Yang Baxter and anisotropic sigma and lambda models, cyclic RG and exact S-matrices

Journal of High Energy Physics, Sep 2017

Integrable deformation of SU(2) sigma and lambda models are considered at the classical and quantum levels. These are the Yang-Baxter and XXZ-type anisotropic deformations. The XXZ type deformations are UV safe in one regime, while in another regime, like the Yang-Baxter deformations, they exhibit cyclic RG behaviour. The associ-ated affine quantum group symmetry, realized classically at the Poisson bracket level, has q a complex phase in the UV safe regime and q real in the cyclic RG regime, where q is an RG invariant. Based on the symmetries and RG flow we propose exact factorizable S-matrices to describe the scattering of states in the lambda models, from which the sigma models follow by taking a limit and non-abelian T-duality. In the cyclic RG regimes, the S-matrices are periodic functions of rapidity, at large rapidity, and in the Yang-Baxter case violate parity.

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Yang Baxter and anisotropic sigma and lambda models, cyclic RG and exact S-matrices

HJE Baxter and anisotropic sigma and lambda models, cyclic RG and exact S-matrices Calan Appadu 0 1 Timothy J. Hollowood 0 1 Dafydd Price 0 1 Daniel C. Thompson 0 1 0 Swansea , SA2 8PP , U.K 1 Department of Physics, Swansea University Integrable deformation of SU(2) sigma and lambda models are considered at the classical and quantum levels. These are the Yang-Baxter and XXZ-type anisotropic deformations. The XXZ type deformations are UV safe in one regime, while in another regime, like the Yang-Baxter deformations, they exhibit cyclic RG behaviour. The associated a ne quantum group symmetry, realized classically at the Poisson bracket level, has q a complex phase in the UV safe regime and q real in the cyclic RG regime, where q is an RG invariant. Based on the symmetries and RG S-matrices to describe the scattering of states in the lambda models, from which the sigma models follow by taking a limit and non-abelian T-duality. In the cyclic RG regimes, the S-matrices are periodic functions of rapidity, at large rapidity, and in the Yang-Baxter case violate parity. Integrable Field Theories; Quantum Groups; Sigma Models - Yang 1 Introduction 2 3 2.1 2.2 3.1 3.2 3.3 3.4 4.1 4.2 4.3 5.1 5.2 5.3 5.4 5.5 Classical SU(2) sigma models Lax connection and Poisson brackets Non-local charges and in nite symmetries Classical SU(2) lambda models Target spaces Lax formalism Poisson structure and symmetries Non-local charges and in nite symmetries 4 Renormalization group ow ow The sigma model RG Yang Baxter lambda model RG ow Anisotropic XXZ lambda model RG ow 5 Quantum group S-matrix: q complex phase Quantum group S-matrix: q real The RSOS S-matrix High energy limit The S-matrix proposals 6 Discussion A Lambda spacetimes 1 Introduction Sigma models are fascinating because they are the building blocks of string worldsheet theories but also they share many of the features of QFTs in higher dimensions in a simpler context. And within the space of sigma models, the ones that are integrable have the additional lure of tractability. The key examples are the Principal Chiral Models (PCM), whose target spaces are group manifolds G. There is a G-valued eld f and the action can be written1 S = 2 (1.1) 1We take x = t x and so for vectors A = A0 A1 along with A = 12 (A0 A1). { 1 { outgoing particles as well as the internal quantum numbers i; j; k; l. The PCM can appear as a bosonic sub-sector of a consistent string theory CFT background, e.g. the D1-D5 near horizon geometry, providing a modern holographic motivation for studying this theory. The more prosaic view, which we adopt here, is that PCMs are an exceptionally informative 1 + 1-dimensional QFTs exhibiting asymptotic freedom in the running coupling ( ) and a dynamically generated mass gap. The action given in eq. (1.1) manifests a GL GR global symmetry, f ! U f V . A feature that makes the PCM tractable is that it is classically integrable and the GL GR symmetry is part of a much larger classical Yangian Y (gL) Y (gR) symmetry generated by non-local charges.2 At the quantum level this integrability persists leading to the factorization of its Smatrix [2, 3]. This means that it is completely determined by 2 ! 2 body processes which preserve the individual momenta, as illustrated in gure 1. The states are labelled by their rapidity and by internal quantum numbers i; j; : : :. For example, in the SU(N ) PCM, there are N 1 particle multiplets with mass ma = m sin( a=N ), a =; 1; 2; : : : ; N 1, and each multiplet transforms in the [!a] [!a] representation of the GL GR symmetry, where !a are the highest weight vectors of the ath fundamental representation.3 The 2-body S-matrix has the characteristic product form [4]: S( ) = SGL ( ) SGR ( ) ; (1.2) where = 1 2. The product form re ects the fact that the states transform in a product of representations of GL and GR. The S-matrix building block SG( ) is G-invariant, in fact Yangian invariant, and is built from a rational solution of the Yang-Baxter Equation.4 Since the PCM is asymptotically free and its spectrum is massive and dynamically generated, directly connecting the conjectured quantum S-matrix picture to the Lagrangian description in eq. (1.1) is subtle. Nonetheless, consistency checks can be made by studying the theory in a regime in which perturbation theory can be employed and compared against the factorized S-matrix. The study of the exact solution of the model was initiated in the classic works [4{7]. As a byproduct of the successful comparison of Thermodynamic Bethe 2A concise introduction to this symmetry can be found in [1]. There are also an in nite number of local conserved charges which include and energy and momentum. 3For the groups SO(N ) the representations are actually reducible combinations. 4For the higher rank groups, the product form of the S-matrix must be multiplied by a scalar factor to provide the bound state poles. { 2 { Ansatz and perturbative calculations of the free energy in a background charge one obtains an exact expression for the mass gap.5 A natural question to ask, is whether the PCM can be deformed in a way that preserves integrability? For the case SU(2) | which we will concentrate on in this work | there are several ways to do this, while for higher rank groups the possibilities appear to be more limited. We will concentrate on the deformations that preserve one of the chiral symmetries, SU(2)L, say. Deformation which preserve the SU(2)L symmetry can be written where is endomorphism of the Lie algebra, T a = abT b. A fascinating problem is to determine systematically which choices of lead to integrable models both classically and quantum mechanically. For the particular case of G = SU(2), there are anisoptropic type deformations that involve in the most general case three di erent couplings components of the SU(2)L current J general deformation of this type as [12, 13] = P a J aT a,6 we can write the action for the most T a = a 1T a. Introducing the S = (1.3) HJEP09(217)35 S = where we have de ned the R-Lie bracket [Ra; Rb] R[a; b]R = c2[a; b] ; [a; b]R = [Ra; b] + [a; Rb] ; { 3 { for all a; b in the Lie algebra and where c is a free parameter. The action of the deformed theory is de ned by taking in (1.3) = 1(1 R) 1 ; 5The case of G = SU(2) viewed as the O(4) -model was done in [8], with the extension to SU(N ) in [9, 10] and other Lie algebras in [11]. iT 2)=p2 and the alternative decomposition J = J3T 3 + J+T + J T +. ab. So for SU(2), T a = i a p = 2 where 6Throughout the paper we use a basis fT ag that are anti-hermitian and normalized so that Tr(T aT b) = a are the Pauli matrices. In addition, we de ne T = (T 1 where is the real deformation parameter. YB deformations of this type can be de ned for an arbitrary group and in general the deformed theories have a Kalb-Ramond eld which correspond to the terms odd in R when the operator (1 R) 1 is expanded in powers of R. generality, can be written as For SU(2) there is a single class of deformations of this type which, without loss of (1.8) (1.9) (1.10) 1. In this case, one can show that the Kalb-Ramond eld is a total derivative and | at least with periodic boundary conditions | the YB and XXZ sigma theories are equivalent with in the regime with two parameter Fateev model [18].7 We will not consider this more general deformation any further and focus on deformations that preserve the SU(2)L symmetry because these cases have an associated lambda model. The lambda models are a completely di erent class of integrable deformations of the PCM. In fact of each of the sigma models, whether PCM, XXZ, XYZ or YB, i.e. all having an SU(2)L symmetry, have an associated lambda model that inherits the integrability of the parent sigma model. Motivated by the process of non-abelian T-duality in string theory, each sigma model whose target space is a G group manifold with GL global symmetry has an associated lambda model.8 The de nition of the lambda model associated to the SU(2) PCM go back to [19] but in a more general context are best constructed by Sfetsos's gauging procedure [20]: 1. Write down a theory which is the sum of the actions of the sigma model eq. (1.3) and a WZW model for a G-valued eld F . 