider the additional consequences when rotation-invariance is taken intoaccount as well.
B.1 Rotation-invariance for vanishing rapidities
We shall compare the consequences of using two distinct representations for the infinitesimalgenerator of rotations, namely (also recall that a ∧ b = ǫ ij a i b j ) J := − ǫ ij x i ∂∂x j − ξ χ i χ i = − x ∧ ∂ x − ξ χ R := − ǫ ij x i ∂∂x j − ǫ ij ν i ∂∂ν j = − x ∧ ∂ x − ν ∧ ∂ ν (B.1)Martelli and Tachikawa [27] advocated in favour of the generator J , because it naturallyappears in the contraction procedure they used in deriving the ecga . Here, we wish tocompare with the results found for the naturally-looking generator R , also mentioned as apossible alternative in [27]. 21rom a purely algebraic point of view, there is no criterion which would lead one toprefer one of these two choices. Both obey the same commutation relations with the othergenerators of the ecga and both commute with the Schr¨odinger operator S defined insection 3.Here, we shall show that physically distinct results are found, depending on the use of J (which we shall refer to as ‘ J -invariance ’) or R (which we shall refer to as ‘ R -invariance ’).Namely, the rapidity-less ecga -covariant two-point function F = h φ φ i has the form F = f ( u − ν + ) t − e − ξ ν − ∧ (2 u − ν + ) δ ∆ , ∆ δ ξ + ξ , (B.2)where ν ± = ( ν ± ν ) and still contains an undetermined differentiable function f = f ( w )of the single variable w = u − ν + . Any explicit dependence on the single variables ν ± of thetwo-point function F is already contained in (B.2). The additional requirement of rotation-invariance leads to a clear distinction (cid:26) f is arbitrary ; J -invariance f = f ( w + w ) ; R -invariance (B.3) Proof:
Begin by writing down the two-particle form of the generators J and R (where inview of the coming application to F , the Bargman super-selection rule ξ = − ξ has alreadybeen used) J = − u ∧ ∂ u − ξ (cid:0) ∂ ν + · ∂ ν − + ξ ν + · ν − (cid:1) − (cid:0) ν + ∧ ∂ ν + + ν − ∧ ∂ ν − (cid:1) R = − u ∧ ∂ u − ν ∧ ∂ ν − ν ∧ ∂ ν = − u ∧ ∂ u − ν + ∧ ∂ ν + − ν − ∧ ∂ ν − (B.4)In working out the condition of rotation-invariance J F ! = 0 or RF ! = 0, respectively, weshall need the following auxiliary formulæ, with b ǫ = (cid:18) − (cid:19) and a , b ∈ R such that a ∧ b = a · ( b ǫ b ) ∂ a e − a ∧ b = − ( b ǫ b ) e − a ∧ b ( b ǫ a ) · b = − a ∧ b (B.5)( b ǫ a ) · ( b ǫ b ) = a · ba ∧ ( b ǫ b ) = − a · b Then, application of the two distinct rotation generators to the two-point function (B.2) leadsstraightforwardly in the case of J -invariance simply to the identity J F = 0, whereas in thecase of R -invariance we obtain w ∧ ∂ w f ( w ) = 0. Hence, given the form (B.2), J -invarianceis automatic and does not impose any further condition on the function f . On the otherhand, in the case of R -invariance, f can only be a function of the scalar | w | = w + w .This proves the assertion (B.3). q.e.d.As we shall see below, this distinction between the generators J and R will lead to furtherconsequences in the case of non-vanishing rapidities.22 .2 Rotation-invariance for non-vanishing rapidities: non-logarithmiccase In the non-logarithmic case, the ecga -covariant two-point function reads F = f ( u − ν + ) t − e − γ + · u − γ − · ν − e − ξ ν − ∧ (2 u − ν + ) δ ∆ , ∆ δ ξ + ξ , (B.6)where in addition to the previous conventions, we also defined γ ± := ( γ ± γ ). Therefore,the two rotation-generators must be generalised to include rapidity terms and read for asingle particle J = − x ∧ ∂ x − γ ∧ ∂ γ − ξ χ · χ , R = − x ∧ ∂ x − γ ∧ ∂ γ − ν ∧ ∂ ν (B.7)and for two particles J = − u ∧ ∂ u − ξ (cid:0) ∂ ν + · ∂ ν − + ξ ν + · ν − (cid:1) − (cid:0) ν + ∧ ∂ ν + + ν − ∧ ∂ ν − (cid:1) − γ + ∧ ∂ γ + − γ − ∧ ∂ γ − R = − u ∧ ∂ u − γ + ∧ ∂ γ + − γ − ∧ ∂ γ − − ν + ∧ ∂ ν + − ν − ∧ ∂ ν − (B.8)Again, the undetermined function f = f ( γ + , γ − , w ) depends on the single variable w = u − ν + . However, since the rotations can also transform the rapidities γ ± , f can in additionalso depend explicitly on them. Hence, f = f ( γ + , γ − , w ) will be a function of 6 variables,subject to a single condition coming form rotation-invariance.Using the same auxiliary identities (B.5) as before, straightforward but slightly tediouscalculations lead to J -invariance : γ + ∧ ∂f ∂ γ + + γ − ∧ ∂f ∂ γ − + γ − ξ · ∂f ∂ w = 0 R -invariance : γ + ∧ ∂f ∂ γ + + γ − ∧ ∂f ∂ γ − + w ∧ ∂f ∂ w = 0 (B.9)In order to find the general solutions of these, in the case of J -invariance, one introducesa new variable v := γ − − ξ b ǫ w and takes f as a function f ( γ + , γ − , v ). Then the firstof eqs. (B.9) reduces to (cid:0) γ + ∧ ∂ γ + + γ − ∧ ∂ γ − (cid:1) f = 0. Three obvious and independentsolutions of this are γ , γ − and γ + · γ − , from which the general