Classically integrable σ-models are known to be solutions of the 1-loop RG equations, or "Ricci flow", with only a few couplings running. In some of the simplest examples of integrable deformations we find that in order to preserve this property at 2 (and higher) loops the classical σ-model should be corrected by quantum counterterms. The pattern is similar to that of effective σ-models associated to gauged WZW theories. We consider in detail the examples of the η-deformation of S 2 ("sausage model") and H 2 , as well as the closely related λ-deformation of the SO(1, 2)/SO(2) coset. We also point out that similar counterterms are required in order for non-abelian duality to commute with RG flow beyond the 1-loop order. 1 bhoare@ethz.ch 2 n.

Following arXiv:1907.04737, we continue our investigation of the relation between the renormalizability (with finitely many couplings) and integrability in 2d σ-models. We focus on the "λ-model," an integrable model associated to a group or symmetric space and containing as special limits a (gauged) WZW model and an "interpolating model" for non-abelian duality. The parameters are the WZ level k and the coupling λ, and the fields are g, valued in a group G, and a 2d vector A ± in the corresponding algebra. We formulate the λ-model as a σ-model on an extended G×G×G configuration space (g, h,h), defining h andh byOur central observation is that the model on this extended configuration space is renormalizable without any deformation, with only λ running. This is in contrast to the standard σ-model found by integrating out A ± , whose 2-loop renormalizability is only obtained after the addition of specific finite local counterterms, resulting in a quantum deformation of the target space geometry. We compute the 2-loop β-function of the λ-model for general group and symmetric spaces, and illustrate our results on the examples of SU (2)/U (1) and SU (2). Similar conclusions apply in the non-abelian dual limit implying that non-abelian duality commutes with the RG flow. We also find the 2-loop β-function of a "squashed" principal chiral model. 1 bhoare@ethz.ch 2 n.levine17@imperial.ac.uk 3 Also at the Institute of Theoretical and Mathematical Physics, MSU and Lebedev Institute, Moscow. tseytlin@imperial.ac.uk 1 Our notation and conventions are summarized in Appendix A. In particular, we use hermitian generators T a of the Lie algebra so that if g = e v ∈ G then v = i T a v a ∈ Lie(G) is anti-hermitian. The action is defined as S = 1 4π d 2 σL so that L has extra factor of 2 compared to the "conventional" normalization.

Motivated by the search for new integrable string models, we study the properties of massless tree-level S-matrices for 2d σ-models expanded near the trivial vacuum. We find that, in contrast to the standard massive case, there is no apparent link between massless S-matrices and integrability: in well-known integrable models the tree-level massless S-matrix fails to factorize and exhibits particle production. Such tree-level particle production is found in several classically integrable models: the principal chiral model, its classically equivalent "pseudo-dual" model, its non-abelian dual model and also the SO(N +1)/SO(N ) coset model. The connection to integrability may, in principle, be restored if one expands near a nontrivial vacuum with massive excitations. We discuss IR ambiguities in 2d massless tree-level amplitudes and their resolution using either a small mass parameter or the i -regularization. In general, these ambiguities can lead to anomalies in the equivalence of the S-matrix under field redefinitions, and may be linked to the observed particle production in integrable models. We also comment on the transformation of massless S-matrices under σ-model T-duality, comparing the standard and the "doubled" formulations (with T-duality covariance built into the latter). 1 bhoare@ethz.ch 2 n.

We consider several classes of σ-models (on groups and symmetric spaces, η-models, ⋋-models) with local couplings that may depend on the 2d coordinates, e.g. on time τ . We observe that (i) starting with a classically integrable 2d σ-model, (ii) formally promoting its couplings hα to functions hα(τ ) of 2d time, and (iii) demanding that the resulting time-dependent model also admits a Lax connection implies that hα(τ ) must solve the 1-loop RG equations of the original theory with τ interpreted as RG time. This provides a novel example of an ‘integrability-RG flow’ connection. The existence of a Lax connection suggests that these time-dependent σ-models may themselves be understood as integrable. We investigate this question by studying the possibility of constructing non-local and local conserved charges. Such σ-models with D-dimensional target space and time-dependent couplings subject to the RG flow naturally appear in string theory upon fixing the light-cone gauge in a (D + 2)-dimensional conformal σ-model with a metric admitting a covariantly constant null Killing vector and a dilaton linear in the null coordinate.

We consider a class of 2d σ-models on products of group spaces that provide new examples of a close connection between integrability and stability under the RG flow. We first study the integrable G × G model derived from the affine Gaudin construction (for which the 1-loop β-functions were found in arXiv:2010.07879) and show that its condition of integrability is preserved also by the 2-loop RG flow. We then investigate the RG flow in the gauged G × G/H model, in particular the integrable T1,1 model found in arXiv:2010.05573. We also construct a new class of integrable G × G/H models in the case when the subgroup H is abelian. In the simplest case of G = SU2, H = U1 this leads to an integrable σ-model on the T1,q space (with a particular B-field). This model is also shown to be stable under the 2-loop RG flow, and we relate this property to its invariance under T-duality in an isometric U1 direction. This T1,q model may be interpreted as an integrable deformation of the GMM model (of two coupled WZW theories with generic levels) away from the conformal point.