7The matching of parameters (de ned after eq. (76) of [18]) is given by r u 2 = (`u 1 + 1) ; 2 = (ru 1 + 1) ; = u : 8There are also examples associated to symmetric space quotients G=H that we will not consider here. ` u { 4 { 2. Gauge the joint G symmetry, which acts on the WZW U F U 1 and the sigma model eld by left action f ! U f . 3. Gauge x the G symmetry by setting the sigma model eld f = 1. eld by vector action F ! Applied to the deformed PCM de ned in eq. (1.3), the result of this procedure leads to a deformation of a G WZW model written in the following way: eld that can be integrated out. The rst term is the gauged WZW model action [21{25] for a G-valued eld F , where the whole vector G symmetry is gauged, and k 2 Z is the level. What is crucial for us is that if the original sigma model is integrable then so is the associated lambda model. There is also a sense that the original sigma model is recovered in the limit k ! 1 along with a non-abelian T-duality [20]. It is noteworthy that this relation is also seen quantum mechanically at the level of the S-matrix where nonabelian T-duality manifests as an IRF-to-vertex transformation on the space of asymptotic states [26].9 A fascinating question is to understand whether these integrable deformations persist in the quantum theory and if so, what are their factorizable S-matrices. We have already remarked that the PCM S-matrix takes the product form of two rational factors (1.2) that manifest the Yangian Y (su(2)L) Y (su(2)R) symmetry. This form seems to generalize: the XXZ models in the regime < lie in the class of \SS models" considered by Fateev [18],10 which have an S-matrix of the form [12] In this expression, S( ; 0) is the S-matrix of the sine-Gordon theory with coupling11 S -XXZ( ) = SSU(2)L ( ) S( ; 0) : 8 2 = 0 1 + 0 : (1.13) (1.14) The tensor product form of the S-matrix in (1.13) will prove ubiquitous and deserves some comment. Like the PCM S-matrix (1.2) it re ects the factor that the particle states carry two sets of quantum numbers which under scattering are completely independent. 9It is worth remarking that at the classical level non-abelian T duality can be thought of as a canonical transformation [27] while at the quantum level the IRF-to-vertex transformation can be thought of as a change of basis in the Hilbert space [28{30]. It would be interesting to make the connection between the two phenomena more explicit. 10In terms of Fateev's more general model with U( 1 ) U( 1 ) symmetry and parameters (a; b; c; d), we have a2 = u(u + `), b = 0 and c = d = `=2 and ? = (u + `) 1 and 3 = u 1. Then ? = ( ) 1 ; 3 = ( ) 1 ; 3 < ? : (1.12) 11Our 0 is 0=8 of Zamolodchikov and Zamolodchikov [55]. For us the breather spectrum is mn = 2M sin( n 0=2), n = 1; 2; : : : < 0 1. { 5 { The XXZ deformation has broken the SU(2)R Yangian symmetry but rather than disappearing it is deformed to an a ne quantum group Uq(s[u(2)) symmetry, where the deformation parameter in section 3. Note that for 0 < 1, the model has bound states that correspond to the breathers of the sine-Gordon theory. In the present context, the nth breather transforms as a singlet under Uq(su(2)) but as a reducible representation of SU(2)L corresponding to the tensor product of n spin 12 representations. The XXZ model of Fateev displays an important general feature of the integrable deformations: Yangian symmetries generally get deformed into a ne quantum group symmetries. The label \quantum" here might be thought a misnomer because the quantum group symmetries are manifest in the classical theory at the Poisson bracket level [31{33]. This point deserves some comment. We shall show that the deformation parameter q does indeed depend on ~ (or more precisely the coupling that plays the role of ~) as q = exp[ ~]. However, there is a consistent classical limit, where ~ ! 0 but the coupling constant dependent quantity ! 1 such that q is xed. In addition, as part of the overall consistency we will show that the q is an Renormalization Group (RG) invariant and so the quantum group symmetries are well de ned in the quantum theory and the classical limit where it becomes realized at the Poisson bracket level. The lambda deformations also have a characteristic e ect on the S-matrix [26]. For the PCM itself, the deformation changes the S-matrix block for the SU(2)L symmetry into an a ne quantum group invariant block, but realized in the Interaction-Round-a-Face (IRF), or Restricted-Solid-On-Solid (RSOS), form:12 The original PCM S-matrix is recovered in the limit k ! 1, where the kink factor becomes unrestricted, and then an IRF-to-vertex transformation which is the S-matrix manifestation of non-abelian T-duality: will describe all the integrable deformations of the PCM and this intuition will turn out to be true. In this paper, we will concentrate on the XXZ and YB deformations of the 12This type of S-matrix block appears in the context of the \restricted sine-Gordon theory'" [28{30] and also perturbed WZW models [34]. (1.16) (1.17) { 6 { SU(2) PCM and their associated lambda models at the quantum level and map out their renormalization structure and their S-matrices and con rm the ubiquity of the product form. Speci cally in this paper we: 1. Review the classical integrability of the deformed sigma models and establish some new results for the Poisson brackets of the associated lambda models. 1. We show that the lambda models have quantum group symmetries in the classical theory realized at the level of the Poisson brackets. 2. We then consider the RG ow of the sigma and lambda models at one loop order (so in the lambda models to leading order in 1=k). We show that the XXZ models, both sigma and lambda, have one regime which has UV safe ows, whereas in the other regime there are cyclic RG type ows. The YB lambda model also has cyclic RG ows. tum groups are RG invariants. 3. We show that the quantum deformation parameters q of the classically-realized quan4. Using the RG ow and the structure of the classical symmetries, we propose Smatrices to describe all the lambda models. For the examples with cyclic RG ow, the S-matrix has periodicity in the rapidity when the rapidity is large. 5. We then argue that S-matrices of the sigma models are obtained in the large k limit after an IRF-to-vertex transformation. In a follow up paper, we will address the question of whether the theories that we nd with cyclic RG behaviour actually exists as QFTs in the continuum limit [35]. We will nd that the continuum theories can be formulated as a Heisenberg XXZ spin chain. When the RG ow of the theory has a UV safe limit, the spin chain is critical and a continuum limit can be de ned. On the contrary in the regime with cyclic RG ows, the spin chain has a gap and a continuum limit does not exist. Th conclusion would be that the theories with cyclic RG behaviour only exist as e ective theories with an explicit cut o . 2 Classical SU(2) sigma models In this section, we consider some of the aspects of the sigma models and in particular the symmetries, that will inform our S-matrix hypotheses. 2.1 Lax connection and Poisson brackets The most direct way to prove classical integrability is to write down the equations of motion in Lax form, that is as the atness condition on an auxiliary connection that depends on an additional free parameter, the spectral parameter, + L (z)] = 0 ; (2.1) for arbitrary z. { 7 { If we de ne the SU(2)L invariant current J motion along with the Cartan-Maurer identity of the YB deformed sigma models can be written in Lax form with a gauge transformation [32, 33, 36]. We note in passing that the Lax connection is valued in the loop algebra s[u(2) = su(2) C[z; z 1], the untwisted a nization of su(2) (with vanishing centre). This can also be described as the a ne algebra with the homogeneous gradation and we will denote it as s[u(2)h. For the anisotropic models, the Lax connection take a characteristic form that generalizes nicely as one goes through the hierarchy from PCM to XXZ to XYZ: For the PCM, the functions wa(z) are rational while for the XXZ case, with a = ( ; ; ), the functions wa(z) are trigonometric (or hyperbolic) [13], where For these theories, if we transform to a multiplicative spectral parameter z ! log z and then expand in powers of z, it is noteworthy that the Lax connection takes values in the twisted loop algebra, where the twist is an automorphism : (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) (T 1;2) ! T 1;2 ; (T 3) ! T 3 : The Lie algebra splits into its eigen-spaces under and in the twisted loop algebra each eigen-space receives a di erent scaling of the spectral parameter. The twisted loop algebra thus has elements T 3z2n, T 1z2n+1 and T 2z2n+1, with n 2 Z. Since the automorphism inner the twisted loop algebra is simply equal to original in another gradation, in this