Classical and quantum aspects of Yang-Baxter Wess-Zumino models

Journal of High Energy Physics, Mar 2018

Abstract We investigate the integrable Yang-Baxter deformation of the 2d Principal Chiral Model with a Wess-Zumino term. For arbitrary groups, the one-loop β-functions are calculated and display a surprising connection between classical and quantum physics: the classical integrability condition is necessary to prevent new couplings being generated by renormalisation. We show these theories admit an elegant realisation of Poisson-Lie T-duality acting as a simple inversion of coupling constants. The self-dual point corresponds to the Wess-Zumino-Witten model and is the IR fixed point under RG. We address the possibility of having supersymmetric extensions of these models showing that extended supersymmetry is not possible in general.

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Classical and quantum aspects of Yang-Baxter Wess-Zumino models

HJE Classical and quantum aspects of Yang-Baxter Saskia Demulder 0 1 2 5 6 Sibylle Driezen 0 1 2 5 6 Alexander Sevrin 0 1 2 3 5 6 Daniel C. Thompson 0 1 2 4 5 6 0 Singleton Park , Swansea SA2 8PP , U.K 1 Campus Groenenborger , 2020 Antwerpen , Belgium 2 Pleinlaan 2 , B-1050 Brussels , Belgium 3 Physics Department , Universiteit Antwerpen 4 Department of Physics, Swansea University 5 Theoretische Natuurkunde, Vrije Universiteit Brussel & The International Solvay Institutes 6 xed point under RG. We address the We investigate the integrable Yang-Baxter deformation of the 2d Principal Chiral Model with a Wess-Zumino term. For arbitrary groups, the one-loop are calculated and display a surprising connection between classical and quantum physics: the classical integrability condition is necessary to prevent new couplings being generated by renormalisation. We show these theories admit an elegant realisation of Poisson-Lie Tduality acting as a simple inversion of coupling constants. The self-dual point corresponds to the Wess-Zumino-Witten model and is the IR possibility of having supersymmetric extensions of these models showing that extended supersymmetry is not possible in general. Integrable Field Theories; Sigma Models; String Duality 1 Introduction 2 3 4 5 6 Yang-Baxter and Yang-Baxter Wess-Zumino models Renormalisation of the YB-WZ model 3.1 3.2 Case I: general group G and restriction to the integrable locus Case II: simply laced groups and general parameters Poisson-Lie T-duality of the YB-WZ model The supersymmetric YB-WZ model Summary, conclusions and outlook A Conventions B Charges in SU(2) C Properties of R 1 Introduction D Geometry in the non-orthonormal frame to transcend the usual perturbative tool kit. A rather long standing question has been to establish the complete landscape of integrable sigma models. A substantial breakthrough was made by Klimcik with the explicit demonstration that the Yang-Baxter sigma models [1] are integrable [2]; thereby providing a one-parameter integrable deformation of the principal chiral theory associated to any semi-simple Lie algebra. These theories, now often called -deformations, have taken great prominence since they provide a Lagrangian description of a theory whose symmetry is deformed to a quantum group [3]. When extended to theories on symmetric spaces and to super-cosets, this has yielded a remarkable quantum group deformation of the AdS5 opening the door to an intriguing interpretation within holography. S5 superstring [4] { 1 { A surprising feature of the -deformed theory in the context of the AdS5 string is that it appears to describe a scale invariant but not Weyl invariant theory. This is seen directly by the target spacetime's failure to satisfy the equations of Type IIB supergravity but instead to obey a set of \generalised" supergravity equations [5]. Recent work has started to place these -theories, and the generalised supergravity that govern their target spacetimes, in the context of double/exceptional eld theory [6, 7] and make explicit the link to T-folds and non-geometric con gurations [8]. A link between the r-matrix, satisfying a (modi ed) classical Yang-Baxter equation, that de nes Yang-Baxter sigma models and the spacetime non-commutativity parameter has been developed in [9, 10] using the open-closed map. Notably, the -theory displays a so-called Poisson-Lie (PL) symmetry. This means that it possesses a generalised T-dual in the Poisson-Lie sense proposed by Klimcik and Severa [11]. The Poisson-Lie dual model, modulo an analytic continuation, has been established to be a well-known integrable deformation called the -deformation. Introduced by Sfetsos [12] these theories interpolate between a Wess-Zumino-Witten (WZW) [13] or a gauged WZW model and the non-abelian T-dual of the principal chiral model on a group manifold or symmetric coset space respectively. The connection between the - and the -theories was rst shown for explicit SU(2) based examples [14, 15] and established in generality by [16, 17]. Like the -theories, -models can also be applied to cosets [18] and semi-symmetric spaces [19] and are thought to encapsulate quantum group deformations with q a root of unity. In contrast to the -theory, the target spacetimes associated to the -model provide genuine solutions of supergravity (with no modi cation) [20{24]. Given these successes a natural recent focus has been to understand potential generalisations of these approaches to include multi-parameter families of integrable models. On the side of the -deformation (or Yang-Baxter model) notable are the two-parameter bi-YangBaxter deformations [25], the inclusion of a Wess-Zumino term [26] and indeed the recent synthesis of these [27]. On the side, multi-parameter deformations have been constructed and studied in [15, 28{30]. There is also some evidence that a Poisson-Lie connection should be present between multi-parameter - and -models; for example the bi-Yang Baxter model has been shown to be related to a generalised -model [31]. The Yang-Baxter theory with a WZ term (YB-WZ) appears amenable to similar treatment since it can be written as an E -model [32] (though the corresponding theory is not clearly spelt out as yet). The construction of Lax pairs directly from the E -model has recently been studied in [33]. In this work we will provide further study of the multi-parameter YB-WZ model. For the case of SU(2) this system was studied in [34, 35]. Speci cally we shall, Study the one-loop renormalisation of the general YB+WZ model extending results in the literature from SU(2) [34] to arbitrary groups. We will nd that the conditions placed on a sigma model by integrability have an interesting interplay with renormalisation. The condition required of classical integrability is preserved by RG ow. Second, when dealing with non-simply laced algebras one nds the classical integrability condition is necessary for the renormalisation of the model not to introduce new { 2 { HJEP03(218)4 couplings in addition to those of the bare theory. That a classical property seems to be so tied to a very quantum calculation is notable. We will clarify some details of the quantum group symmetries in these models and in particular show that the parameters de ning the symmetry algebra are invariants of the RG ow. We comment on the role of Poisson-Lie dualisation for the YB-WZ model. Considered within the framework of the E -model [32], the YB-WZ can be seen as being part of a pair of Poisson-Lie dual models. In particular, it admits a formulation as an E -model associated to the Drinfeld double d = gC. When the integrability condition is satis ed, the Poisson-Lie T-duality transformation preserves the structure of the action (2.2) while the coupling parameters follow very simple \radial inversion" transformation rules. We will examine the possible worldsheet supersymmetrisation of the YB-WZ model associated to SU(2) U(1) which is the simplest but non-trivial example that allows N = (2; 2) in the undeformed (WZW) case. While N = (1; 1) supersymmetry is always possible, going beyond that requires the introduction of additional geometric structures. We show that N = (2; 2) is forbidden for generic values of the deformation parameters while N = (2; 0) or N = (2; 1) is possible only for speci c values. This leads us to conjecture that an N = (2; 2) YB-WZ model is not possible in general. The paper is organised as follows. Section 2 introduces the Yang-Baxter Wess-Zumino model together with its integrability properties relevant to the subsequent discussions. In section 3 we give an explicit derivation of the one-loop -functions of the YB-WZ model in the case of arbitrary groups. Given the result, we nd that one needs to carefully distinguish between two cases: when the group is simply-laced or not. In the former case, a consistent renormalisation does not require the model to be integrable. For the latter case, the classical integrability condition turns out to be necessary to prevent the creation of new couplings in the theory by renormalisation. A detailed discussion of the RG behaviour is given in both cases. Section 4 formulates the YB-WZ action (2.2) within the framework of the E -model and derives the Poisson-Lie T-dual model. In section 5 we study the possibility of extended supersymmetry of the YB-WZ model. We end with a summary and conclusions in section 6. The conventions used throughout this paper are given in appendix A. Appendix B reviews the construction [35] of the charges of the SU(2) YB-WZ model paying particular care to the overall normalisations required to expose the correct RG properties. In appendix C and D we collate a set of useful expressions which were used in the calculations of the -functions. 