solution can be constructed.On the other hand, in the case of R -invariance one easily lists 5 independent solutions sothat finally J -invariance : f = f (cid:0) γ , γ − , γ + · γ − , w + b ǫ γ − ξ − (cid:1) R -invariance : f = f (cid:0) w , γ , γ − , w · γ + , w · γ − (cid:1) (B.10)Reverting to the original variables γ , gives the expressions in the main text or appendix A.A comparison of the two distinct eqs. (B.9) shows the origin of these two distinct formsof the function f : while in the case of R -invariance, the habitual form the of the rotationgenerator guarantees that formal scalar products of the vectors w , γ + and γ − are alwaysrotation-invariant, this holds no longer true in the case of J -invariance, where only scalar23roducts between the rapidity vectors γ ± have this property. Model-specific calculations willpermit to distinguish between these possibilities.We also observe that in the case of J -invariance, taking the non-exotic limit ξ → γ − → , in order to maintain a finite value in the last argument of the function f .In this way, one can understand how the constraint γ = γ in non-exotic CGA is recovered.No such limit argument can be made in the case of R -invariance. B.3 Rotation-invariance for non-vanishing rapidities: logarithmiccase
One must must take further into account that the γ ’s become Jordan matrices, such thatthe rotation generators have to be generalised to the forms J = − x ∧ ∂ x − γ ∧ ∂ γ − γ ′ ∧ ∂ γ ′ − ξ χ · χ , R = − x ∧ ∂ x − γ ∧ ∂ γ − γ ′ ∧ ∂ γ ′ − ν ∧ ∂ ν (B.11)for a single particle and with analogous extensions in the two-particle case. In addition tothe two-point function F = h φ φ i already analysed in the non-logarithmic representations,one now has to consider the additional two-point functions G = h φ ψ i and H = h ψ ψ i .Furthermore, we have already seen in the main text that two distinct cases have to bedistinguished, depending on whether the matrices b ∆ , have Jordan form (case 1) or arediagonal (case 2). A) If we consider R -invariance, begin with case 1 (where F = 0 and require co-variance RG ! = 0 ! = RH , one has w ∧ ∂g ∂ w + γ ∧ ∂g ∂ γ + γ ∧ ∂g ∂ γ + γ ′ ∧ ∂g ∂ γ ′ + γ ′ ∧ ∂g ∂ γ ′ = 0 (B.12)and analogously for h , since all individual terms in (A.17) are explicitly rotation-invariant.In principle, the functions g , h depend on the 10 variables w , γ , , γ ′ , . Since rotation-invariance imposes a single extra condition, there remains a function of 9 rotation-invariantvariables, for example g = g (cid:16) w , γ , γ , γ ′ , γ ′ , w · γ , w · γ , w · γ ′ , w · γ ′ (cid:17) h = h (cid:16) w , γ , γ , γ ′ , γ ′ , w · γ , w · γ , w · γ ′ , w · γ ′ (cid:17) (B.13)For case 2, an analogous argument applies and one recovers (B.13) along with an analogousform for f . B) A very different result is found for J -invariance. If one considers case 1 first, one hasagain F = h φ φ i = 0, whereas G = h φ ψ i has the same form as the two-point function F treated above in the non-logarithmic case. It remains to consider the two-point function H = h ψ ψ i = t − e − γ + · u − γ − · ν − h (B.14)where the scaling function h can be written as h = h ( u − ν + ) − g ( u − ν + ) (cid:0) ′ ln | t | + u · ( γ ′ + γ ′ ) + ν − · ( γ ′ − γ ′ ) (cid:1) (B.15)24n complete analogy to the previous sub-section, J -invariance implies the conditions D h = 0 , D g = 0 (B.16)with the differential operator D := γ + ∧ ∂∂ γ + + γ − ∧ ∂∂ γ − + γ ′ ∧ ∂∂ γ ′ + γ ′ ∧ ∂∂ γ ′ + γ − ξ · ∂∂ w (B.17)This gives the following equation for h : D h − g D (cid:0) ′ ln | t | + u · ( γ ′ + γ ′ ) + ν − · ( γ ′ − γ ′ ) (cid:1) = 0 (B.18)Working out the differential operator and taking the condition w = u − ν + into account,leads to D h − (cid:18) ( γ ′ + γ ′ ) · (cid:18) ξ γ − + u (cid:19) + ( γ ′ − γ ′ ) ∧ ν − (cid:19) g = 0 (B.19)However, since h depends only on w and not on u or ν ± separately, this condition is onlycompatible with our previous results if γ ′ = γ ′ = (B.20)Then only the matrix b ∆ of the conformal weight can have a Jordan form and g = g (cid:0) γ , γ − , γ + · γ − , w + b ǫ γ − ξ − (cid:1) h = h (cid:0) γ , γ − , γ + · γ − , w + b ǫ γ − ξ − (cid:1) (B.21)Reverting to the original γ , gives the expressions in the main text or in appendix A.Similar arguments apply to case 2: since now F = h φ φ i 6 = 0, consideration of G leadsto γ ′ = and the other mixed two-point function G gives γ ′ = . Then no logarithmicstructure remains. For J -invariance, there is but a single case, see table 1. References [1] H. Saleur, “Polymers and percolation in two dimensions and twisted N = 2 supersym-metry,” Nucl. Phys. B382 , 486 (1992) [arXiv:9111007[hep-th]] .[2] V. Gurarie, “Logarithmic operators in conformal field theory,” Nucl. Phys.
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