case is it called the principal gradation and we denote it s[u(2)p. Finally, for the XYZ case, the functions wa(z) are elliptic functions { 8 { where the Jacobi elliptic functions have an elliptic modulus In addition, Whilst these theories still have the SU(2)L symmetry, the SU(2)R symmetry is broken to a nite Z4 subgroup. The question as to whether the associated Yangian symmetry becomes deformed is an interesting one that we do not tackle here. Note that since the SU(2)L symmetry is preserved these theories are distinct from the general two-parameter HJEP09(217)35 deformations considered in [18]. Note that the XXZ model in the regime > has the same equation of motion as the YB model but the Lax connections are completely di erent. The relation between the two formulations was considered in detail in [33]. As part of the standard formalism of integrability (e.g. see the book [38]), a key structure is the Poisson bracket of the spatial component of the Lax connection L L+ L . This is sometimes called the Maillet algebra [39] and in general takes the form fL1(x; z); L2(y; w)g = [r(z; w); L1(x; z) + L2(x; w)] (x y) [s(z; w); L1(x; z) L2(y; w)] (x y) 2s(z; w) 0(x The notation means that the bracket acts on a product of su(2) modules V the subscripts indicate which of the copies a quantity acts on: L1(z) = L (z) L2(z) = 1 L (z). The tensor kernels r(z; w) and s(z; w) act on V V . In many cases, the kernels r and s can be written in the form where (z) is known as the twist function and in many cases = Casimir tensor. For example, for the YB deformation with the de nition of the Lax connection in [33], the kernels r and s take precisely this form with a twist function Pa T a T a is the Note that here we include the factor 2 which plays the role of ~ in the quantum theory. For the XXZ model in the trigonometric formulation, the r=s kernels take a similar form, except that depends on z and w, and the twist function (z) = 2 1 p 2 sinh2 z cosh2 z : { 9 { (2.9) (2.10) y) : (2.11) V and 1 and (2.12) (2.13) (2.14) (2.15) Z 1 1 T (z) = Pexp dx L (x; z) (2.16) Integrable eld theories have an in nite sets of both local (integrals of expressions local in the elds and their derivatives) and non-local conserved charges. All these charges can be extracted from the Lax connection. The local conserved charges include the energy and momentum but the non-local ones are our central focus here because they generate some remarkable in nite symmetries in the form of Yangians and quantum groups. The non-local charges are encoded in the monodromy matrix, the parallel transport of the Lax connection, along the spatial direction which is conserved in time (in the in nite volume limit with appropriate fall o assumed). We can think of T (z) as a generating function for the charges. It is natural to lift the Poisson bracket on L (x; z) to the monodromy matrix. However, this is where a problem arises as a result of the non ultra-locality of the Poisson bracket: when the kernel s is non-trivial the Poisson bracket of the monodromy matrix is ill-de ned due to the 0(x y) term in (2.11). This non ultra-locality can lead to ordering ambiguities when considering nested integrals in the expansion of the monodromy matrix and a violation of the Jacobi identity for the monodromy matrix. One way to deal with the ambiguities is to use Maillet's prescription [39]. This corresponds to lifting the Poisson bracket to the monodromy matrix in the form fT1(z); T2(w)g = [r(z; w); T1(z)T2(w)] + T1(z)s(z; w)T2(w) T2(w)s(z; w)T1(w) : It is remarkable that the non ultra-locality and its associated ambiguities generally turn out not to a ect the discussion of the Yangian and quantum group symmetries when they are manifested at the classical level [36]. As we will see, there can also be quantum group symmetries that can only be seen consistently at the quantum level. The in nite symmetries are associated to the expansion of the monodromy matrix T (z) around special points z which de ne non-local charges that generate Yangian or quantum group symmetries. The general idea is as follows: generically the kernel r has a pole as z ! w; however, there are special points z in the neighbourhood of which, z = z + ; w = z + ~; the Poisson bracket algebra has a nite limit as and ~ are scaled to 0. The special points can also be at in nity in which case one takes z = 1 and w = ~ 1. If the r=s kernels take the form (2.12), then poles of the twist function are special points (see [37] for a general analysis for these cases). For example, for the YB deformation with twist function (2.13), there are poles at z = i , around which i i 1 + 2 1 + 2 + ~ ~ 3 X T a a=1 T a + O( ) : T a + O( ) ; (2.17) (2.18) (2.19) z = +i z = i +2 +1 0 . . . 1 2 Q2+ Q~+ Q+ Q+1 Q+2 . . . Q32 Q31 Q2 Q1 Q ~ Q Q 2 . . . i . The blue/red and positive/negative graded charges are associated to i , respectively. The red and blue charges generate the a ne quantum group in homogenous gradation and all the other charges are obtained by repeated Poisson brackets of these charges. It has been shown that the charges de ned by expansing the monodromy matrix around these special points generate a classical version of an a ne quantum group symmetry Uq(s[u(2)) with a deformation parameter [32, 33, 36] comes from the overall normalization of the action and plays the role of ~, and so is usually set to 1 in a classical analysis [32, 33, 36]. For us, pursuing a quantum analysis, having the correct overall normalization is crucial because the correctly de ned q is then an RG invariant. For Yang-Baxter deformations a similar result was obtained for arbitrary groups and also symmetric space coset -models in a now seminal paper [40].13 In the expansion of T (z) around z = i , the charges are naturally are classi ed by the order in which they appear [33] (positive/negative grade for z = i ): see gure 2. The U( 1 )R charge Q3, local in elds, is supplemented with non-local conserved charges Q that obey a (classical) quantum group Uq(su(2)) symmetry under the Poisson bracket fQ+; Q g = i qQ3 q q Q3 q 1 ; fQ ; Q3g = iQ : (2.21) In addition to these, one obtains generators Q~ symmetry (the extension is centreless since Q~ 3 = associated to the a ne extension14 of this Q3 and so the a ne algebra is actually 13For group case the result of [40] is that q = exp[ (1 2)3=2] with the above after taking into account that the overall tension has been set as = = 1 p 2 which matches 1 = (1 + 2)2. Although not present focus it would be remiss not to mention that that a Yang-Baxter deformation of the MetsaevTseytlin action for strings in AdS5 S 5 was constructed in [41, 42]. 14Recall that the a ne extension s[u(2) supplements the Chevalley generators fE1; F1; H0g of su(2) with an additional root and corresponding generators fE0; F0; H0g obeying the standard relations [Hi; Ej] = aijEj, [Hi; Fj] = aijFj and [Ei; Fj] = ijHj together with the Serre relations. Here the generalised Cartan matrix aij has o diagonal elements equal 2. K = H0 + H1 is central and in what follows we will a loop algebra). There are an in nite series of higher charges, but these can be recovered by taking repeated Poisson brackets of the charges shown. The grading that is imposed on the algebra by the order of the expansion that the charges appear around the special points is precisely the homogeneous gradation s[u(2)h. The other important point is that the full set of charges that generate the a ne quantum group are associated to a pair of special points. w = ~ 1, the kernels have the expansion In the YB sigma model there is also a special point at in nity. Setting z = In this case, the non-vanishing contribution is at O( ). The charges that are de ned by the expansion of monodromy matrix around in nity generate an in nite Yangian symmetry Y (su(2)L) that includes the global SU(2)L symmetry. Now we turn to the anisotropic XXZ deformed sigma model with twist function (2.15). In this case, the in nite symmetries are associated to the pole of the twist function at z = 0 and to the behaviour at 1. Before proceeding it is more convenient to transform to multiplicative spectral parameter z ! log z in which case the twist function takes the form The pole is now at z = 1, and expanding around it, we have (z) = 2 1 p 2 The leading behaviour here is O( ) and so it indicative of a Yangian symmetry. In fact, expanding around this pole gives the Yangian symmetry Y (su(2)L) in the trigonometric formulation. The special points at 1 map to z = 0; 1, around which consider modules where K = 0, i.e. centreless representations for which s[u(2) becomes the loop algebra. Note that we will not distinguish the real form sl(2) from su(2) where appropriate. This being the case, representations are the tensor of an su(2) representation and functions of a variable z. There is a choice, known as gradation, to be made as to the relative action in su(2) space and z-space. In the homogenous gradation is E1 = T +, F1 = T , E0 = z2T , F0 = z 2 T +, H1 = H0 = T 3. In the principal gradation E1 = zT +, F1 = z 1 T , E0 = zT , F0 = z 1T +, H1 = H0 = T 3. These