2 Yang-Baxter and Yang-Baxter Wess-Zumino models In this rst section, we present the Yang-Baxter Wess-Zumino model (YB-WZ) as constructed in [26], which will be the main topic of the remainder of this paper. Given a Lie algebra g, we introduce an endomorphism R : g ! g skew symmetric with respect to { 3 { hx; Ryi) and obeying the modi ed classical Yang-Baxter (mCYBE) equation, R. The canonical realization of R is most easily seen in a Cartan-Weyl basis for the Lie algebra where it maps generators belonging to the CSA to zero and where it acts diagonally on generators corresponding to positive (negative) roots with eigenvalue +i ( i). Equipped with this structure, we de ne the YB-WZ action in worldsheet light-cone coordinates as, + R + R i h g 1dg; [g 1dg; g 1dg]i : Here as usual the coe cient of the Wess-Zumino term, k, is an integer, quantised such that the path integral based on this action is insensitive to the choice of the extension A short calculation yields, after integration by parts and discarding the total derivative, S = 2 K = 1 Z ( k) Rg + Rg2 v ; in which we recall v = dgg 1 are the right invariant one-forms and, Rg = adg R adg 1 ; which, like R, obeys the mCYBE and is skew symmetric with respect to the ad-invariant Cartan-Killing form h ; i. Using the inverse of eq. (2.4), v = ( ) 1 k + ( 2 + ( ( 2 k)( 2 + ( k k k ) )2 Rg 2 )2) Rg K ; in dv v ^ v = 0, one easily gets, if and only if the coe cients are related via [26], 2 = 2 k 2 : { 4 { (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) HJEP03(218)4 So we conclude that the currents K are on-shell at provided eq. (2.8) holds. This is su cient to guarantee classical integrability as the equations of motion follow then from the atness of the standard gC-valued Zakharov-Mikhailov Lax connection [42], L (z) = 1 1 z K : 2 [jkj ; 1[ and nd the WZW point [13]: = 2 h0 ; We call the solutions to eq. (2.8) the integrable locus.1 From eqs. (2.7) and (2.4) one deduces the further conditions 6= 0 and 6 = , ensuring that the kinetic term is properly de ned. In addition, as all parameters , , , k are real and the kinetic term should have the right sign ( > 0), we conclude from eq. (2.8) that the allowed values of 2 ]0 ; jkj] and 2 h k2 2 i ; 0 , where on and are = jkj we For the particular subset of the integrable locus given by [1], the action eq. (2.2) reduces to what has become known as the -deformed principal chiral model which is integrable [2] with the dynamics encoded in the atness of a gC-valued Lax connection L (z) depending on a spectral parameter z 2 C. This theory displays a fascinating structure of in nite symmetries [3]. At the Lagrangian level the left acting G symmetry is preserved and is complemented, as in the undeformed principal chiral model, with nonlocal charges furnishing a Yangian Y(g). The right acting G symmetry is broken to its Cartan in the action eq. (2.2), but is enhanced by non-local charges to form a classical version of for a given simple root there exists a local charge QH and non-local charges Q a quantum group Uq(g) [3] (actually further extended to an a ne Uq(g^) [36]). Schematically, that obey, under the renormalisation group ow of couplings [15]. The charges that generate these symmetries can be obtained by expansions around suitable values of the spectral parameter of the monodromy matrix, Z U (z) = P exp d L (z) ; which is conserved by virtue of the atness of L . The Yangian left acting symmetries are found through expansions around z = 1 whereas the right acting quantum group 1To translate to [26] we have the dictionary of parameters ( ; ; ; k) ! ( 2; R; k0; K) A = ; 2 = k ; k0 = ; K = 4 ; however we shall continue with the ( ; ; ; k) such that k gives the level of the WZW model that will appear at IR xed points. { 5 { (2.9) (2.11) (2.12) (2.13) (2.10) symmetries are found [3] via the expansion of the gauge transformed Lax around special points corresponding to poles in the twist function of the Maillet algebra [37]. Much of the story for the general -deformed model was rst established for the case of g = su(2) which corresponds to the sigma model on a squashed S3 (the Kalb-Ramond potential encoded by eq. (2.2) is pure gauge in this case and though it doesn't e ect the equation of motions it corresponds to an improvement term to ensuring atness of currents). The integrability was established many years ago by Cherednik [38]. Somewhat later the classical Yangian symmetry was shown in [39] and the (a ne) quantum group symmetry in [40, 41].2 Now we turn to the case where k 6= 0 which is the main focus of this paper. Again historically this was rst well explored for the case of g = su(2). The left acting symmetry is still a Yangian [34] but the right acting symmetry is more mysterious [35] (we review the construction of the charges generating these generalised symmetries in appendix B). One nds a structure similar to an a ne quantum group Uq(s[u(2)) with,3 but with a modi cation in how the a ne tower of charges is build up. Namely, instead of taking successively the Poisson bracket to access the next charges in the tower, the Poisson bracket is multiplied at each step by an additional factor, To move down, the Poisson bracket needs to be multiplied by its inverse (see gure 2 of [35] for further details). Here the combination, q = exp 8 model in the particular case where g = su(2). In this paper we mostly focus on the case where g is arbitrary. As we will see the general case shows several features which are absent when g = su(2).4 3 Renormalisation of the YB-WZ model Our aim is to calculate the -functions for the couplings f ; ; g in the theory de ned by eq. (2.2) without rst assuming that the couplings lie on the integrable locus eq. (2.8). 2There is a small but potentially important subtlety here. In [41] the a ne charges are constructed from the expansion of a trigonometric Lax at in nity and appear in the principal gradation. When the charges are extracted from the gauge transformation of the rational Lax evaluated around the poles in the twist function as in [36] they appear in the homogeneous gradation; to go between the two gradations requires a spectral parameter dependent rede nition of generators. 3Here we restore the overall normalisations to the results in [35] and map to our conventions. 4Mathematically all di erences between the general case and the simpler case where g = su(2) arise from the fact that su(2) is the only simple Lie algebra where all roots are simple roots. { 6 { The coupling k being integer quantised evidently does not run. To do so we will proceed geometrically; for a general two-dimensional non-linear sigma model the -function for the metric G at one-loop, and Kalb-Ramond two-form potential B in local coordinates x are given by, d d d d G B = ^G = 0 R = ^B = 0 1 4 1 2 r H H2 + O( 0)2 ; + O( 0)2 ; G = ^G + r( W ) ; B = ^B + ( W H) + (d ) ; where H = dB is the torsion 3-form and the connections and curvatures are to be calculated using G. However, the di eomorphism and gauge covariance of G and B means that these -functions are ambiguous (even at one-loop order) [43, 44] allowing us to modify them by,5 HJEP03(218)4 with W and arbitrary target space one-forms. For the sigma model de ned in eq. (2.2), of which the left acting G symmetry is unaltered by the deformation, the target space data is most naturally expressed in a non-orthonormal frame formalism with frames de ned by the left-invariant one-forms u = g 1dg = iuATA as, GAB = AB + R2AB ; BATB and algebra indices out of position are lowered AB. To completely x things one should set 0 = 2 so that the standard normalisation of the WZW models is recovered in the case sions of the geometry in the non-orthonormal frame listed in appendix D one nds that (3.1) (3.2) (3.3) (3.4) 2 RAB (3.5) (3.6) the -functions are given by, can be discarded within the action. The terms in blue involve tensor structures that are not present in the metric ansatz. If these terms are not removed it would mean that under the RG ow the metric would ow out of the ansatz speci ed by eq. (3.3). Let us exploit the di eomorphism symmetry to 5Note that the Lie derivative acts on LW B = W H + d W B and the latter term being a total derivative )2 2( )2 1 2 AB cG 2 1 ( 2 + 2 )2 FADC