arXiv:1008.0776v2 [hep-th] 11 Aug 2010 KUNS-2286 Hidden Yangian symmetry in sigma model on squashed sphere Io Kawaguchi∗and Kentaroh Yoshida† Department of Physics, Kyoto University Kyoto 606-8502, Japan Abstract We discuss a hidden symmetry of a two-dimensional sigma model on a squashed S3. TheSU(2) current can be improved so that it can be regarded as a flat con- nection. Then we can obtain an infinite number of conserved non-local charges and show the Yangian algebra by directly checking the Serre relation. This symmetry is also deduced from the coset structure of the squashed sphere. The same argument is applicable to the warped AdS3spaces via double Wick rotations. ∗E-mail: io@gauge.scphys.kyoto-u.ac.jp †E-mail: kyoshida@gauge.scphys.kyoto-u.ac.jp
1Introduction It is well known that two-dimensional sigma models onsymmetricspaces are classically integrable [1]. Simple examples are theO(3) non-linear sigma model and theSU(2) prin- cipal chiral model. The integrability of the sigma models survives at quantum level and physical quantities such as S-matrix can be computed exactly [2,3]. Thus the integrability allows us to study non-perturbative aspects of quantum field theory. There is a hidden symmetry [4, 5], called the Yangian symmetry [6], behind the integrable structure. The integrable structure on symmetric spaces may appear as classical integrability of type IIB superstring on AdS5×S5[7] and now it has a new perspective and potential applications in the AdS/CFT correspondence [8]. The integrable string backgrounds have recently been classified based on symmetric coset structure [9]. Then an interesting issue is to consider integrability and hidden symmetry of the sigma models onnon-symmetric spaces and discuss a generalization of the AdS/CFT dictionary. Simple examples of non- symmetric spaces are squashed spheres and warped AdS spaces. In this letter, as a simple example, we will discuss a hidden symmetry of a sigma model on a squashedS3. This squashed geometry is realized as aU(1)-fibration (ψ) over S2(θ, φ) and the metric is given by ds2=L2 4 [dθ2+ cos2θ dφ2+ (1 +C)(dψ+ sinθ dφ)2].(1.1) The constant parameterCmeasures a deformation ofS3andC= 0 just describes the roundS3with the radiusL. TheSU(2)L×SU(2)Rsymmetry for the roundS3is broken toSU(2)L×U(1)RforC6= 0 . ForC= 0 the Yangian symmetry is well known as a hidden symmetry. But it remains to be clarified forC6= 0 whether or not theSU(2)L×U(1)R can enhance to an infinite-dimenisonal symmetry. 2Yangian symmetry We start from the classical action of the sigma model: S=−1 2 ∫ ∫ dtdx [ (∂µθ)2+ cos2θ(∂µφ)2+ (1 +C)(∂µψ+ sinθ∂µφ)2 ] .(2.1) For simplicity, we will not take the Virasoro conditions into account and hence the base space is assumed to be a two-dimensional Minkowski spacetime with the coordinates xµ= (t, x) and the metricηµν= (−1,1) . 1
The isometry of the metric (1.1) isSU(2)L×U(1)Rand hence the action (2.1) is invariant under the following transformations: δ1(φ, ψ, θ) =ǫ(−1,0,0), δ2(φ, ψ, θ) =ǫ ( −sinφtanθ ,sinφ cosθ ,−cosφ ) , δ3(φ, ψ, θ) =ǫ ( cosφtanθ ,−cosφ cosθ ,−sinφ ) , δ4(φ, ψ, θ) =ǫ(0,−1,0). Hereǫis an infinitesimal constant.Note that the transformations are independent of the squashing parameterC.The first three are theSU(2)Lgenerators and the last is theU(1)Rone. By following the Noether’s theorem, we can construct the four conserved currents j1 µ=∂µφ+ sinθ ∂µψ+Csinθ(∂µψ+ sinθ ∂µφ), j2 µ= cosφ ∂µθ−sinφcosθ ∂µψ −Csinφcosθ(∂µψ+ sinθ ∂µφ), j3 µ= sinφ ∂µθ+ cosφcosθ ∂µψ +Ccosφcosθ(∂µψ+ sinθ ∂µφ), j4 µ= (1 +C)(∂µψ+ sinθ ∂µφ). After some algebra, for theSU(2)LcurrentjA µ(A= 1,2,3), we obtain that ǫµν ( ∂µjA ν−1 2εA BCjB µjC ν ) =CnAǫµν∂µ(sinθ)∂νφ with an anti-symmetric tensorǫµνnormalized asǫtx= +1 andnAis a unit vector on the S2 nA≡(sinθ,−sinφcosθ,cosφcosθ). Thus theSU(2)Lcurrent does not satisfy the flatness condition. But it can be improved with the ambiguity of the Noether current so that it satisfies the flatness condition. To improve the current, the following topological currentIA µhas to be added tojA µ: IA µ≡ ±√Cǫµν∂νnA.(2.2) This term is obviously conserved:∂µIA µ= 0 . We will discuss the geometrical meaning of (2.2) later. 2
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