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Conformal Invariance and Quantum Integrability of Sigma Models on Symmetric Superspaces

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Conformal Invariance and Quantum Integrability of Sigma Models on Symmetric Superspaces Physics Letters B 648 (2007) 254–261 www.elsevier.com/locate/physletb Conformal invariance and quantum integrability of sigma models on symmetric superspaces A. Babichenko a,b a Racah Institute of Physics, the Hebrew University, Jerusalem 91904, Israel b Department of Particle Physics, Weizmann Institute of Science, Rehovot 76100, Israel Received 7 December 2006; accepted 2 March 2007 Available online 7 March 2007 Editor: L. Alvarez-Gaumé Abstract We consider two dimensional nonlinear sigma models on few symmetric superspaces, which are supergroup manifolds of coset type. For those spaces where one loop beta function vanishes, two loop beta function is calculated and is shown to be zero. Vanishing of beta function in all orders of perturbation theory is shown for the principal chiral models on group supermanifolds with zero Killing form. Sigma models on symmetric (super) spaces on supergroup manifold G/H are known to be classically integrable. We investigate a possibility to extend an argument of absence of quantum anomalies in nonlocal current conservation from nonsuper case to the case of supergroup manifolds which are asymptotically free in one loop. © 2007 Elsevier B.V. All rights reserved. 1. Introduction super coset PSU(2, 2|4)/SO(1, 4) × SO(5). Hyperactivity in at- tempts to exploit integrability and methods of Bethe ansatz as a Two dimensional (2d) nonlinear sigma models (NLSM) on calculational tool in checks of ADS/CFT correspondence (see, supermanifolds, with and without WZ term, seemed to be ex- e.g., reviews [12] and references therein), also supports this in- otic objects, when they appeared in condensed matter physics terest, since both spin chains (on the gauge theory side) and 2d twenty years ago as an elegant calculational tool in problems NLSM (on the ADS side) appearing there, usually have a su- of self avoiding walks [1] and disordered metals [2]. Later on pergroup symmetry. Some more examples of this kind appear they appeared in string theory context [3–5]. A progress in their in the context of noncritical strings ADS/CFT correspondence understanding might be especially important for the theory of [13–15]. integer quantum Hall plateau transition [6,7] and disordered In this Letter we try to investigate aspects of conformal in- systems [8], but this progress is very slow. Many difficulties variance and quantum integrability of 2d NLSM (without WZ prevent a usage of standard technique in investigation of 2d terms) on some symmetric supergroup manifolds. List of the NLSM. One of them is unavoidable noncompactness of rele- models we are interested in is the following. It starts from the vant target space supermanifolds. Another one is a complicated principal chiral models (PCM) on the basic supergroups Lie: representation theory of the supergroups (their superalgebras), G = A(m|n), B(m|n), D(m|n), D(2, 1; α),G(3), F (4). In ad- where so called atypical representations play important, if not dition we consider the following coset superspaces: the main, role [9]. An interest to 2d NLSM on supermani- B(m|n) B(m|n) D(m|n) D(m|n) , , , , folds was renewed recently in string theory in the context of B(k|l) × B(i|j) D(m|n) D(k|l) × D(i|j) A(m|n) ADS/CFT correspondence, when it was understood that some D(2, 1; α) G(3) F(4) , , , ADS backgrounds can be described in terms of supercosets [10, × × ; (1) 5 A(1) A(1) A(1) D(2, 1 3) C(3) 11]. For example, ADS5 ×S is nothing but (bosonic part of) the where m = k +i, n = l +j. In all these cosets the factor algebra H is a maximal regular subalgebra of G. Regular subalgebras E-mail address: [email protected]. of the basic Lie superalgebras were classified in [16]. All the 0370-2693/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2007.03.003 A. Babichenko / Physics Letters B 648 (2007) 254–261 255 superspaces in (1) are symmetric. We hope that these toy mod- cosets seems problematical, but two loops beta function calcu- els will serve as a laboratory in investigation of more realistic lation confirms that it is equal to zero. We calculate the central ones, appearing both in condensed matter physics, and in string charges of these cosets. Calculation of one loop beta functions theory. for the rest of the superspaces selects asymptotically free ones. One of the most interesting observations in this subject was For them we analyze the quantum anomaly in the first non- done in the paper [17], where it was shown that 2d NLSM trivial nonlocal current conservation, and