FBC E (R2)DE ; ) 2 RAB : { 7 { try and ameliorate the situation. With this in mind, note that for a one-form W whose components WA are constant in frame indices we have: r(A W B) = 1 2 FADC (R2)DB + FBDC (R2)DA WC ; (iW H)AB = 3 F[AB D R C]DGCEWE kFABC GCDWD : First, we try to use an appropriate choice of W to remove the o ending blue term in ^G. However, using the properties listed in appendix C, one can show that the only sensible choice of G 1W involving the structure constants and the R-matrix will always be Killing. Nevertheless, by taking the components WA proportional to FABC BC , one can show that it is again Killing but can now in fact absorb the o ending rst term in ^B. Finally, we R remark that for the case of g = su(2) (cG = 4 in our conventions) the contribution of the parameter cancels exactly in ^G and can be gauged away by an appropriate gauge choice in B eq. (3.2) since RABuA ^ uB is a pure gauge improvement term for su(2). We now consider the remaining o ending term in ^G eq. (3.5). Using a Cartan-Weyl basis for the Lie algebra and calling Lie algebra indices corresponding to positive (negative) roots as a, b, . . . (a, b, . . . ) and those corresponding to directions in the CSA by m, n, . . . one gets, FADC FBC E(R2)DE = cG AB + FAmC FBC m : The second term is non-vanishing only if the index A corresponds to a positive root and the index B to the corresponding negative root (or vice-versa) so one would expect it to be proportional to R2AB. Explicit computation gives, FamC FaC m = aa ~a ~a = ~a ~a R2aa ; (3.7) (3.8) (3.9) (3.10) where ~a ~a is the length squared of the root a. In our normalization it is always equal to 2 for simply laced groups (g = An, Dn, E6, E7 and E8). For the non-simply laced groups its either 2 or 1 (for g = Bn, Cn and F4) or 2 or 1/3 (for g = G2). So the term in blue in eq. (3.5) can be rewritten as, FADC FBC E(R2)DE = cG AB + 2 x(A; B) R2AB ; where for simply laced groups x(A; B) = 1 holds. For non-simply laced groups x(A; B) assumes two di erent values pending the values of the indices A and B. This implies that only for simply laced groups the RG stays within the ansatz speci ed by eq. (3.3). However, there is a second way to remain within the ansatz eq. (3.3). Till now we did not impose any restriction on the parameters , , and k. Looking at the bothersome term in the last line of eq. (3.5) we see that it precisely vanishes at the integrable locus eq. (2.8) and we remain within the ansatz eq. (3.3) for any group (simply laced and nonsimply laced)! So we should distinguish two cases: case I, a restriction to the integrable locus for general groups, and case II, a restriction to simply laced groups where we can keep the parameters general. { 8 { trace of the stress tensor, hT i = with D the dimension of g. The quantity ~ , which one recognises in the spacetime e ective Lagrangian for bosonic strings, can serve as a c-function for the models we are considering [45].6 Here we nd for arbitrary groups in general, = D 6 0 1 4 2 H2 + O( 0)2 : (3.11) Before analysing both cases, we will consider a useful quantity to understand the RG ow: the Weyl anomaly coe cient ~ . It is de ned through the expectation value of the ~ = D 6 + cGD 8 where l is the rank of g. Focusing on the particular case of the integrable locus (i.e. case I) this equation reduces to, Whilst perhaps not so elegant, after applying the RG equations for this case we have, ( 2 + 2)( 4 + k2( 2 d ~ dt D 6 = ~ 1 ( + cGD 24 c2GD k 2 6In general one would need to average, i.e. integrate this over spacetime coordinates but the special form of the metric on a group manifold means that is not needed here. { 9 { : of 16 times the central charge c = kkd+imhvG ). Notice that because 3 4 + 3 4 2 2 2 has no real roots for 2 explicitly see the monotonicity of the ow dt ~ required ~ jUV > ~ jIR. The IR is no more than the WZW CFT at Dhv + O( k1 )2 in accordance with the large level expansion 6k > 0 with t ! 1 in the UV giving as 2 R and 2 2 R we 3.1 Case I: general group G and restriction to the integrable locus We will now restrict ourselves to the integrable locus, i.e. the coupling constant satis es eq. (2.8), whilst keeping the group G arbitrary. Eqs. (3.5) and (3.6) now become, k2 ; )2 1 2 AB 3 2 + k2 Note that eq. (3.18) is simply a rescaling of that obtained for su(2) in [34]. Therefore, the group dependence in the ow equations is limited to the rate of the ow. Indeed, absorbing the factor cG in the RG time, t ! cGt, the ow can be made independent of the Lie group G. Eq. (3.17) also consistently yields the ow of the dependent parameter , HJEP03(218)4 Using these equations one immediately gets, d dt 2 2 showing that the integrable locus is preserved by the RG! Moreover, this system has an RG invariant aside from the coe cient of the WZ term, 2 k 2 ; d dt = cG k 2 = 2cG 2 = 2 : 2 k 2 ; Eq. (3.16) yields the RG equations for the independent coupling constants and , d dt d dt = = cG k 2 ( cG 2 2 ( 2 )2 ; )2 2 in terms of which we have a single independent RG equation, d dt = cG ( 2 2 k2) 2 + 2 (k2 + 2)2 Returning to the discussion in section 2 we see that the parameters entering the charge algebra are RG invariants since they are functions of and k alone. Discussion of the RG behaviour at the integrable locus. The case of SU(2) was already considered in [34], where at rst sight it appears to be di erent because the coupling is a total derivative in the Lagrangian and serves merely as an improvement term in the currents. The renormalisation of this coupling in the case of SU(2) can be absorbed by a gauge transformation generated by of eq. (3.2). So in fact the analysis of the RG phase portraits performed in [34] is equally valid here, corroborating the group dependence of the ow. However, for completeness and later discussion we present in gure 1 the RG behaviour of the G = SU(3) YB-WZ model at level k = 4 restricted to the integrable locus. In this case, we have an RG invariant given by eq. (3.21) which labels the RG trajectories. The only xed point is now the WZW at = jkj = 4, = = 0 in the IR. Again, on the = line the one-loop result blows up and the metric is degenerate. Since we are restricted to the integrable locus, where satis es eq. (2.8), the physically allowed theories are located in the regions where , or equivalently the RG invariant , is real. There are two such regions indicated in green. (3.18) (3.19) (3.20) (3.21) (3.22) γ t WZW Θ2 0 Θ2 < 0 0 2 4 6 8 10 α t integrable locus of the couplings vs. . The arrows point towards the IR. The red line = depicts the points where the one-loop result blows up and it is labeled by 2 = k 2 = irrelevant directions. The green regions are those where the coupling ( ; ) and the RG invariant are real. The yellow line portrays 2 = 10, the cyan line 2 = 10 an the purple line 2 = A physically allowed trajectory is portrayed by the yellow line in gure 1, along which the RG invariant has the constant real value. By varying the value of the RG invariant 2 R, we can cover the full region of physically allowed trajectories. In the green region where < 0, we start from the trivial xed point at = = 0 in the UV and end up at the WZW in the IR in a nite RG time. In the green region where > 0, the WZW is again an IR xed point but the asymptotic behaviour is not yet apparent. However, as we will see in the coming section, these two green regions are to be physically identi ed by a duality. In the other regions, we have either 0 > 2 > k2, represented by the cyan line, or 2 < k2, represented by the purple line. The crossover is given by 2 = k 2 which corresponds to the red = line. In any case, we ow either to the WZW or to a strongly -coupled theory in the IR. Conversely, owing towards the UV leads either to the trivial xed point or to an unsafe theory. Let us analyse the behaviour around the IR WZW xed point. If we linearise the ow around the xed point, i.e. let = k + and = 0 + = 0 + , we see from eqs. (3.18), d dt = cG k ; d dt = cG k ; d dt = cG k : (3.23) Since they all have positive sign's on the right-hand side we conclude that these are indeed irrelevant. Making use of the RG invariant eq. (3.21) and the integrable locus eq. (2.8) we can express the action as, S = 1 + ( 2 2 + k2) 2 R + ( 2 2 + + IW Z ; (3.24) HJEP03(218)4 where we choose the positive sign for the -coupling. Now expanding around the IR xed point to leading order in we have, 2k 2 R + 2k2 2 g 1 To interpret this let us now go to the Euclidean setting and de ne the usual WZW CFT currents, J (z) = J a(z)ta = ; J (z) = k 2 k 2 which obey a current algebra, and are Virasoro primary with weights (1; 0) and (0; 1) with respect to the Sugawara stress tensor. Consider a composite eld ``(z; z) transforming in representations labelled by ` and ` under the a ne GL GR symmetry. This eld will also be Virasoro primary and will be have an anomalous dimensions ( ` ; `). As explained in [46] the associated representation of the full Virasoro o KM algebra is degenerate with a null vector. Because of this the anomalous dimension can be extracted as, ` = c ` cG + k ; Dab(z; z) = tr(g 1tagtb) ; where c`I = t`at`a. Examples of such primaries are g(z; z), the group element itself, but also composites including the adjoint action, that transforms in the adjoint of GL on the rst index and the adjoint of GR on the second. This operator has anomalous dimension \wrong" currents i.e., cG D = D = cG+k and can be used to de ne the k 2 K = Kata = with dimensions (1 + ` ; `). Now we can see that the deforming operator is of the form, O(z; z) Ka(z; z)MabJ b(z) ; M = 1 + 2k k2 + 2 R + 2k2 ; and has total dimension 2+2 Suppose that we send D > 2 and is irrelevant even without any further corrections. ! 