gradations lift to the quantum group deformation Uq(s[u(2)). (2.22) (2.23) (2.24) (2.25) +2 +1 0 Q3 2 Q3 = Q~ 3 ~ Q Q 1 (or 0; 1 with a multiplicative spectral parameter). The blue/red and positive/negative graded charges are associated to 1, respectively. The red and blue charges generate the a ne quantum group in principal gradation and all the other charges are obtained by repeated Poisson brackets of these charges. The expansions in this case are associated to a quantum group symmetry with the same deformation parameter (2.20) as in the YB case, once we identify parameters as in (1.9). The charges emerge as illustrated in gure 3 [33]. Once again there are an in nite set of charges but the ones shown generate the a ne algebra and the higher charges are then obtained by repeated Poisson brackets of the lower charges. The a ne algebra is now revealed to be associated to the principal gradation s[u(2)p. So although the YB and XXZ sigma models have the same equations of motion and what seems like identical symmetries, a Yangian and an a ne quantum group, there is a subtle di erence. The a ne quantum group for the YB is in the homogeneous gradation while in the XXZ case it is in the principle gradation. This interpretation is consistent with the spectral parameter rescaling of SU(2) generators found in [33] required to go between the two expansion. For the YB deformation of arbitrary rank groups, for which only a rational Lax description exists, the same homogenous gradation shows itself [36]. The existence of these symmetries at the classical level is important because they will inform our search for the quantum S-matrices that describe the quantum versions of these theories. The symmetries are summarized in table 1. 3 Classical SU(2) lambda models The lambda model associated to a sigma model have been de ned in (1.11). The second term in (1.11) vitiates the gauge symmetry and A becomes an auxiliary Gaussian eld. Correspondingly, the equations of motion of A change from rst class to second class constraints [43]: F T A+ ; A ; Model -YB PCM -XXZ Left symm. Right symm. Y (su(2)) Y (su(2)) Y (su(2)) Uq(s[u(2)h) Y (su(2)) Uq(s[u(2)p) is given in (2.20) in terms of the underlying coupling constants. The only (subtle) di erence between the symmetries is that in the YB case, the a ne quantum group is naturally in homogeneous grade, while in the anisotropic XXZ case it is in principal grade. where After integrating out the auxiliary eld A , we can write the resulting theory as k Z 2 Sk; [F ] = k SWZW[F ] + d 2 Tr F This form makes it clear that as an expansion in current-current deformation of the WZW model: 1 the theory can be interpreted as a k Z 2 Sk; [F ] = k SWZW[F ] + d 2 Tr F 1 1 + The implication is that if the couplings ow into the UV in such a way that lambda model can be interpreted as a perturbed WZW CFT. The equations of motion of the theory have a simple form when written in terms of the auxiliary eld A :15 : 1 ! 0, the The isotropic lambda model associated to the PCM for which = 1I, gives = I + k 1 : = 1I ; = k k = B 0 0 1 0 C ; 0 1 1 A This is the model constructed and studied in [19]. The XXZ version of the model has 15Note that the transpose is de ned with respect to the trace: Tr(a b) = Tr( T a b). (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) This should be compared with the YB version of the model for which = k k + 1 ; = I + (I 1 k R) 1 = = 1 k k where where and where the original sigma model couplings are (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) HJEP09(217)35 Now we can see that the XXZ lambda model, even in the regime > , i.e. > , where the associated sigma models are equivalent up to a boundary term, is distinct from the YB lambda model. In particular, the YB lambda model breaks parity symmetry explicitly as can be seen from the fact that , entering the de nition eq. (1.11), is not symmetric. There is also a XYZ lambda model for which 1 = diag( i ) with all i distinct, rst constructed in [43]. This will be considered in more detail elsewhere [35]. 3.1 Target spaces With the group element parametrized as F = C + iS C S S e i S S e i C iS C ! ; where we have de ned Sx sin x and Cx models with target spaces of the following form cos x, the lambda theories can viewed as sigma k A0 A5 0 1 2 ^ d ; 1 2 A1 d 2 + A2 d 2 + A3d 2 + A4 d d ; = where Ai = Ai( ; ). The non-trivial dilaton is due to a determinant arising from performing the Gaussian integration on the non-propagating ex-gauge elds A in the path integral. The exact functional forms are not particularly enlightening but are recorded in appendix A. Here we note the feature, seen in other lambda deformations, that all the coordinate dependence cancels in the expression for the dilaton beta function,16 ~ = R + 4r2 2 (H3)2 : Explicitly we nd that for the XXZ lambda model 1 12 1) ~XXZ = 2 4 ( + 1) in comparison to the result obtained for the YB lambda model in [44] and as It is noteworthy that these come out as constant despite that fact, as we will discuss later, the couplings ; ; run under RG. This is a feature of lambda models and was observed in the generalised gauged WZW models of Tseytlin [48]. This strongly suggests that, like isotropic lambda deformations, both of these can give rise to complete solutions of type II supergravity (i.e. de ne conformally invariant world sheet theories) when the theory is complemented by a similarly deformed non-compact SL(2) WZW together with Evidently since we have two functions of three variables one can force ~ XXZ and ~Y B 2 = Later we will see this relation arising form identifying the RG invariants of the two models. However, a more discerning comparison of ~XXZ and ~Y B can be made by recasting them in their common sigma model variables ( and ) making use of eq. (1.9). The result is striking: they do not match! This indicates that the XXZ and YB lambda theories are not completely equivalent. This may be surprising since the XXZ and YB sigma models di ered only by a gauge transformation of the NS two-form. Under a conventional Buscher T-dualization, one would expect this di erence to give rise to theories related by a combination of di eomorphism and gauge transformations after dualization. However the Sfetsos procedure we employed is not a dualization but instead a deformation and so there is no reason a priori to expect such a relationship to be the case. The exception is in the limit k ! 1, in which case the Sfetsos procedure reduces to non-abelian T-dualization; indeed, in this limit we nd that the two expressions coincide 1 2 ~YB ~XXZ 4 + O(k 1) : (3.18) One may recognise this as the being exactly the expected scalar curvature of the anistropic XXZ sigma model on the squashed sphere. 16To be precise ~ = stress tensor 2 hTaai = ~ R(2) + : : : and phism generated at leading order by the derivative of the dilaton [45{47]. 1 G 1 G appears as a coe cient of the expectation value of the trace of the 4 i are related to the beta-functions of couplings via a di eomor3.2 Both the XXZ and YB lambda models inherit the integrability of their mother sigma models. This can be shown by constructing Lax representations of their equations of motion. For the YB lambda model, the Lax connection was established in [44]. Let us rst de ne and functions of the spectral parameter z: (z) ! z z 2 1 ; In terms of the auxiliary gauge eld A , the Lax connection equals L (z) = ( (z) R)(1 R) 1A : The sigma model limit is obtained by restoring = k =(k other constants xed. In this limit we have + 1) and taking k ! 1 with A becomes identi ed with J and the Lax connection reduces to that of the YB sigma model (2.2). Having made this connection, in order to facilitate an easier comparison to the standard form Maillet algebra, it suits us henceforward to rede ne z ! 1=z for the YB lambda model. For the anisotropic XXZ lambda model, the Lax operator takes the form 3 X wa( a=1 and Note in the sigma model limit k ! 1, wa(z) and equivalents (2.3) and A becomes identi ed with J . reduce to their XXZ sigma model cosh2 = (1 )( + ) 2 ( 1 ) : 3.3 The Poisson brackets of the lambda models are inherited from the underlying WZW model where the Kac-Moody (KM) currents are J+ = J = k 2 k 2 F 1A+F A ; 1 F A F 1 + A+ ; and whose Poisson brackets take the form of two commuting classical KM algebras [49] J a(x); J b (y) = f abcJ c (y) (x y) J+a(x); J b (y) = 0 : 2 k ab 0(x y) ; In the present context, the f abc are the structure constants of the su(2) Lie algebra. In the YB lambda model, the spatial component of the Lax connection is written in terms of the Kac-Moody currents as [44] L (x; z) = (c+(z) + d(z)R) J+(x) + (c (z) + d(z)R) J (x) ; (3.26) (3.27) (3.28) (3.29) (3.30) (3.31) (3.32) HJEP09(217)35 where where This is the twist function quoted in [26] for the isotropic lambda model. d(z) = 2 k(1 2 k(1 2) ( 2) ( (z) +(z) + (z)) ; (z) 1) : The way to extract the Maillet form of the Poisson bracket of L (x; z) is to think of a change of variables on phase space from the KM currents J to the Lax operator L (z) and L (w), for a pair of generic points z and w. This yields precisely the form (2.11) with kernels b ; # b ; g(z; w) = d(z)d(w) + c (z)c (w) c (z)d(w) c (w)d(z) ; and either sign on the right-hand side can be taken. recover the simpler form (2.12) with a twist function There are two relevant limits to consider. The rst is ! 