conclude that there without WZ term (a PCM model) on the supermanifold with is no anomaly with a proper choice of regularization. So the PSL(n|n) symmetry is conformal in all orders of perturbation 2d NLSM on the superspaces (1) are quantum integrable, and theory. In [11] this result was obtained for PSU(2|2). The au- moreover, those from the list (2) are conformal invariant. thors of [17] pointed out the existence of a Casimir like chiral algebra of the model, but a principal difficulties did not allow to 2. Beta function in one and two loops investigate the full spectrum of its representations. All the ma- chinery of CFT is hardly applicable for these nonstandard 2d We start from a geometrical approach to background field CFTs, although in some cases CFT methods were successfully perturbation theory calculations of beta function for 2d NLSM applied [9,18,19]. on a Riemannian supermanifold. We are going to discuss the As it is well known, any 2d NLSM on a symmetric space action is classically integrable (see, e.g., [20] and references therein). 1 1 − Classical integrability expresses itself, in particular, in the pres- = d2x G 1∂ G 2 S 2 Str μ (3) ence of conserved nonlocal charges, or, in a more rigorous way, 4π λ in the presence of Backlund transform and spectral parameter where G is an element of supergroup (supercoset) manifold, dependent Lax pairs. Generalization of the standard procedure and Str is the supertrace. A review of the method and main re- of nonlocal current construction to the symmetric superspace sults for nonsuper case one can find in [25]. Recall that usual case seems straightforward. It was shown for ordinary sym- QFT background field methods should be modified being ap- metric (nonsuper) spaces that on the quantum level, absence plied to 2d NLSM, if we wish to preserve target manifold Rie- of anomaly in these nonlocal current conservation is guaran- mannian covariance of calculations. One should expand the ac- teed only if the factor group of a coset is either simple [21,22] tion around the classical geodesic trajectory ρa on the manifold. or consists of a product of identical simple group by itself [23]. Then a result of calculations is expressed in terms of the basic a One can expect that the same feature will remain in the case of covariant object—curvature tensor Rbcd , their covariant deriva- symmetric superspaces. In this sense, the list of cosets above tives, and products with different kind of indices constructions. represents a good candidates for quantum integrable models. In particular, the one loop beta function is proportional to the (one should consider the first and the third cosets with k = i Ricci tensor and l = j). (1) = 1 2 c = 1 2 Since the argument about presence/absence of anomaly in βab λ R acb λ Rab. (4) nonlocal currents is based on the dimensions of operators cal- 2π 2π culated as engineering dimensions, one should be sure these In general, only the one loop result is regularization scheme dimensions are correct in the UV limit. In the ordinary (non- independent, higher loops depend on regularization. In dimen- super) case this is guaranteed by asymptotical freedom (posi- sional regularization there exists the choice, for which the two tiveness of the beta function, at least in one loop) of 2d NLSM loop result looks in the simplest way: on symmetric spaces. As we will see below, in general it is not 2 the case for symmetric superspaces. Requirement of asymptotic β(2) =− λ4R Re(cd) ab 2 a(cd)e b (5) freedom which we are going to impose in order to preserve an 3(2π) ability to talk about naively calculated dimensions of the op- where the parenthesis means the symmetrization over the in- erators, will restrict possible values of m in the list above to dices, and lowering/raising of indices is made by the manifold be grater then n. So we start from calculation of one loop beta metric/its inverse. In principle, all this technology of beta func- functions for the above cosets. As we will see, part of them tion calculation may be extended to the supermanifolds. For D(2n + 1|2n) definitions of the main objects of Riemannian geometry on su- D(n + 1|n), D(2, 1; α), , permanifolds see for example [26]. On the mathematical level D(n + 1|n) × D(n|n) of rigorosity, there are some principal difficulties in basic defi- D(n + 1|n) D(2, 1; α) , (2) nitions of supermanifolds (even on the level of charts self con- A(n + 1|n) A(1) × A(1) × A(1) sistency [27]). But there is a way to overcome these difficulties have zero one loop beta function. We extended our calcula- in such a way that usual objects of Riemannian geometry will tions to two loops and got zero. As we will show, the beautiful be well defined [28,29].
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