1 then we are in exactly the situation considered in [13, 46] of the ow of the PCM plus a Wess-Zumino term with the WZW as the IR Now recall that the Callan-Symanzik equation can be used to relate the beta function to the anomalous dimension and indeed we see that in the large k limit (in which loop corrections are suppressed) the anomalous dimension of O, O ! ckG precisely in agreement with the leading order of the beta functions eq. (3.23). It would be interesting to develop this line further and to try and ascertain all loop summation of the anomalous dimension following similar techniques to those adopted in the context of -models in [47]. There is however an added complexity that the deforming operator is not diagonal in the algebra indices but mixed with the inclusion of the M matrix. (3.25) (3.26) (3.27) (3.28) (3.29) (3.30) Although it is outside the primary purpose of this paper | which is to study integrable deformations | it is intriguing to look at the case of simply laced groups for which a consistent renormalisation did not require the model to lie on the integrable locus. It is then possible to rewrite eqs. (3.5) and (3.6) as, cG k R2AB ; This gives the following RG equations for the coupling constants , and (with RG time RAB : (3.31) ^AGB = G 6= SU(2). We call this point FP2. When G = SU(2) the RG equations blow up on the FP2 values (since then = ) and the second xed point is removed. Furthermore, for SU(2) one sees that the terms involving cancel in the ow equation for _ and the general RG equations of the remaining and will coincide with the corresponding RG equations when restricted to the integrable locus (see above). The RG behaviour when not restricted to the integrable locus. To illustrate the existence of the second xed point FP2, we consider the RG ow for the case of the group G = SU(3), setting k = 4. We plot the ow in two slices of the three-dimensional coupling space ( ; ; ) in order to visualise various directions around the xed points. Figure 2a shows the ow of vs. in the = 0 slice and gure 2b the ow of vs. in the From the above gures 2a and 2b, we see qualitatively that the WZW exhibits three independent irrelevant directions and the FP2 xed point one irrelevant and two relevant independent directions. This can be made precise by again analysing the linearlised ows in the neighbourhood of the xed points. In a more compact notation, denoting i = f ; ; g with i 2 f1; 2; 3g, the linearlised ow can be written as, In the neighbourhood of the WZW point we nd, = Aij i + O( i2) : AiWj ZW = cG j j k ij ; -10 0 FP2 WZW -10 β(t) 0 WZW FP2 2 4 6 8 10 (t) 2 (t) -4 -2 0 4 6 8 10 (a) The RG ow in the = 0 slice. (b) The RG ow in the depicts the points where the one-loop result blows up. The black dots represent the RG xed points WZW and FP2. We see that FP2 exhibits two relevant (orange lines) and one irrelevant (black line) direction. The WZW is a true IR xed point with only irrelevant directions. which gives indeed three independent irrelevant directions (they have positive eigenvalues). On the other hand, in the neighbourhood of the second xed point we nd, (3.35) (3.36) HJEP03(218)4 for which the eigenvalues read: AiFjP 2 = cG(cG + 4) jkj(cG 0 ccGG+44 B 0 cG 4 0 1 0 0 ccGG+44 A 1 C ; cG(cG + 4)2 jkj(cG 4)2 ; cG(cG + 4) jkj(cG 4) ; cG(cG + 4)2 jkj(cG 4)2 : Thus, from the second xed point indeed two relevant and one irrelevant independent directions emerge. At the two xed points the Ricci curvature R (D.12) evaluates to, RWZW = RFP2 = 4jkj ; cG 4(cG 4)2 c 2 G 16 D + 2cG (cG + 4) l 1 k j j ; with D the dimension and l the rank of G such that the target spaces are weakly curved for large enough k and for which the one-loop result is trustworthy. Whilst there is no reason to believe that the location of FP2 (i.e. the value of at the xed point) is one-loop exact, it seems likely that its existence is robust to loop corrections. It is then conceivable that FP2 may de ne a CFT. This being the case, from the general dilaton -function eq. (3.12) we can read o the e ective central charge ce of FP2 at one-loop to nd, c e = D + cG(cG + 4) (16 + (cG 16)cG)D + 6c2Gl 2jkj(cG 4)3 Before further discussing this possibility let us explore other aspects of the RG ow. At rst sight, and consistent with cUV > cIR, is that FP2 de nes a UV xed point from which in the deep IR one arrives at the WZW theory. However, care has to be taken when traveling over the line = , displayed in red in the gures. In the vicinity of this line the one-loop approximation is evidently not trustworthy; the target space curvatures blow up for small values of j j as is clear from the curvature R eq. (D.12) and indeed the metric GAB (3.3) becomes degenerate. In light of the apparent singularity of the one-loop ow equations where appears in denominators, it is then quite surprising that numerically a global picture emerges with ows that transgress the red line. We are then led to ask if such an RG trajectory can cross the = line in a nite RG time. To show that this is possible we concentrate on the slice of = jkj illustrated in gure 2b and for further simplicity consider going backwards along the orange direction = 0 starting near to the WZW point. Along this trajectory we can calculate the RG time t, with t = log , by evaluating, t = Z = f dt = i d d : (3.38) One can show that there is no pathology associated with = Given this, one is encouraged to take seriously the quantity ~ = jkj in this quantity. de ned in eq. (3.12) as a would-be c-function for the ow connecting FP2 and WZW. For simplicity we again consider this quantity along the orange direction = 0; in gure 3. What we see is that ~ is sensitive, unsurprisingly, to the singularity at = k. Whilst ~ jUV > ~ jIR and its derivative is strictly positive, it is not a positive de nite quantity and diverges at = k. Of course one should not read too much into this; the singularity is just symptomatic of the breakdown of the perturbative approximation. One could still expect that a correct strictly positive monotonic function exists and it agrees = jkj in gure 2b and plot the result with this one-loop approximate result where the one-loop result is valid. Combining the observation that the one-loop approximation is robust around the xed points (for large k) and the unexpected global continuity of the numerical solutions in gures 2a and 2b leads us to tentatively suggest that there is indeed an RG ow between a new xed point FP2 and an IR WZW model but that the sigma model description may not be the correct variables to reveal this. There are several points that merit investigation: At FP2, det(GAB) < 0. This means that some currents occur in the action with a negative coe cient of their kinetic term. A conservative viewpoint would be to regard this as non-physical but this then begs the question what is the UV completion of the model? Let us instead take FP2 seriously. Should FP2 de ne a CFT, it is presumably non-unitary. In this case we would have an RG ow between a non-unitary UV theory and an unitary IR theory. Perhaps this suggestion is not β˜Φ(t) 5 -10 15 10 5 0 -5 -10 t = jkj shown in gures 2a and 2b that connects the UV FP2 to the WZW as outlandish as might rst seem. By way of example we could consider RG ows in minimal models. It is well known [48, 49] that the IR.7 Less familiar perhaps are the RG minimal model M(p;p+1) triggers an RG ow resulting in the (p ows involving non-unitary minimal models, 1)th minimal model i.e. M(p;q) with p; q co-prime and q 6= p + 1, whose study was initiated in [50{52]. More generally [53], chains of non-unitary minimal models can be connected by RG (1;3) deformation of the pth ows triggered by alternating deformations of (1;5) and in an unitary minimal model. One example from [53] is,8 (2;1) which then terminate : : : M(5;12) (1;5) (2;1) ! M(5;8) ! M(3;8) (1;5) ! M(3;4) : (3.39) An other example in [53] terminates in a ow from the Yang-Lee edge singularity to the trivial c = 0 theory. Recently in [54] it was shown that it is possible to relax the requirement of unitary and still show the existence of a monotonic decreasing cfunction along such ows. So the learning here is that it is not a manifest impossibility to conceive an RG ow between a non-unitary UV CFT and a unitary IR CFT. The fate of FP2 with regard to higher loop corrections needs to be established; does it persist? Is the postulated FP2 both scale and Weyl invariant?9 What are the corresponding a ne symmetries and the exact value of the central charge at FP2? What is the spectrum of primaries for this postulated CFT at FP2? 