0, for which the r=s kernels (z) = k(1 2)(1 + ) 2 2 (1 1 )2 z 2 (1 + )2z2 : The other interesting limit, is the sigma model limit for which k ! 1, = k =(k +1), with and xed: T a + T a + b ; These kernels provide a di erent realization of the Poisson bracket algebra of the YB sigma model compared with [33] whose twist function we quoted in (2.13). For the anisotropic XXZ lambda model, the spatial component of the Lax connection is where where 3 a=1 L (x; z) = X fa(z)J+b (x) ga(z)J a(x) T a ; f1(z) = f2(z) = f3(z) = g3(z) = k(1 k(1 2 2 g1(z) = g2(z) = 2 2 1 k(1 s 2 2) 2) k(1 s 2 1 r 2 r 2 1 1 2) 2) 2 2 2 2 2 2 2 2 coth( + z) coth( z) ; coth( z) coth( + z) ; csch( + z) csch( z) ; csch( z) csch( + z) ; and one nds that the r=s kernels are b ; (3.33) (3.34) (3.35) (3.37) (3.38) and where the twist function is (z) = k(1 + )p1 2 p 2 2 2 2 + + (1 ) cosh(2z) (1 + ) cosh(2z) : The isotropic limit, involves taking and z 1 and one can verify that this gives (3.32). The sigma model limit yields (2.15). ! In this case, the RG ow appears to follow a cycle. However, the cycle passes outside the perturbative regime (small and ) and so it is not clear that the one-loop result can be trusted. Theories with RG limit cycles have been the subject of a lot of interest and there are several physical applications (see the review [57] and references therein). The relation between the reality of q and RG behaviour seems to be quite general: In the YB lambda model, the one loop beta functions are [44] The dilaton beta function eq. (3.16) is invariant under this mapping. We can use the RG invariant to eliminate to get a single equation for : ! 1 2 log (1 + 2) 3 1 + 2 2 : (k2( 1 )2 + 4 2)(k2( 1 )2 + 4 2 2 ) : d d = 1 4k 2 Integrating gives implicitly in terms of the RG scale 2 tan 1 k( 1 ) 2 + tan 1 k ( 2( 2 + 1)( 2 2 + 1) It is important to note that in the lambda model, the loop counting parameter is the inverse WZW model level k 1 and the beta function is exact at this order in k 1 as a function of the couplings and . One can readily verify that there is an RG invariant combination Once again we see that the quantum deformation parameter that we established at the classical level (3.43) is an RG invariant. In the YB lambda model we note an important duality symmetry on couplings and elds constant shift: ! 1 ; k ! k ; F ! F This leaves the worldsheet action eq. (3.3) classically invariant and extends the similar duality symmetry seen in the isotropic case. Under this transformation we note that the target space metric and two form are necessarily invariant however the dilaton receives a 4(1 + )2 1) + 4 2 2 k k2( + k log k2( 1 )2 + 4 2 1)2 + 4 2 2 = 4 log : (4.8) (4.9) (4.10) (4.11) (4.12) (4.13) 1 0 -1 -2 -2 -1 0 1 2 HJEP09(217)35 the blue dot in the middle. The red curved is an example of a cyclic trajectory which has a jump from = +1 to 1 at = 0 and a jump from = 1 to = +1. gure 4. Apart from discontinuities at in nity, the ow follows a cycle. The jumps are seen to be continuous in terms of the dual couplings in (4.10) and so we interpret the ows as following a physically continuous set of theories. In addition, the beta function (4.19) has a pole at = 1 but the ow is perfectly well de ned through it. For a cyclic RG ow, a key quantity is change in the energy scale as the ow goes around one cycle [51{53]. This follows easily from (4.13): around a cycle each of the arctan functions jump by and so around a complete cycle the energy scale changes by a factor ! exp[ ] : (4.14) Given the famous c-theorem of Zamolodchikov [58], the presence of RG cycles may come as some surprise since navely these seem to forbid the existence of a monotonic function along the ow. To assuage anxiety we note the couplings as functions of scale are multi-sheeted can this can allow for a (unbounded) monotonic function that jumps sheets as a cycle is traversed (see [59] for a toy model exhibiting this fact). One may further wonder about the robustness of these cycles as the one-loop RG equations are employed in domains where the couplings are not small; however one should keep in mind that the loop counting parameter k 1 does remain small. Nonetheless, further study is required to de nitively conclude the existence of such behaviour; it may be that the theory in this domain should be viewed only as an e ective theory with a cut-o that is necessarily encountered before an RG cycle can be completed. We will return to this in the next section and comment further about this possibility in the conclusion. ow in the XXZ lambda model. The RG ow of the two couplings follows from the general formula in [60]: (4.15) ) 2(1 (1 2)( + 1) ) 2 2)2 : ; 02 = k ) 2 : 0 2 ! 4 ( 1 ) ; Note in the sigma model limit, k ! 1 we get precisely the sigma model RG ow (4.1) when we use (3.8). ow in this case also has an invariant We have used the same notation 0 for the RG invariant here because in the sigma model limit, (3.8) with k ! 1, we have precisely, as it must be, the RG invariant of the sigma model (4.2). The XXZ lambda model also has a duality symmetry that takes ! 1 ; ! 1 ; k ! k ; F ! F The RG invariant is also invariant under this symmetry. These kinds of duality symmetries have previously been investigated in the context of current-current deformations of WZW models in [61, 62]. equation for , There are two distinct types of RG ow that depend on whether 0 is real or imaginary which are \UV safe" and \cyclic", respectively. The RG ows are shown in gure 5. We can use the RG invariant to solve for and substituting into (4.15), we can write a single d d = 1 4k (k2( 1 ) 2 4 02)(k2( 1 ) 2 ) 04(1 + )2 : (4.19) We will soon exploit the fact that this is identical with the RG ow equation for the YB lambda model (4.12) with 0 ! i . integral Se = R p Ge 2 ~ Since ~ [63, 64] can be thought of as a generalised central charge function (and its the central charge action) is natural to study its property along the RG ows. In principle one simply needs to substitute the solution of the RG equations into the expression (3.15). In practice given the implicit form for the solutions to 2.0 1.5 by the blue blob. The blue line is a line of UV xed points. The green curve is a UV safe trajectory that has 0 2 R. The red curve is a cyclic RG trajectory with 0 = i , 2 R. The trajectory has a jump in the coupling from 1 to 1, but is continuous in the dual coupling 1= . eq. (4.15) it is expedient to proceed numerically and study the evolution along for instance the green and red trajectories of gure 5. On the UV safe trajectory one nds ~ decreases monotonically except at one point (the saddle point in gure 5 where = 1, = 1 ) where ~ jumps from 1 to +1. Similarly on the UV cyclic red trajectory ~ decreases monotonically except at two points ( where = 1 and = 1 ). Being a function of cyclic functions of RG time in this case ~ returns to itself after a complete cycle. Thus with the exception of isolated points in which ~ is discontinuous, it is elsewhere monotonic. Although these points look rather innocuous in the RG ow | they are saddles in the ; plane | they are distinguished from the sigma model perspective as locations in which the determinant of the target space metric changes sign. The UV safe regime, corresponds to 0 2 R, so quantum group parameter q a complex phase. In this region, as the ow runs backwards towards the UV, goes to zero while goes to a constant that we denote which is determined by the RG invariant via These ows have a safe UV limit and in the UV, we can expand the couplings in powers of q = ( = ) , where = k k + 2 0 : = 4 k(1 + ) = 2 0 + k ; (4.20) (4.21) and | the lambda parameter | is the dynamically generated mass scale. The series are R = r 1 2k : p R2 + R2 : R2 The scalar eld determines the one of the components of the currents via The critical line emerges because the model remains critical as we change the radius. This corresponds to adding the term J+3J 3 to the action which is clearly equivalent to the coupling at = 0 for small . Adding J+1J 1 + J+2J 2 on top of this, gives an integrable massive deformation corresponding to turning on (for > 0). So in the UV limit, ! 