Deformed integrableσ-models, classicalR-matrices and classical exchange algebra on Drinfel’d doubles

Abstract

We describe a unifying framework for the systematic construction of integrable deformations of integrable σ-models within the Hamiltonian formalism. It applies equally to both the 'Yang-Baxter' type as well as 'gauged WZW' type deformations which were considered recently in the literature. As a byproduct, these two families of integrable deformations are shown to be Poisson-Lie T -dual of one another.

A new class of integrable deformations of CFTs

“…Indeed, the computation of the corresponding RG flow equations using gravitational methods was performed in [2,3] and is in agreement with results from field theoretical methods [8][9][10]. Furthermore, it was shown that these models, named generically as λ-deformed, can be embedded in specific cases of low dimensionality to supergravity [11][12][13][14][15], while their relation to η-deformations for group and coset spaces introduced in [16][17][18] and [19][20][21] respectively, via Poisson-Lie T-duality [22] and appropriate analytic continuations, was uncovered in [23][24][25][26][27]. More recently, the computation of the all-loop correlators for the…”

“…It was shown in [23][24][25][26][27] that, the λ-and η-deformations are related via Poisson-Lie T-duality and appropriate analytic continuations. Our construction and results suggest that there likely exist new integrable σ-models of the η-type to be constructed.…”

Poisson-Lie duals of the η deformed symmetric space sigma model

“…A duality of this type was considered in [33], which relates an η s deformation (the superscript denoting it is based on a split solution of the modified classical Yang-Baxter equation) to the corresponding λ deformation. In both cases the duality is with respect to the full symmetry group of the undeformed model.…”

“…Starting from the corresponding model on the Drinfel'd double (2.20) this allows us to investigate in which subgroups of G the η deformed model can be Poisson-Lie dualised and construct the corresponding backgrounds. Our starting point for the construction of the η deformed model will be the compact simple Lie group G, with Lie algebra g = Lie(G), and the complex Drinfel'd double D ≡ G C , with Lie algebra d ≡ g C = Lie(D) [10,12,33,41].…”

Yang-Baxter deformations of Minkowski spacetime

“…In the latter case, there are a lot of classical r-matrices satisfying the cybe and some of them are associated with well-known gravitational backgrounds, such as Lunin-Maldacena-Frolov backgrounds [39,40], gravity duals for non-commutative gauge theories [41,42], Schrödinger spacetimes [43][44][45][46][47] and gravity duals for dipole theories [48][49][50][51][52], as shown in a series of works [53][54][55][56][57]. Very recently, the reality of the classical action has been revisited in [58] and a unified picture of deformed integrable sigma models has been provided in [59].…”