7More precisely this occurs when the deformation parameter is negative, when the deformation parameter is positive the ow results in a massive theory. 8Notice that this terminates in the unitary critical Ising model with c = 12 and, just as with the ow between tri-critical Ising and critical Ising, the single massless Majorana fermion of the nal IR theory can be interpreted as the goldstino for the spontaneous breaking of the supersymmetry present in M(3;8). 9For instance in -deformed PSU(2; 2j4) current understanding is that only scale invariance holds [5]. These are evidently interesting challenges that we hope to return to in a future paper. For the present we continue with our principle concern; the YB-WZ model on the integrable locus. 4 Poisson-Lie T-duality of the YB-WZ model Motivated by the Poisson-Lie symmetric structure of the -deformation, one could wonder how the YB-WZ action (2.2) behaves under Poisson-Lie symmetry. Remarkably the YBWZ model at the integrable locus features an example of the most simple realisation of PL. The Poisson-Lie duality transformation preserves the structure of the action, reshu ing the coupling constant in a surprisingly Busher-rule like manner. At the RG (the WZW) the action is self-dual. This section, being somewhat technical, can safely be omitted at a rst reading and the reader can jump directly to the resulting \e ective" transformation rules of the Poisson-Lie transformation of the YB-WZ model in equations (4.10). When restricted to the integrable locus, the YB-WZ model admits a 1st order formulation as an E -model [32]. We refer the reader to the original paper for full details of this construction but note here the essential ingredients of an E -model, and its connection to sigma models, are: (i) An even dimensional real Lie-algebra d (ii) An ad-invariant inner product ( ; )d : d d ! R (iii) An idempotent involution E that is self-adjoint with respect to the inner product (iv) A maximally isotropic subalgebra h (i.e. (z1; z2)d = 0 8z1;2 2 h and dim h = 12 dim d). Given the data of (i)-(iii) one can construct a 1st order action known as the E -model. Given further (iv) one can integrate out auxiliary elds from the E -model to arrive at a non-linear sigma model. The eld variables of this sigma model are sections (de ned patchwise if needed) of the coset D=H (with D; H the groups corresponding to d and h). If a second maximally isotropic subalgebra h~ can be found then the procedure can be repeated to yield a second non-linear sigma model on D=H~ | this is the Poisson-Lie dual. For both the YB ( -theory) and the present case of interest, the YB-WZ theory, the relevant algebra is d = gC, viewed as a real Lie algebra with elements z = x + iy with x; y 2 g. The addition of the WZ term requires that the inner product be modi ed to [32], where the parameters used in [32] translate to, (z1; z2)d = CIm hei z1; z2i ; C = k2 + 8 2 ; e i = k + i k i = + ; (4.1) (4.2) which are both RG invariant and match the parameters determining the (a ne tower) charge algebra in the case of SU(2) established in [35], see also appendix B. The involution E , whose precise de nition will not be illuminating for us and can be found in eq. (3.8) dependance on and also on ep = of [32], dresses up the swapping of real and imaginary parts of z 2 gC with parametric . So unlike the innerproduct, E is RG variant. We have two maximal isotropics given by the embeddings of g: h = h~ = R R tan 2 (R2 + 1) cot 2 (R2 + 1) + i g : i g ; That these are subalgebras follows immediately since R satis es the mCYBE and R projects into the Cartan. That they are isotropic with respect to (4.1) xes the trigonometric functions. Since h = ei a + n where a and n are the corresponding algebras in the Iwasawa decomposition D = KAN , we can think of h as a twisted upper triangular subalgebra and the other, h~ , as lower triangular. Locally at least we can decompose, HJEP03(218)4 where we emphasise that the deformed inner product on gC of eq. (4.1) is used to de ne the WZW models and that the term depending on the projectors has coe cient 2 times 10There is a slight simpli cation here of the general formulas of [32] since g 2 G the adjoint action adg commutes in this case with the idempotent E. D = H H ~ = H~ H ; such that,10 and because the standard Iwasawa decomposition can be modi ed to incorporate the twisting by as in [32] we can identify D GC = G H ~ = G H . Thus the cosets D=H and D=H~ can be identi ed with G and so g 2 G can serve as eld variables on either of the two dual models. To extract the sigma models one needs to specify projectors P and P~ ImP = h ; KerP = (1 + E )d ; ImP~ = h~ ; KerP~ = (1 + E )d : Explicitly if we let (making use of the de nition of E in eqs. (3.7,3.8) of [32]), and de ne, then, w = e i + i cosh(p) + ie i sinh(p) = w1 + iw2 ; x = w1 w2 ; O = R tan 2 (R2 + 1) ; O~ = R cot 2 (R2 + 1) ; P(g 1dg) = (O i) (O + x) 1 g 1dg ; P~(g 1dg) = O~ + i ~ O x 1 g 1dg : (4.8) Equipped with all of this we can now simply specify the non-linear sigma models obtained after integrating out the auxiliary elds from the E models. They read, d S = SW ZW;k[g] + S~ = SW ZW;k[g] + d k Z k Z d d d d d ; ; (4.3) the inner product eq. (4.1) one T-duality acts on the parameters as, that of the kinetic term of the WZW model. Using R that the rst of these actions matches the general model in eq. (2.2) with parameters , and obeying the integrable locus relation. What of the Poisson-Lie dual theory? After some tedious trigonometry and using the relations eq. (4.2) together with the de nition of nds the action S~ is also of the form of eq. (2.2) but the ! ~ = ! ~ = ! ~ = k 2 ; ; k2 + 2 = 2 : (4.10) This is a truly elegant result; recall that k1 plays the role of 0 so that these Poisson-Lie Tduality rules really do resemble the radial inversion of abelian T-duality. Being canonically equivalent it must be the case that the T-dual model is also integrable, and indeed one sees that ~; ~; ~ also sit on the integrable locus; this serves as a check of the T-duality rules. We can see that the WZW point is rather special; it is the self-dual point of the duality transformation.11 As remarked earlier in the RG portrait gure 1 there are two regions that corresponding to a real action, shaded in green and for which 2 > 0. The Poisson-Lie duality action simply maps the region for < k one-to-one with that of > k; these two regions of course touch at the self-dual WZW The action of this T-duality on the charge algebra is also of note. It follows immediately that the RG invariant combination is transformed as, ! ~ = k 2 : (4.11) Then we see that the quantum group parameter q = exp k2+ 2 i is invariant under Th 8 duality. However recall that the a ne tower of charges (at least in the su(2) where it has been established explicitly) di ers from the standard a ne quantum group by a multiplicative factor between gradations of kk+ii . This factor undergoes an S-transformation, i.e it is mapped to negative its inverse. This illustrates that whilst the T-duality rules look quite trivial, at the level of charges the canonical transformation that maps the two T-dual theories can have quite an involved action. 5 The supersymmetric YB-WZ model This section falls a bit outside the main line of the paper but is motivated by the following observation. It is clear from the previous discussion that starting from a generic d = 2 non-linear -model and requiring (classical) integrability, imposes severe restrictions on the target manifold and its metric and torsion 3-form. However another way to restrict the allowed background geometries is by requiring the existence of extended worldsheet supersymmetries. Indeed asking that the non-linear -model exhibits N > (1; 1) supersymmetry 11Self-duality under PL of WZW models (with no deformations) was exhibited already in [55]. introduces additional geometric structure which only exists for particular background geometries. A hitherto unexplored terrain is the eventual relationship between integrable models on the one hand and extended supersymmetry on the other (however see [56] for some early work in this direction). In this section we explore the possibility of having N > (1; 1) supersymmetry in the YBWZ models studied in this paper. This is an interesting point in itself because if one thinks about the potential use of these integrable models as backgrounds for type II superstrings in the NS worldsheet formulation then the existence of an N = (2; 2) supersymmetric extension is necessary as well. As we will see, the integrable deformations of the WZWmodel studied in this paper do generically not allow for an extended supersymmetry. Given