On integrability of 2-dimensional σ-models of Poisson-Lie type

Journal of High Energy Physics, Nov 2017

We describe a simple procedure for constructing a Lax pair for suitable 2- dimensional σ-models appearing in Poisson-Lie T-duality

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On integrability of 2-dimensional σ-models of Poisson-Lie type

HJE On integrability of 2-dimensional -models of Poisson-Lie type Pavol Severa 0 0 Section of Mathematics, University of Geneva We describe a simple procedure for constructing a Lax pair for suitable 2dimensional -models appearing in Poisson-Lie T-duality 1Supported in part by the grant MODFLAT of the European Research Council and the NCCR SwissMAP of the Swiss National Science Foundation. Integrable Field Theories; Sigma Models; String Duality 1 Introduction 2 3 4 5 terms of d-valued 1-forms A 2 1 ( ; d) satisfying There is a class of 2-dimensional -models, introduced in the context of Poisson-Lie Tduality [5], whose solutions are naturally described in terms of certain at connections. The target space of such a -model is D=H, where D is a Lie group and H D a subgroup. The -model is de ned by the following data: an invariant symmetric non-degenerate pairing h; i on the Lie algebra d such that the Lie subalgebra h d is Lagrangian, i.e. h? = h, and a subspace V+ d such that dim V+ = (dim d)=2 and such that h; ijV+ is positive de nite. The construction and properties of these -models are recalled in section 2 (including the Poisson-Lie T-duality, which says that the -model, seen as a Hamiltonian system, is essentially independent of H). Let us call them -models of Poisson-Lie type. The solutions ! D=H of equations of motion of such a -model can be encoded in dA + [A; A]=2 = 0 A 2 1;0( ; V+) 0;1( ; V ); (1.1a) (1.1b) where V := (V+)? d. Namely, the atness (1.1a) of A implies that there is a map ` : ~ ! D (where ~ is the universal cover of ) such that A = of A is in H then ` gives us a well-de ned map in this way are exactly the solutions of equations of motion. d` ` 1. If the holonomy ! D=H. The maps ! D=H obtained As rst observed by Klimc k [3], and later by Sfetsos [12], and Delduc, Magro, and Vicedo [2], some -models of Poisson-Lie type are integrable. Their integrability is proven by nding a Lax pair, i.e. a 1-parameter family of at connections (with parameter ) A 2 1 ( ; g) dA + [A ; A ]=2 = 0 of the phase space, i.e. for every A 2 1 ( ; d) satisfying (1.1). { 1 { The aim of this note is to make the construction of A transparent. We simply observe 1 that if A 2 ( ; d) satis es (1.1) and if p : d ! g is a linear map such that A suitable family p : d ! g will then give us a family of at connections leave this question open. 2 -models of Poisson-Lie type and Poisson-Lie T-duality In this section we review the properties of the \2-dimensional -models of Poisson-Lie type" introduced in [5] (together with their Hamiltonian picture from [6] and using the target spaces of the form D=H, as introduced in [7]). Let d be a Lie algebra with an invariant non-degenerate symmetric bilinear form h; i of symmetric signature and let V+ d be a linear subspace with dim V+ = (dim d)=2, such that h; ijV+ is positive-de nite. Let M = D=H where D is a connected Lie group integrating d and H D is a closed connected subgroup such that its Lie algebra h d is Lagrangian in d. This data de nes a Riemannian metric g and a closed 3-form on M . They are given by g( (X); (Y )) = p = 1 2 hX; Y i 1 2 D + 8X; Y 2 V+ 1 2 dhA; Li right-translates of V+. 1 Here is the action of d on M = D=H, p : D ! D=H is the projection, D 2 the Cartan 3-form (given by D(XL; Y L; ZL) = h[X; Y ]; Zi (8X; Y; Z 2 d)), L 2 is the left-invariant Maurer-Cartan form on D (i.e. L(XL) = X), and A 2 the connection on the principal H-bundle p : D ! D=H whose horizontal spaces are the 3(D) is 1(D; d) 1(D; h) is 1The conceptual de nition of g and is as follows: the trivial vector bundle d M ! M is naturally an exact Courant algebroid, with the anchor given by and the Courant bracket of its constant sections being the Lie bracket on d. Then V+ M d M is a generalized metric, which is equivalent to the metric g then de ne a -model with the standard action functional Z S(f ) = Z Y f where is (say) the cylinder with the usual metric d 2 d 2 and f : extended to the solid cylinder Y with boundary . ! M is a map For our purposes, the main properties of these -models are the following: The solutions of the equations of motion are in (almost) 1-1 correspondence with 1-forms A 2 1 i it admits a lift ` : ~ ! D such that A := ( ; d) satisfying (1.1). More precisely, a map f : uniquely speci ed by f (the lift ` is not unique | it can be multiplied by an element d` ` 1 satis es (1.1). Notice that A is inverse of the matrix of hea; ebi). The Hamiltonian of the -model is (written using a basis ea of d, with facb being the structure constants of d and tab the H = of maps ` : R ! D which are quasi-periodic in the sense that for some h 2 H we have `( + 2 ) = `( )h, modulo the action of H by right multiplication. The reduced phase space is (LD)=D (i.e. periodic maps modulo the action of D); it is the subspace of 1(S1; d) given by the unit holonomy constraint.) 3 Constructing new at connections As we have seen, the solutions of our -model give rise to (...truncated)


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Pavol Ševera. On integrability of 2-dimensional σ-models of Poisson-Lie type, Journal of High Energy Physics, 2017, pp. 15, Volume 2017, Issue 11, DOI: 10.1007/JHEP11(2017)015