0 and goes to a constant and one has Bernard and LeClair [66] identify the S-matrix of the perturbed theory, the so-called \fractional sine-Gordon" theory, as S -XXZ( ) = SRSOS( ; k) S( ; 0) ; (4.22) (4.24) (4.25) (4.26) (4.27) (4.28) where the second block is the sine-Gordon soliton S-matrix and the rst factor describes additional kink quantum numbers of the states. The sine-Gordon S-matrix with coupling is kR2 0 = R2 R2 = k(1 2 ) : This is exactly the RG invariant we de ned in (4.16) and explains our earlier notation. Now we turn to the regime of imaginary 0 = i , for 2 R. Note that the RG equation (4.19) with 0 ! i is precisely the same as the RG equation (4.12) in the YB = k=(2 )@ FF 1. = 1 + X n=1 nq2n ; = 1 X n=1 nq2n 1 : The points = varying and = 0 parametrize a line of UV xed points shown in blue in gure 5. For small couplings the action takes the form of a current-current perturbation of the WZW model,17 k SWZW[F ] d2x J+1J 1 + J+2J 2 + J+3J 3 : (4.23) The xed line corresponds to just turning on the J+3J 3 perturbation. It is known that the SU(2) WZW model does lie on a line of xed points. In order to see this, one uses the fact that the SU(2) WZW model at level k can be realized as a compact scalar on a circle of radius R coupled to Zk parafermions [65, 66]. The WZW point has the critical radius lambda model. This is signi cant and suggests that YB lambda model and XXZ in the cyclic regime are closely related. The solution for in terms of is he same as (4.13) and there is an RG cycle. A typical trajectory is shown in gure 5 in red. Just as in the YB lambda model, the trajectory follows a closed cycle which involves a jump from +1 to 1 in which is continuous in the dual coupling 1= . 5 In this section, we make informed conjectures for the S-matrices of the generalized lambda and sigma models. In order to pin down the S-matrix there are some important pieces of information to take into account: 1. The S-matrix of the isotropic lambda model associated to the PCM takes the form of a product of the rational, i.e. Y (su(2)) invariant S-matrix, and an a ne quantum group Uq(s[u(2)) RSOS kink S-matrix [19, 26]: In the limit, k ! 1 the RSOS factor becomes the rational limit of the unrestricted SSOS( ) which is itself the vertex-to-IRF transform of the SU(2) invariant S-matrix block. This manifests at the S-matrix level that the k ! 1 limit of the lambda model is the non-abelian T-dual of the PCM: (5.1) (5.2) -model 2. The XXZ sigma model with 0 2 R lies in the class of SS models of Fateev [18]. The S-matrix is then known to have the product form (1.13) where 0 is the RG invariant related to the UV limit of the coupling as in (4.16). 3. The YB lambda model breaks parity while the XXZ model preserves parity. 4. As described in sections 2 and 3, the classical sigma and lambda models have Poisson bracket realizations of the a ne quantum group Uq(s[u(2)) where q is related to the RG invariants as in (2.20), (3.43) and (3.62). 5. For the theories with cyclic RG ow with a periodicity e , it is expected that ! the S-matrices at high energy have a periodicity in rapidity to match [51]: S( + ) = S( ) ; 1) : (5.3) The intuition here is that in the UV at energy scales E m, the RG cycle behaviour requires that the theory has a discrete scaling symmetry E ! E exp( ). But for a particle state with E m, i.e. 1, we have E me =2 and so the scaling symmetry corresponds to a rapidity shift ! 5.1 Quantum group S-matrix: q complex phase Before making our S-matrix conjectures, there are some general features of S-matrix theory in the integrable context to take into account. S-matrices for relativistic integrable QFTs with degenerate particle multiplets are built out of solutions to the Yang-Baxter equation, for which quantum groups provide an algebraic framework. For present purposes, we will be interested in the quantum group deformation of the a ne (loop) Lie algebra Uq(s[u(2)). We start with the case when q is a complex phase in which case the S-matrix describes the scattering of solitons in the sine-Gordon theory [55]. The S-matrix in an must satisfy some important identities (described, for example, in the lectures [56]): 1. Factorization. Due to integrability, there is no particle production and the complete S-matrix is determined by the 2 ! 2 body S-matrix elements, as illustrated in gure 1. 2. Analyticity. The S-matrix is an analytic function of the complexi ed rapidity with poles along the imaginary axis 0 < Im associated to stable bound states. Since there is no particle creation in an integrable eld theory there are no particle thresholds, however, there can be anomalous thresholds in the form of additional, usually higher order, poles 3. Hermitian analyticity HJEP09(217)35 4. Unitarity 5. Crossing where C is the charge conjugation matrix. Unitarity is implied by Hermitian analyticity and the braiding relation which is more natural in the context of quantum groups. Sikjl( ) = Skijl( ) : X Sikjl( )Smkln( ) = im jn ; kl 2 R : Sikjl( ) = Ckk0 Sklj00i(i )Cj0j1 = Sklji(i ) ; X Sikjl( )Skml n( kl ) = im jn ; (5.4) (5.5) (5.6) (5.7) ei q)ei ej + ej ei i > j ; i = j ; i < j ; group Uq(s[u(2)) where, on a basis for V , e1 is a generator of the Hecke algebra (the commutant of the quantum group acting on tensor products) and obeys (T + q) (T q 1) = 0 : In (5.9), f ( ) is a scalar factor which is needed to ensure that the S-matrix satis es the S-matrix constraints of crossing and unitarity. Based on matrix form of R, there are four basic processes; identical particle, transmission and two kinds of re ection: In the present context, the basis states jmi transform in the spin 12 representation of su(2), or the quantum group Uq(su(2)), with m = . The 2-body S-matrix is a map, or 1 2 intertwiner, S( ) : V ( 1 ) V ( 2) ! V ( 2) V ( 1 ) ; where V ( ) is the vector space spanned by the states j ; i. Here, the rapidity of the states is i, and = 1 2 is the rapidity di erence. The S-matrix takes the form 1 2 where x( ) = ec , c to be determined, and R(x) is the R-matrix of the a ne quantum HJEP09(217)35 The braiding relation (5.7) is automatically satis ed because the Hecke algebra relation (5.12) implies (5.8) (5.9) (5.10) (5.11) (5.12) SI ( ) = ST ( ) = R S ( ) = 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 = f ( )(xq q 1 x 1) ; = f ( )(x x 1) ; (5.13) = f ( )x 1(q as long as the scalar factor obeys f ( )f ( ) = 1 (xq x 1q 1)(x 1q xq 1) : Unitarity then follows if the S-matrix is Hermitian analytic SI ( ) = SI ( ) ; ST ( ) = ST ( ) ; R R ) ; providing the scalar factor satis es Crossing symmetry requires that either f ( ) = f ( ) : x = q =(i ) or x = ( q) =(i ) : (5.15) (5.16) (5.17) (5.18) (5.19) (5.20) (5.21) (5.22) (5.23) HJEP09(217)35 It turns out that the resulting S-matrices are physically equivalent and so we choose the former. However, with this choice some extra factors of 1 appear in the crossing symmetry relation and charge conjugation operator; however, these are unobservable.18 Crossing symmetry implies where the charge conjugation operator acts as and there is a further constraint on the scalar factor: SI ( ) = ST (i ) ; R S+( ) = q 1 SR (i ) ; Cj 1 2 ; i = iq 1=2 1 2 ; i f ( ) = f (i ) : R R In addition, if the theory is parity symmetric then one has an additional constraint on the re ection amplitudes19 in (5.18), we have For the sine-Gordon theory, the S-matrix was originally constructed in the seminal work of Zamolodchikov and Zamolodchikov [55]. In this case, with the former choice q = exp[ i = 0] ; x( ) = exp[ = 0] : 18See the discussion in appendix C of [29] for details. 19Since parity ips the spatial coordinate, the ordering of particles is interchanged. Parity also ips momenta pi = m sinh i and so sends i ! i. However the rapidity in the S-matrix is the rapidity di erence of particles and so remains unchanged under the combined action of ipping the order and momenta of individual particles. clearly violating (5.19). formation on the states of the form [28{30] Hermitian analyticity can, however, be restored by a simple rapidity-dependent trans R R ) ; 1 ! x( ) 1=2 j 1 ; i : However, there is a problem: the S-matrix as written is not Hermitian analytic: the re ection amplitudes are non-compliant because they satisfy (5.24) (5.25) : (5.28) 12 i, in This transformation removes the factors of x 1 from the re ection amplitudes and restores Hermitian analyticity.20 It has an algebraic interpretation of moving from the homogeneous to the principal gradation of the a ne algebra s[u(2).21 The transmission and identical amplitudes are insensitive to this change whereas in the principal gradation the re ection amplitudes become R R S+( ) = S ( ) = f ( )(q q 1) ; (5.26) such that the resulting S-matrix now also describes a parity symmetric theory. Due to the change in gradation, the S-matrix is now invariant under the a ne quantum group in principal gradation Uq(s[u(2)p). The result S-matrix is precisely the S-matrix of the solitons of the sine-Gordon theory once we specify the scalar function f ( ). This is not determined uniquely by the conditions (5.15), (5.17) and (5.21). However, we can invoke the concept of minimality meaning that the solution has the minimal number of poles on the physical strip: 0 < Im < . The signi cance of this is that poles on the physical strip along the imaginary axis are usually interpreted in terms of bound states propagating in either the direct or crossed channels.22 The minimal expression can be written in various ways, for example as f ( ; 0) = 1 2 i 1 Y ( 2n0 + i 0 ) (1 + 2n 0 2 + i 0 ) ( 2n 0 1 n=1 ( 2n +01 + i 0 ) (1 + 2n 0 1 + i 0 ) ( 2n0 i 0 ) (1 + 2n 0 3 i 0 ) (1 + 2n 0 2 i ) i 0 ) 0 : (5.27) It is simple to show that this solves the conditions by computing its divisor. The other important condition that this expression satis es is the Hermitian analyticity condition (5.17). Another way to write the results that will be useful later is as the integral expression (valid for 0 > 2) f ( ; 0) = q 1 q 1 exp 2 0 w Z 1 dw cosh[ w( 0 2)=2] sin[w(i )=2] sin[w =2] 20To ensure crossing symmetry charge conjugation needs to be modi ed so that Cj j agreement with the original construction of [55] but with the additional factors of needed for the choice made for q explained in [29]. a(z) ! U a(z2)U 1 where U = z the transformation (5.25) on states. 