is a non-linear sigma model with target manifold M endowed with a metric G and a closed 3-form H (the torsion). Locally we write H = dB. Passing to an N = (1; 1) supersymmetric extension of the model does not require any further geometric structure. Indeed the action for the N = (1; 1) supersymmetric non-linear sigma model written in N = (1; 1) superspace is remarkably similar to the non-supersymmetric one,12 S = Z d are some local coordinates on the target manifold. However asking for more supersymmetry does introduce additional geometrical structure. E.g. N = (2; 2) supersymmetry requires the existence of two complex structures J+ and J which are endomorphisms of the tangent space T M and which are such that (M; G; H; J ) is a bihermitian structure [57, 58], i.e. M is even-dimensional and the complex structures J satisfy, 1. J 2 = J X; J Y ] = 0 for all X; Y 2 T M, which is the integrability condition for the complex structures, 3. G(J X; Y ) = G(X; J Y ) for all X; Y 2 T M, so G is a hermitian metric with respect to both complex structures, (+)J+ = r ( ) J = 0 with r ( ) covariant derivatives which use the Bismut conr J = G H G H J ; where the covariant derivative r in the above is taken with the Christo el symbol as connection. This condition is equivalent to the requirement that the exterior derivative of the two-forms ! (X; Y ) = G(X; J Y ) are given by: nections: such that in a local coordinate bases, 1 2 ( ) = fg G 1H ; 1 2 J d! (X; Y; Z) = H(J X; J Y; J Z): 12A brief summary of our superspace conventions can be found in appendix A. (5.2) (5.3) (5.4) Using the covariant constancy of the complex structure one can rewrite the integrability condition (condition 2) as, H(X; J Y; J Z) + H(Y; J Z; J X) + H(Z; J X; J Y ) = H(X; Y; Z) : (5.5) Note that demanding N = (2; 0) or N = (2; 1) instead of N = (2; 2) supersymmetry only requires the existence of J+ satisfying the above conditions. We now rewrite these conditions for the deformed models studied in this paper. Since at the level of the action the deformation preserves the left acting G symmetry while it breaks the right acting G symmetry (to its Cartan subgroup), the geometry and the N = (2; 2) conditions are most naturally presented in the basis of left-invariant one-forms uA. Given the deformed metric GAB eq. (3.3) and the torsion HABC eq. (3.4), we nd that the above conditions for N = (2; 2) supersymmetry translate in this basis to the following: 1. The rst condition is simply, (5.6) (5.7) (5.8) (5.9) (5.10) (5.11) (5.12) 2. The second condition (using the form in eq. (5.5)) results in: J AC J C B = BA : 3J D[AJ EBHC]DE = HABC ; where HABC is given in eq. (3.4). 3. The third condition yields, GAB = J C AJ D BGCD or using eq. (3.3): J C A CB = C J B CA J C BR2CA + J C AR2CB : 4. After a little bit of work, the covariant constancy of the complex structures (the fourth condition), translates to, and where the spin connection integrability condition, DB ; M A BC = A BC GADHDBC ; 1 2 J AE REBCD = RAECD J EB ; where the curvature tensors R are given by, The integrability condition eq. (5.11) is the requirement that the complex structures commute with the generators of the holonomy group de ned by the connections in eq. (5.10). A BC is given by eq. (D.7). Eq. (5.9) implies an While the rst three conditions are given by algebraic equations eqs. (5.6){(5.7), which can be analyzed in a way similar to what was done in [59, 60], the last one, eq. (5.9), is involved. However, the integrabilty conditions for the latter are algebraic again and can be explicitly analyzed. In [59, 60] these conditions were analyzed for the undeformed case, = = 0, i.e the standard WZW model and it was found that on any even-dimensional group manifolds there exist solutions to the above equations. Let us brie y review those results. In the undeformed case the connections in eq. (5.10) are simply, M+ABC = 0 ; M A BC = FBC A : (5.13) With this one veri es that the curvature tensors in eq. (5.12) vanish (re ecting the fact group manifolds are parallelizable), either trivially or by virtue of the Jacobi identities, and as a consequence the integrabilty conditions, eq. (5.11), are automatically satis ed. Turning then to eq. (5.9) one nds that J+AB is constant while J AB satis es, D FCDA J DB : In order to analyze the latter one introduces a group element in the adjoint representation, which can easily be shown to satisfy, SAB = vA uB ; D SAD : Using this and eq. (5.14) one shows that SAC J C D(S 1 D ) B (which is J in the right invariant frame) is constant as well. In this way the remaining conditions for N = (2; 2) supersymmetry, eq. (5.6){(5.8), all reduce to algebraic equations on the Lie algebra which were solved in [59, 60]. The result is remarkably simple: any complex structure pulled back to the Lie algebra is almost completely equivalent to a choice for a Cartan decomposition. Indeed the complex structure acts diagonally on generators corresponding to positive (negative roots) with eigenvalue +i ( i). It maps the CSA to itself so that it squares to minus one and so that the Cartan-Killing metric restricted to the CSA is hermitian. In the deformed case the integrability conditions eq. (5.11) become non-trivial and need to be investigated rst. While in principle this can be done for general groups (resulting in not particularly illuminating complex expressions) we limit ourselves in this paper to a detailed analysis of the simplest case: SU(2) U(1). A more systematic analysis of the relation extended supersymmetry and integrability in general is currently underway and will be reported on elsewhere. For SU(2) U(1) the deformation is a total derivative and can be ignored in the present analysis. We choose a basis for the Lie algebra where t0 = ( 3 + i 0)=2, t0 = ( 3 i 0)=2, t1 = ( 1 + i 2)=2 and t1 = ( 1 i 2)=2. In this basis the non-vanishing components of the Cartan-Killing metric are given by 00 = 11 = 1 and those of R by R11 = R11 = i. The non-vanishing components of the deformed metric in the left invariant frame, eq. (3.3), are G00 = and G11 = . For the torsion, eq. (3.4), we get H011 = H011 = i k. The hermiticity condition eq. (5.8) and the integrability condition eq. (5.11) are both linear in the complex structures and as a consequence are easily analyzed. The hermiticity condition eliminates 10 of the 16 components of each complex structure. A straightforward but somewhat tedious calculation shows the following result for the integrability condition: 1. It is identically satis ed without any further conditions if = 0. This is just the undeformed SU(2) U(1) WZW model known to be N = (2; 2) supersymmetric (in fact it is even N = (4; 4) supersymmetric ). 2. It is satis ed if = jkj and only J 00 = J 00 and J 11 = J 11 are non-vanishing. 3. Otherwise, for generic values of , and k it has no solutions. So we can conclude that in general the deformed SU(2) U(1) YB-WZ model does not allow for an N = (2; 2) supersymmetric extension. Remains of course case 2 in the above. From now on we take = jkj. Checking eq. (5.14) one nds that only a vanishing J is consistent with eqs. (5.6) and (5.7) while J+ is constant and its non-vanishing components 0 are given by e.g. J+0 = 0 1 J+0 = J+1 = 1 J+1 = i. This choice for J+ also satis es eqs. (5.6) and (5.7). So we conclude that the model is indeed N = (2; 1) or N = (2; 0) supersymmetric but does not allow for N = (2; 2) supersymmetry. To end this section we formulate this model in N = (2; 1) superspace thereby making the N = (2; 1) supersymmetry explicit. In general one starts with a set of N = (2; 1) super elds z and z satisfying the constraints D^ +z = +i D+z and D^ +z = i D+z which are a consequence of the fact that the non-vanishing components of the complex structure J+ are J+ = +i and J+ vector on the target manifold (i V ; i V ), = i . The action is expressed in terms of a d 2 d 2 d^+ i V D z i V D z : G = G = B : 1 2 1 2 S = S = Z d 2 d 2 Passing to N = (1; 1) superspace, one identi es the metric, and the Kalb-Ramond 2-form, Now let us apply this to the deformed SU(2) U(1) model where = jkj. The group element g 2 SU(2) U(1) is parameterized in a standard way by, g = e 2 i e 2i ('1+'2) cos 2 e 2i ('1 '2) sin 2 e 2i ('1 '2) sin 2 e 2i ('1+'2) cos 2 ! ; G +B (D+z D z + D+z D z ) + (D+z D z D+z D z ) ; (5.17) (5.18) (5.19) (5.20) (5.21) '1 2 R mod 2 ; '2 ; 2 R mod 4 : (5.22) We introduce complex coordinates z = (z; w) and z = (z; w) with ; = f1; 2g such that J+ acts as +i on dz and dw. The complex structure in this case is exactly the same as the one studied originally in [61, 62] (for a more detailed treatment see [63, 64]), so we can use the results obtained there to write the group element in terms of the complex coordinates as, g = (zz + ww) 21 (1+i) w z ! z w ; where the complex coordinates are related to the original coordinates as, z = e 21 e 2i ('1 '2) sin w = e 21 e 2i ('1+'2) cos 2 : Note that in the undeformed case, which allows for N = (2; 2) supersymmetry, z