Integrable deformations of sigma models

Abstract

In this pedagogical review we introduce systematic approaches to deforming integrable 2-dimensional sigma models. We use the integrable principal chiral model and the conformal Wess-Zumino-Witten model as our starting points and explore their Yang-Baxter and current-current deformations. There is an intricate web of relations between these models based on underlying algebraic structures and worldsheet dualities, which is highlighted throughout. We finish with a discussion of the generalisation to other symmetric integrable models, including some original results related to ZT cosets and their deformations, and the application to string theory. This review is based on notes written for lectures delivered at the school "Integrability, Dualities and Deformations," which ran from 23 to 27 August 2021 in Santiago de Compostela and virtually.

Homogeneous Yang-Baxter deformations as undeformed yet twisted models

“…The case of the supercoset is formally identical. We now consider a real Lie supergroup G whose algebra admits the Z 4 grading g = g (0) + g (1) + g (2) + g (3) so that [[g (i) , g (j) ]] ⊂ g (i+j mod 4) , where [[ , ]] is the superbracket (that coincides with the anti-commutator when both g (i) , g (j) have odd grading, and the commutator otherwise). Notice that if x, y belong to the Grassmann enveloping algebra of g, then the bracket that must be used is always given by the standard commutator [x, y], because the Grassmann variables take care of the additional minus signs.…”

“…Notice that if x, y belong to the Grassmann enveloping algebra of g, then the bracket that must be used is always given by the standard commutator [x, y], because the Grassmann variables take care of the additional minus signs. If we identify H with the Lie group of h = g (0) , then the action of the σ-model on the supercoset G \ H is still given by (3.35), with the exception that now d = 1 2 P (1) + P (2) − 1 2 P (3) and that the trace should be substituted by the supertrace [71]. With these minor substitutions, the hYB deformation of the supercoset action is still given by (3.36) [9], and the Lax connection is the obvious generalisation of the supercoset Lax connection…”

“…In recent years there has been a lot of activity in the study of integrability-preserving deformations of 2-dimensional σ-models, see [1][2][3] for recent reviews. In this paper we will deal with the so-called "homogeneous Yang-Baxter" (hYB) deformation [4,5], which is a variant of the "inhomogeneous Yang-Baxter" deformation (sometimes called η-deformation) of [6][7][8][9].…”

Novel aspects of integrability for NLSMs in symmetric spaces

“…Finally, let us mention that integrability is not restricted to symmetric spaces. Recently there has been a lot of activity in the construction of integrable NLSMs [9][10][11][12][13], see also [14,15] and references therein.…”

Deformed WZW Models and Hodge Theory -- Part I

“…Such deformations are of interest, since they reduce the amount of supersymmetry while retaining integrability, allowing for a more general study of the AdS/CFT correspondence beyond N = 4 SYM [27][28][29], see also [30] for an elaborate list of references. Nevertheless, even in the bosonic setting integrable deformations of two-dimensional σ-models have received much attention [31][32][33][34][35][36][37][38][39][40][41][42][43] and a vast web of connections and dualities between them has been uncovered over the years [44][45][46][47][48], see also [49][50][51] for recent reviews.…”