21In more detail the change of grade can be achieved by a re-scaling and conjugation on the loop algebra ip2T 3 . Then with the identi cation z 2 = x, the conjugation is precisely 22There are also double poles which are explicable as anomalous thresholds. The sine-Gordon S-matrix also has an RSOS cousin, the restricted sine-Gordon Smatrix [28{30] which is associated to case when q is a root-of-unity. Details of this will emerge in section 5.3. Quantum group S-matrix: q real Given that the YB and XXZ in the regime have a quantum group parameter = ] that is real, implies that we also need an S-matrix that will be a close cousin of the sine-Gordon S-matrix but with this real value of q. On top of this, since the resulting theories have a cyclic RG behaviour, heuristic arguments suggest that the S-matrix should have a periodicity in real rapidity [51]:23 This periodicity requires more than a simple analytic continuation of couplings. Such an S-matrix was constructed in [51] built on the same quantum group R-matrix as the sine-Gordon S-matrix but now with real q. Crossing symmetry now requires that S( + ) = S( ) : x = exp[ i = ] : such that equal: The R-matrix now has a periodicity under shifts (more precisely up to ! some minus signs). This periodicity can be inherited by the S-matrix if the scalar factor is f ( + ; ) = f ( ; ) : The situation with Hermitian analyticity is di erent from the real 0 regime: both the S-matrix in the principal and homogeneous gradations are Hermitian analytic as long as the scalar factor satis es (5.17). In principle grade, the two refection amplitudes S While in homogeneous grade, the two refection amplitudes di er: R S ( ) SR( ) = f ( ; )(q R S ( ) = f ( ; )x 1(q To complete the construction we must specify the scalar factor. Note that simply taking the analytic continuation of the sine-Gordon scalar factor (5.27) from 0 ! i would not have the requisite periodicity (5.31) or satisfy the Hermitian analyticity constraint (5.17). On the contrary, the minimal solution to the constraints can be written as the convergent product [51] f ( ; ) = q Y1 (1 n=1 (1 q4nx 2)(1 q4nx2)(1 q4n+2x2) q4n 2x 2) : Note that this immediately satis es (5.17) and is manifestly periodic under ! . 23We follow the convention of [51] and allow the S-matrix to actually change up to some minus signs over a period. Note that S-matrices with a real periodicity in rapidity cannot have bound states but can have an in nite set of resonance poles [51, 54]. (5.29) (5.30) (5.31) R are (5.32) (5.33) (5.34) So there are two consistent S-matrices Sh( ; ) and Sp( ; ), associated to the homogeneous and principal gradations, respectively. It is important that the S-matrix that uses the homogeneous gradation of the a ne quantum group, breaks parity SR+( ) 6= SR ( ), whereas the principal gradation case preserves parity. The other important point to emphasize here is that when q is real, the S-matrix associated to the a ne quantum group Uq(s[u(2)) automatically has the periodicity in real rapidity that matches the heuristic proposal of [51] that theories with cyclic RG behaviour should have just such a periodicity at high centre-of-mass energy. But note that the Smatrix goes beyond this because it has the periodicity for any centre-of-mass energy. In order construct our S-matrices we will also need a piece to handle the kink quantum numbers of the states. This is precisely the RSOS kink S-matrix of the restricted sineGordon theory [28{30]. It is built out of a solution of the Yang-Baxter Equation, or more precisely the star-triangle relation, that plays the role of Boltzmann weights in an Interaction Round a Face (IFR) statistical model, e.g. see [67]. In the IRF S-matrix, the states are kinks Kab( ) and states are labelled by the vacua a; b on either side. The vacua (the local heights of the statistical model) are associated to representations of Uq(su(2)) so to spins a; b; : : : 2 f0; 21 ; 1; 32 ; : : :g. When q, the quantum group a parameter is a root of unit, q = exp i =(k + 2) ; (5.35) HJEP09(217)35 there is a restricted model, where the spins are restricted to lie in the set of integrable representations of level k, so a; b; : : : 2 f0; 12 ; 1; : : : ; k2 g. A basis of states in the Hilbert space with N kinks is labelled by a sequence faN+1; aN ; : : : ; a1g, which has the interpretation of a fusion path, so the spin aj+1 representation must appear in the tensor product of the aj representation with the spin 12 representation (truncated by the level restriction). This means that there is an adajency condition aj+1 = aj The analogue of the R-matrix, is an intertwiner W between 2-kink states [67]: X W a b jKab( 1 )Kbc( 2)i c u jKad( 2)Kdc( 1 )i ; (5.36) where u = =(i ) and = 1 2. These intertwiners satisfy the star triangle relation [67]. The solution of the star triangle relation W (u) is the raw fodder from which one fashions the RSOS S-matrix for kinks states. There are 3 basic types of non-vanishing elements that take the form W Ba 1 a uC = W Ba a uC = 0 a a a 1 2 1 2 1 2 1 2 1 A 1 A [1] u] [1 [u] [1] p[2a + 2][2a] [2a + 1] ; W Ba 0 a a uC A = [ (2a + 1) + u] [ (2a + 1)] ; (5.37) where we have de ned The W intertwiner satis es some identities that are important for the S-matrix that we going to build [67]: (i) the initial condition u=(k + 2) : ! W c 0 = bd ; e (ii) rotational symmetry and (iii) inversion relation W c 1 u s [2b + 1][2d + 1] c b u X d W c u W c u [1 u][1 + u] [1]2 be : The alert reader will recognize that the rotational symmetry and inversion relation as proto-identities for crossing symmetry and braiding unitarity, respectively. When k is generic (i.e. not an integer), the local heights a; b; : : : are valued in 12 Z and the Boltzmann weights W (u) de ne the SOS statistical model. However, when k is an integer there is consistent restriction of the local heights to the nite set f0; 12 ; 1; : : : ; k2 g. The restriction is consistent because [0] = [k + 2] = 0 so consequently W (u) cannot propagate a kink state with admissible local heights jKab( 1 )Kbc( 2)i with a; b; c 2 f0; 12 ; 1; : : : ; k2 g into one with an inadmissible local height jKad( 2)Kdc( 1 )i with d 62 f0; 12 ; 1; : : : ; k2 g, in practice d = 0 or k2 + 1, due to the adjacency condition. This is guaranteed if [0] = [k + 2] = 0. In order to make a consistent S-matrix, b SRSOS( ; k) = v( )W c u( ) ; (5.42) one has to construct a suitable scalar factor v( ) in order that the S-matrix is unitary and crossing symmetric. The scalar factor must satisfy v( ) = v(i ) ; v( )v( ) = sin2( =(k + 2)) sin(( + i )=(k + 2)) sin(( i )=(k + 2)) : (5.43) One can readily verify that the solution to these conditions can be expressed in terms of the usual sine-Gordon scalar factor in (5.28) with 0 = k + 2, up to a constant factor: v( ; k) = (q q 1)f ( ; k + 2) : (5.44) where f ( ; 0) is de ned in (5.27). (5.38) (5.39) (5.40) (5.41) HJEP09(217)35 The RSOS kink S-matrix has a good limit k ! 1, the SOS limit, as long as the local heights are suitably shifted, a ! k4 + a, etc, before the limit is taken. So the idea is that one takes the local heights well away from the end points a = 0 and a = k2 as k ! 1. In that limit, one can easily verify that the S-matrix becomes identical to the rational SU(2) S-matrix with a simple mapping between the kinks of the SOS picture and states of the spin 12 representation: Ka+ 12 :a( ) ! j "; i ; K a 12 ;a( ) ! j #; i : (5.45) This is an IRF-to-vertex transformation which relies on the fact that the N -kink Hilbert space of unrestricted paths of length N faN+1; aN ; : : : ; a1g, is isomorphic to the N spin 12 particle Hilbert space for a xed a1; e.g. 3 2 3 2 1 2 a + 1; a + ; a + 1; a + ; a + 1; a + ; a ! j #"#"""i ; (5.46) etc. Finally, we can compare our S-matrices by writing down an integral representations of the identical particle amplitude, which for the RSOS case means jKa 1;a 12 ( 1 )Ka 12 ;a( 2)i ! jKa 1;a 12 ( 2)Ka 12 ;a( 1 )i : Note that this particular amplitude does not depend on the right vacuum a. For the q a complex phase | the sine-Gordon case | we have For the case q real, the S-matrix of [51], we have SI ( ; 0) = exp i Z 1 dw sin[w ] sinh[ w( 0 1)=2] 0 w SI ( ; ) = exp i = + i X1 2 n=1 sin[2n = ] n 1 + exp[2 n= ] : : SI;RSOS( ; ; k) = exp i Z 1 dw 0 sin[w ] sinh[ w(k + 1)=2] w cosh[ w=2] sinh[ w(k + 2)=2] ; (5.50) Finally for the RSOS case just constructed which is simply (5.48) with 0 ! k + 2. 