and w are the chiral and the twisted chiral N = (2; 2) super eld resp. [61, 62]. In the undeformed case we can readily derive the N = (2; 1) vector Vz and Vw appearing in the action eq. (5.17) as it directly descends from the generalized N = (2; 2) Kahler potential K obtained in [61, 62], K(z; z; w; w) = k (ln ww)2 + Li2 zz ww ; from which we get, ln 1 + Vw0 = ln(zz + ww) ; zz ww ; k z k w where the upper index 0 on V points to the fact that we are dealing with the undeformed case = jkj and = 0. geometry in terms of complex coordinates, In order to extend this to the deformed case, i.e. 6= 0, we rst rewrite the deformed ds2 = H = k k zz + ww (zz + ww)2 (zz + ww)2 dz dz + dw dw w dz z dw w dz z dw ; dz ^ dz ^ (w dw w dw) + dw ^ dw ^ (z dz z dz) ; (5.27) 2 ; 1 2 z w ; where we put = k. From the expression for the torsion one gets the Kalb-Ramond 2-form as well, B = k zz + ww z w dz ^ dw dw ^ dz : From this we obtain Vz and Vw, Vz = Vz0 + ww z zz + ww Vw = Vw0 + zz w zz + ww ; (5.23) (5.24) (5.25) (5.26) (5.28) (5.29) HJEP03(218)4 where Vz0 and Vw0 were given in eq. (5.26). Using eqs. (5.19) and (5.20) one veri es that eq. (5.29) indeed reproduces eq. (5.27) and (5.28). Combining eq. (5.29) with eq. (5.17) gives the action of the deformed theory explicitly in N = (2; 1) superspace. Concluding: as the generic SU(2) U(1) YB-WZ model does not allow for N = (2; 2) supersymmetry, it looks highly improbable that deformed models for other groups would allow for N = (2; 2) supersymmetry. Even when only requiring N = (2; 0) or N = (2; 1) supersymmetry, one nds that this is only possible for speci c values of the deformation parameters. However, it is important to note that the above derivation is based on the canonical form of the R-matrix. There is still an GL(2; C) freedom on the CSA directions of R which, together with the possibility of going beyond a single Yang-Baxter to a bi-Yang-Baxter deformation, could still reveal extended supersymmetry (but since these geometries are more complicated it would seem unlikely that they are more amenable to supersymmetries). 6 Summary, conclusions and outlook In this paper we investigated various properties of the Yang-Baxter deformation of the Principal Chiral model with a Wess-Zumino term introduced in [26]. As the undeformed model, the WZW model, exhibits rather unique features at the quantum level, we made a one-loop renormalisation group analysis of this class of models. For general groups and for generic values of the deformation parameters, the RG ow drives the theory outside the classical sigma model ansatz given in eq. (3.3) and (3.4). However, when the classical integrability condition is invoked, the renormalisation does remain within the sigma model ansatz and moreover the integrability condition is preserved along the RG ow. The fact that a very quantum property | the RG equations | are sensitive to the consideration of classical integrability is rather suggestive. It is therefore natural to conjecture that these models are quantum mechanically integrable. However, the nonultralocal property of such theories precludes a direct application of the Quantum Inverse Scattering method. It would be very interesting to examine how the alleviation approach, used in the context of the related -models [65], might be applied here in order to unravel the quantum S-matrix. Another interesting aspect is that the WZW model is the IR xed point; in comparison the integrable -deformed WZW has the CFT situated as an UV xed point. This model then seems closer in spirit to the irrelevant double trace integrable deformations of 2d CFTs constructed recently in [66]. Recently type deformations have been studied in the context of Gk Gl=Gk+l coset theories [67]; curiously there the CFT is recovered as an IR xed point in the same way as we have here. An unanticipated feature of this class of models is that when restricting to simply laced groups but staying outside of the integrable locus, we found a second xed point of the one-loop -functions which is UV with respect to the IR WZW model. Around this xed point, the curvatures of the target space geometry are small leading us to anticipate that the existence of this xed point is robust to higher loops. However, at this xed point a number of the currents have wrong sign kinetic terms. A conservative view would be to discard this as non-physical but this then begs the question of the UV completion HJEP03(218)4 of the deformation we are considering. Tentatively we might suppose that the xed point corresponds to a non-unitary CFT and that we have an exotic RG ow from this in the UV to the WZW in the IR. Comparable ows have been discovered in the context of minimal models. Needless to say it would be interesting to examine this more robustly. A technique that might help here could be to rephrase the entire discussion of these theories in terms of the free eld representations of WZW models. An obvious exercise which remains to be done is an RG analysis of the integrable models introduced in [27] that incorporate both bi-Yang-Baxter deformations and TST transformations. We expect this to be signi cantly more involved than the analysis performed in the current paper as the deformations in [27] destroy both the left and right acting group symmetry rendering the choice of a good basis to calculate the -functions non-trivial. An appealing feature of the landscape of , Yang-Baxter and deformations is that they provide tractable examples of sigma models that are Poisson-Lie T-dualisable. The theories considered here also share this feature; in fact the Poisson-Lie duality (which normally results in quite convoluted geometries) has a remarkably simple form. It results in a set of \Buscher rules" that resemble Abelian T-duality in that coupling constants are simply inverted. We see quite explicitly the compability of Poisson-Lie duality and RG ow and in particular we nd that the self-dual point of the duality and the xed point of RG are coincident. At this self-dual point the symmetries are enhanced and the theory becomes the WZW CFT. With the understanding that the Heisenberg anti-ferromagnetic XXX k chain has a gapless regime in the same universality class as the SU(2)k WZW model [68] an intriguing question is whether this PL duality can also be given an interpretation in 2 spin-chains. Finally we studied the possibility of supersymmetrising these models. As for any nonlinear sigma model in two dimensions an N = (1; 1) supersymmetric extension is always possible. Going beyond N = (1; 1) requires extra geometric structure, in particular every additional supersymmetry requires the existence of a complex structure satisfying various properties outlined in section 5 of this paper. Compared to the undeformed WZW model these conditions turn out to be rather involved. We solved them explictely in the simplest non-trivial example: SU(2) U(1). For generic values of the deformation parameters no supersymmetry beyond N = (1; 1) is allowed. For the particular case where the deformation , de ned in eq. (2.2), satis es = jkj with k the level of the WZ term an N = (2; 1) extension is still possible while N = (2; 2) is forbidden. We provided the manifest supersymmetric formulation of this model in N = (2; 1) superspace. The above analysis showed no obvious connection between integrability and the existence of extended supersymmetries (perhaps this is not so surprising, see e.g. [56]). A useful exercise in this context would be the following. All bi-hermitian complex surfaces have been classi ed [69, 70]. Those with the topology of SU(2) U(1) are the primary Hopf surfaces. A detailed analysis of the N = (2; 2) superspace formulations of those models combined with their integrability properties would be most interesting, in particular a characterization of the notion of integrability directly in N = (2; 2) superspace would be quite exciting. In view of the results obtained in the current paper we expect that if a connection between extended supersymmetry and integrability can be obtained it would probably not fall in the class of the models introduced in [26], however other possibilities remain, e.g. the models developed in [27] and through the inclusion of an action on the Cartan in the R-matrix. We will come back to this issue in a future publication. Acknowledgments DCT is supported by a Royal Society University Research Fellowship Generalised Dualities in String Theory and Holography URF 150185 and in part by STFC grant ST/P00055X/1. This work is supported in part by the Belgian Federal Science Policy O ce through the Interuniversity Attraction Pole P7/37, and in part by the \FWO-Vlaanderen" through the project G020714N and two \aspirant" fellowships (SD and SD), and by the Vrije Universiteit Brussel through the Strategic Research Program \High-Energy Physics". SD and SD would like to thank Swansea University for hospitality during a visit in which part of this research was conducted. We thank Vestislav