On dual description of the deformed O(N) sigma model

Abstract

We study dual strong coupling description of integrability-preserving deformation of the O(N ) sigma model. Dual theory is described by a coupled theory of Dirac fermions with four-fermion interaction and bosonic fields with exponential interactions. We claim that both theories share the same integrable structure and coincide as quantum field theories. We construct a solution of Ricci flow equation which behaves in the UV as a free theory perturbed by graviton operators and show that it coincides with the metric of the η−deformed O(N ) sigma-model after T −duality transformation.(1.5)Clearly, this procedure breaks the left symmetry, but preserves the right one. It can be shown that the model is still enjoys the zero curvature representation and hence possesses the integrability property. In this paper we consider the case of O(N ) sigma-model, i.e. we take G = SO(N ) and H = SO(N − 1). Integrability of this sigma-model at the quantum level has been first demonstrated by Polyakov [10]. As QFT O(N ) sigma-model corresponds to an asymptotically free theory with a dynamically generated mass scale. It describes scattering of N mesons in the vector representation of the global O(N ) group [11,12]. The scattering matrix for these mesons is strongly constrained by integrability, which implies, in particular, the absence of particle production and factorization of the multi-particles amplitudes into the product of the two-particle scattering. These requirements plus the conditions of crossing invariance and unitarity are so strong that allow one to compute the S-matrix exactly. The two-particle S−matrix for the O(N ) sigma-model has been found by Alexander and Alexey Zamolodchikov in their seminal paper [13]. It has an explicit form S kl ij (θ) = δ ij δ kl S 1 (θ) + δ ik δ jl S 2 (θ) + δ il δ jk S 3 (θ), S 3 (θ) = S 1 (iπ − θ), S 1 (θ) = − 2iπ (N − 2)(iπ − θ) S 2 (θ), S 2 (θ) = Q(θ)Q(iπ − θ), (1.6)

Integrable 2d sigma models: Quantum corrections to geometry from RG flow

“…Finally, it would also be interesting to establish a connection of the quantum-deformed "sausage" σ-model to the massive non-perturbative S-matrix [4] and possibly to the dual massive 2d QFT (cf. [11]).…”

“…For S 2 only the latter case exists as a Yang-Baxter deformation in a strict sense since there is no split R-matrix of Lie(SO(3)) (see, e.g., [52]). 11 Let us note that the metric (2.24) is self T-dual, i.e. invariant under φ →φ andφ → κ −1 φ, r → κ −1 r −1 .…”

“…Indeed, in cases with a UV fixed point and a known underlying quantum S-matrix, the deformed σ-model may be reconstructable from a dual massive model (cf. [11]).…”

Dressing cosets and multi-parametric integrable deformations

“…This η-deformation algorithm, combined also with the coset construction of Refs. [16] and with the alternative λ-deformation one [72], gave rise to various constructions of the deformed integrable σ-models [10,15,17,18,20,30,31,41,73,75,81], many of them exploitable in quantum field theory [2,4,5,8,21,24,25,56,57] and in string theory via the AdS/CFT correspondence [1,7,13,16,19,22,27,32,33,36,37,60,61,64,65,68,69,80].…”

Dual description of η-deformed OSP sigma models

“…In this paper we initiate the detailed investigation of the η-deformed G/H symmetric space sigma model and their Toda QFT duals when G and H are supergroups. Our primary focus will be on superspheres OSP(N |2m)/OSP(N − 1|2m), generalizing the corresponding construction for spheres SO(N )/SO(N − 1) [3,4].…”