5.4 High energy limit The nal information we will need when we establish our S-matrix conjectures is the high centre-of-mass energy limit of the trigonometric S-matrices. This is just the large rapidity limit, i.e. the limit of large x de ned in (5.18) . In order to take the limit, we focus on the identical particle amplitude SI which can be written (by rearranging the arguments of the gamma functions in (5.27)) as SI ( ; 0) = Y 1 i + 2j 0) ( 2 2 i + 1 + j 1 0) 2 ( 2i + 12 + 2j 0) ( 2i + 12 + j 2 1 0) ( i + 12 + 2j 0) ( 2 2i + 12 + j 2 1 0) : (5.47) (5.48) (5.49) (5.51) 0 ) -10 -5 0 5 10 value of 0. The key feature is that for large the amplitude saturates. Note that this amplitude is also valid in the RSOS version of the S-matrix with 0 ! k + 2. The amplitude is a phase which we plot in gure 6. The important point is that for large enough the amplitude saturates. In order calculate the asymptotic value we simply apply Stirling's formula to the expression above: i log SI ( ; 0) ! 4 = 2 ( 0 1 1) X 1 j=1 (2 0j)2 + ( =(2 ))2 + 0) + This means that the while S-matrix has a very simple limit proportional to the Hecke algebra generator S( ; 0) 1 ! ei (1+ 0)=2T 1 : (5.52) (5.53) Note that the RSOS kink S-matrix also has such a universal high energy limit, where now T 1 is realized in the kink Hilbert space. 5.5 The S-matrix proposals In this section, based on all the information and constraints, we make our proposals for the S-matrices of the lambda and sigma models. We begin with the XXZ lambda model in the regime with 0 2 R, i.e. the quantum group parameter a complex phase. Our proposal is that the S-matrix in this regime, is precisely the fractional sine-Gordon S-matrix (4.27) proposed by Bernard and LeClair [66]. The theory in this regime has a pair of a ne quantum group symmetries with q = exp[ i = 0], for the sine-Gordon factor, and q = exp[ i =(k + 2)] for the RSOS factor. The S-matrices for the sigma model follows in the limit k ! 1 and a non-abelian Tduality which has the e ect of replacing the RSOS S-matrix piece with the rational SU(2) S-matrix as shown in (5.2) and one recovered the S-matrix of the anisotropic XXZ sigma model in (1.13). Now we turn to the YB lambda model and the XXZ model in the regime 0 = i , i.e. where the quantum group parameter q = exp[ = ] is real. In these case the RG ows are cyclic. This suggest that the S-matrices are based on the pieces Sh( ; ) and Sp( ; ) constructed in section 5.2. There is also a natural explanation for the existence of the two distinct S-matrices based on the gradation because the YB lambda model is not parity symmetric and this matches the S-matrix for the homogeneous gradation. Correspondingly the principal gradation S-matrix is parity preserving as is the XXZ model. To make a complete S-matrix we need to consider an appropriate RSOS kink S-matrix factor. The only choice consistent with the sigma model and the classical symmetries is the RSOS S-matrix piece SRSOS( ; k). However, this S-matrix does not have the periodicity . The resolution is here is that the periodicity is only expected to appear in the limit of large centre-of-mass energy and we have shown in section 5.4 that the trigonometric S-matrix become constant at high energy. So the heuristic requirement that the S-matrices of theories with cyclic RG behaviour should have a periodicity in rapidity at high energy is actually satis ed. Hence, we make our conjectures; for the YB lambda model while for the XXZ lambda model S -YB( ) = SRSOS( ; k) Sh( ; ) ; S -XXZ( ) = SRSOS( ; k) Sp( ; ) : The sigma model limit, involves taking k ! 1 along with an IRF-to-vertex transformation, S -YB( ) = SSU(2)L ( ) Sh( ; ) ; S -XXZ( ) = SSU(2)L ( ; ) Sp( ; ) ; respectively. These S-matrices exhibit the Yangian Y (su(2)) symmetry and also have the periodicity in rapidity at high energy. 6 Discussion In this work we have considered the deformations of the SU(2) PCM that preserve integrability. The class of deformations focused on, preserved an SU(2) symmetry and so there are associated lambda models. We showed that the lambda models also have a ne quantum group symmetries realized at the classical Poison bracket level. The are many questions remaining. In particular, for the YB deformations and anisotropic ones with for the associated lambda model), the RG ow follows a cycle in coupling constant space. So these theories have a mass gap but no xed point in the UV to de ne a continuum limit. So the main question is: is the UV of these theories well de ned? There are two pieces of evidence to suggest that these theories actually are only de ned with an explicit UV cut o of the order of the mass scale of the particle states. (5.54) (5.55) (5.56) The rst, described in [51] for the case k = 1, comes from de ning the QFT as the continuum limit of a spin chain. The anisotropic XXZlambda models with < , so with q in (3.62) a complex phase, can be regularized by the XXZ Heisenberg spin chain [35] with spins of angular momentum j = k2 and with spin chain anisotropy = cos k + 0 (6.1) where 0 is the RG invariant (4.16). The spin chain in this regime is critical and consequently it is possible to take a continuum limit. The physical excitations and their Smatrix agree precisely with our conjectured S-matrix, the fractional sine-Gordon S-matrix in (4.27). Now if we try a similar spin chain description of the case, then the XXZ spin chain now lies in the 1 regime. In this regime the spin chain has a mass gap and so there is no way to take a continuum limit. Even so, we shall show in [35], that the excitations have an S-matrix that is a close relative of the S-matrix (5.55) It is possible to create a hierarchy between the inverse lattice spacing and excitation mass only in the limit of large . So this suggests that the RG cycle is never actually traversed in the UV before the UV cut o is reached. The second piece of evidence, again for the case k = 1 for the anisotropic XXZ lambda model in the cyclic RG regime, is presented in [52]. The idea is to use nite size e ects to compute the e ective central charge. It is shown that for the case when the theory has a mass gap, the relevant case here, the nite-size e ects do indeed have a periodic behaviour consistent with the beta function analysis but in the deep UV the nite-size central charge either has a singularity or is ill de ned in the very deep UV. Again this suggests that in the cyclic RG regime, the theories only make sense with an explicit UV cut o . The other issue which is interesting to consider is how these issues play out in larger groups. We have already pointed out that the anisotropic models are special to SU(2) and they do not appear to admit generalizations to an arbitrary Lie group. However, the YangBaxter deformation do lift to an arbitrary group and one can speculate that the sigma and lambda models once again have a cyclic RG behaviour. We show this is the case in [35]. We go on to show that there is a natural conjecture for the S-matrix which is rather novel. For the case SU(N ), it is related to the S-matrix constructed in [68] but like the S-matrix S( ; ) considered here is periodic in rapidity. What is novel about the resulting S-matrix is that it exhibits an in nite set of unstable resonance poles thus providing an example of the \Russian Doll" phenomena described in [52]. Unlike the SU(2) example described there, the S-matrix we construct satis es all the S-matrix axioms including hermitian analyticity. Finally there is a generalization of the anisotropic models that we have mentioned in the introduction, namely the XYZ model. The lambda model of this should have an S-matrix that is related to the elliptic S-matrix of Zamolodchikov [69]. Acknowledgments CA and DP are supported by STFC studentships. TJH is supported in part by the STFC grant ST/L000369/1. DCT is supported by a Royal Society University Research Fellowship Generalised Dualities in String Theory and Holography UF 150185. We would like to thank Saskia Demulder, Kostas Sfetsos, Graham Shore, Kostas Siampos and Benoit Vicedo for useful discussions and Arkady Tseytlin for interesting correspondence. A Lambda spacetimes The lambda theories can viewed as sigma models with target spaces of the following form A0 = c( A2 = c( A3 = c( A4 = A5 = 2c2( +2 1)2( + 1)S2 1)2( + 1)S2 S2 2 2( 1 ) A0 = c ( + 1) (C2 ( A1 = c 2C2 ( 1 )C2 ( A2 = c( A3 = c( A4 = c( A5 = 1)S2 (C2 ( 1 ) 1)( 1 S2 S2 )S2 S2 4c2( S 1)S2 C2( (5 ( + 1)( 3 ) + (3 ) + ) + 2C2 2 4 2 3 2 2 + 1 8 2(( 2) + 4 + 2 2 2 2 + 2 + 1 S3 S For the XXZ lambda model we nd (with c = 1 ): 1) + 2C2 ( + 1)C2 ( ( + 1)( + ) + ( + 1)( + 1)2 5 ) ) + ( + 1) 1) 2 1 + 3 + 1) )S3 S 2 This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. k A0 A5 0 ^ d ; ds2 = H3 = k A2 d ^ d log A0 = Ai = Ai( ; ) : The non-trivial dilaton is produced as a result of the determinant in the path integral arising from performing the Gaussian integration on the non-propagating gauge elds. 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Calan Appadu, Timothy J. Hollowood, Dafydd Price, Daniel C. Thompson. Yang Baxter and anisotropic sigma and lambda models, cyclic RG and exact S-matrices, Journal of High Energy Physics, 2017, 35, DOI: 10.1007/JHEP09(2017)035