Apostolov, Benjamin Doyon, Tim Hollowood, Chris Hull, Ctirad Klimcik, Prem Kumar, Martin Rocek and Kostas Sfetsos for useful conversations/communications that aided this project. A Conventions Let us establish our conventions. In this article we consider only semi-simple Lie groups G. For the corresponding Lie algebra g we pick a basis of Hermitian generators, where FABC are the structure constants which satisfy the Jacobi identity: [TA; TB] = i FABC TC ; FABDFDC E + FCADFDB E + FBC DFDA E = 0 : are hTA; TBi = x1R T r(TATB) = particular one gets for the adjoint representation, We denote by h ; i : g g ! R the ad-invariant Cartan-Killing form on g whose components AB (with xR the index of the representation R). In with cG = 2h_ where h_ is the dual Coxeter number of the group. Going now to a Cartan-Weyl basis where we call the generators in the Cartan subalgebra (CSA) Hm, the generators corresponding to positive (negative) roots Ta (Ta), where we have [Hm; Ta] = am Ta and [Hm; Ta] = am Ta. Using this one immediately gets from (A.1) (A.2) (A.3) (A.4) where the sum runs over the positive roots. With this we de ne the length squared of a root ~a by13 ~a ~a = am mnan. With our choice for the normalization of the Cartan-Killing mn = X aman ; a form the length squared of the long roots is always 2 and for the non-simply laced groups the length squared of the short roots is either 1 or 1/3. We de ne left-invariant forms u = iuATA = g 1dg which thus obey duA = 12 FBC AuB ^ uC whilst right-invariant forms v = + 12 FBC AvB ^ vC . The Wess-Zumino-Witten action [13] is, ivATA = dgg 1 obey dvA S = k Z 2 k Z 24 M3 h g 1dg; [g 1dg; g 1dg]i ; (A.5) in which g : ! G and with g the extension of g into M3 such that @M3 = . We adopt light-cone coordinates . For compact G, and demanding that the action is insensitive to the choice of action, requires k 2 Z. In section 5 we deal with non-linear sigma models in N = (1; 1) and N = (2; 1) superspace. Let us brie y review some of the notations appearing there and refer to e.g. [63, 64] for more details. Denoting for this section the bosonic worldsheet light-cone coordinates by, Passing from N = (1; 1) to N = (2; 1) superspace requires the introduction of one more real fermionic coordinates ^+ where the corresponding fermionic derivative satis es, =j = + ; = = ; and the N = (1; 1) (real) fermionic coordinates by + and , we introduce the fermionic derivatives which satisfy, D+2 = D2 = 2 fD+; D g = 0 : The N = (1; 1) integration measure is given by, and all other | except for (A.7) | (anti-)commutators do vanish. The N = (2; 1) Z d 2 d 2 d^+ = d d D+D D^ + : integration measure is, B Charges in SU(2) In this appendix we review the construction [35] of charges satisfying a quantum group algebra for the case of g = su(2) paying rather careful attention to the normalisation of canonical momenta so as to obtain the quantum group parameters expressed in terms of RG invariant quantities. In this appendix we use su(2) generators [T ; T 3] = iT , [T +; T ] = iT 3 and de ne components of the left invariant one-forms via g 1dg u+T + + u T + u3T 3. d 2 d 2 = d d D+D : D^ +2 = i 2 j 2 j Z Z (A.6) (A.7) (A.8) (A.9) (A.10) To orientate ourselves we begin with the Lagrangian eq. (2.2) specialised to the case of the -deformation, i.e. = 1 ; = 1+ 2 1 ; = 1+ 2 2 1 with k = 0 incorporating some of the key points of [3, 41]. Let us de ne some at rst sight non-obvious currents, j = 1 . These have simple Poisson brackets, fj3( 1); j3( 2)g = 0 ; fj ( 1); j3( 2)g = ij ( 2) ( 1 fj ( 1); j ( 2)g = ij3( 2) ( 1 That these are indeed the correct objects to work with becomes evident if we look at the Lax connection of eq. (2.9). Recall that the path-ordered exponential integral of the spatial component of the Lax de nes conserved charges. Expanding around particular values of the spectral parameter gives expressions for the charges. In particular if we expand the gauge transformed Lax L g(z) = g 1L (z)g g 1 @ g around certain points z = i | these correspond to poles in the twist function of the Maillet r/s kernels | we nd that these currents occur naturally as, L g(z = i ) = 4 j T 2i j3T 3 : Using the fact that the Cartan element can be factored in the path ordered exponential occurring in the monodromy matrix [71] one is led to construct (non-local) currents, J+( ; ) = j+( ; ) exp j3(^; )d^ ; J ( ; ) = j ( ; ) exp 2 j3(^; )d^ ; 2 Z 1 J3( ; ) = j3( ; ) : (B.1) (B.2) (B.3) (B.4) (B.6) (B.7) HJEP03(218)4 1 to us and thus that the charges Q = R 1 Jd are conserved subject to standard boundary fall o . The Poisson brackets give, fJ+( 1); J ( 2)g = ( 1 2 j3(^)d^ ; (B.5) 4 where we note that \cross terms" involving the non-local exponentials cancel. Thus one nds that, with suitable normalisation, Now we turn to the full theory including the WZ term. For this case we have the j = k i 8 ( 2 + 2) ( i 2 + k )u + ( ik + )u ; j3 = (ku 3 + u 3) ; 1 8 which obey a non-ultralocal algebra, fj3( 1); j3( 2)g = f j ( 1); j3( 2)g = f j ( 1); j ( 2)g = 4 ij3( 2) ( 1 2) ; 2) ; 2) 2) ; and from which we can build in the same way as above mutatis mutandis (non-local) conserved currents as, J+( ; ) = j+( ; ) exp J ( ; ) = j ( ; ) exp J3( ; ) = j3( ; ) : 8 Z ik 8 + ik j3(^; )d^ ; From this we can derive a related identity, R2x; Ry Rx; R2y = R R2x; y x; R2y + [Rx; y] [x; Ry] ; (B.8) (B.9) (B.10) (B.11) (B.12) (C.1) (C.2) generating the right acting a ne s[u(2). As with the case above these currents appear in the gauge transformed Lax expanded around the poles of its twist function, i.e., = 0) these currents just reduce to the currents L g z = ik j T 8 j3T 3 : For completeness we make the identi cation with the parameters + and used in the + = + = k + i i 2 8i k + i 8 : Even though the currents have a non-ultra-local algebra, the charge algebra is not ambiguous [35] (there is no order of limits problem in regulating the spatial integrals) and the commutator of charges (up to overall normalisations of Q ) still obeys eq. (B.6) with q = e Properties of R We collate here a number of identities used in the massaging of the calculation of the -functions. The strategy of deriving these identities is practically always the same: we expand the mCYBE eq. (2.1) or related versions in the generators TA of the Lie algebra and contract two free indices with two from the structure constants FABC or from FABDRC D. For completeness, we repeat here the mCYBE: [Rx; Ry] R ([x; Ry] + [Rx; y]) = [x; y] 8x; y 2 g : R2x;R2y =R2 R2x;y + x;R2y +(1+2R2)[x;y]; R2x;Ry + Rx;R2y =R R2x;y + x;R2y +2[x;y] +R2([Rx;y]+[x;Ry]); (C.4) for all x;y 2 g. This gives the following (non-exhaustive) list of properties of the R-matrix all of which were used in the derivation of the -functions: G (x) = eA (x)GABeB (x): D Geometry in the non-orthonormal frame Consider a general Riemannian target manifold M with local coordinates x and endowed with a curved metric G. We work in a frame formalism e^A = eA @ where the metric is constant but non-orthonormal: RDAREBFDEC+RDBRECFDEA+RDCREAFDEB FABC =0; (R2)DAREBFDEC+(R2)DARCEFBDE+RDAFBDC+(A$B)=0; (R2)DA(R2)EBFDEC (R2)DC(R2)EAFDEB (R2)DB(R2)ECFDEA 2 E RCERFDFCADFFBE+2RCERFBFCADFDFE cG AB=0; (R2)DAREFFDECFCBF =RCDFAEDFCBE; (R2)ECRFAFEFDFDBC+(R2)ECRFDFAEDFFBC+RCDFAEDFCBE+cGRAB=0; (R2)DFRCEFCBFFADE (R2)DFRCAFEBFFCDE+RCDFAEDFCBE+cGRAB=0; (R2)DF(R2)ECFDACFBEF +2(R2)ECFDACFBED+cG AB=0; (R2)DF(R2)ECFEAFFBDC+2(R2)EC(R2)DAFDEFFBFC 2cG(R2)AB cG AB=0; 2(R ) CFABE FABC =0; (C.7) (C.3) (C.5) (C.6) (C.8) (C.9) (C.10) (C.11) (C.12) (C.13) (C.15) (C.16) (C.17) (C.18) (C.19) (C.20) (C.21) (D.1) Requiring the spin-connection to be metric-compatible and torsion-free gives the following connection coe cients: A BC = GAE EBDGDC + EC DGDB + BC A ; 1 2 where ABC are the anholonomy coe cients determined by, In our case, the target manifold is a Lie manifold G endowed with a deformed geometry. Introducing left-invariant one-forms u = g 1dg = iuATAdx which satisfy, ^ uC ; we go to the frames e^A = uA@ . The deformed geometry in this frame is given by the constant non-orthonormal metric eq. (3.3) and by the torsion eq. (3.4), GAB = AB + The inverse metric is then (using R GAB = 3 = 1 AB + R): ) (R2)AB : For the spin-connection coe cients we nd from eq. (D.4) that ABB = FABC and thus, A BC = GAE FEBDGDC + FEC DGDB + 1 2 FBC A : Noting that the spin-connections are constant, the Riemann tensor can be calculated from, 1 2 1 2 (D.2) (D.3) (D.4) (D.5) (D.6) (D.7) (D.8) (D.9) (D.10) (D.11) and the Ricci tensor from, RABCD = E DB CE A E CB DE A CD E EAB ; RAB = RC ACB = E CA BE C FCB E CEA : With the -functions in mind we end this appendix with a set of useful expressions which are found by plugging in the expressions of the metric eq. (3.3) and the torsion eq. 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Saskia Demulder, Sibylle Driezen, Alexander Sevrin, Daniel C. Thompson. Classical and quantum aspects of Yang-Baxter Wess-Zumino models, Journal of High Energy Physics, 2018, 41, DOI: 10.1007/JHEP03(2018)041