Integrability and RG flow in 2d sigma models

Motivated by the search for solvable string theories, we consider the problem of classifying the integrable bosonic 2d $\sigma$-models. We include non-conformal $\sigma$-models, which have historically been a good arena for discovering integrable models that were later generalized to Weyl-invariant ones. General $\sigma$-models feature a quantum RG flow, given by a 'generalized Ricci flow' of the target-space geometry. This thesis is based on the conjecture that integrable $\sigma$-models are renormalizable, or stable under the RG flow. It is widely understood that classically integrable theories are stable at the leading 1-loop order with only a few parameters running. Here we address what happens at higher-loop orders. We find that integrable $\sigma$-models generally remain RG-stable at higher-loops provided they receive a particular choice of finite counterterms, or quantum (${\alpha }^{\prime }$) corrections to the target-space geometry. We explicitly construct these quantum corrections for examples of integrable $\eta$- and $\lambda$-deformed $\sigma$-models. We then reformulate the $\lambda$-models as $\sigma$-models on a "tripled" $G\times G\times G$ configuration space, where they become automatically renormalizable due to manifest symmetries and a decoupling of some fields. We also consider the integrable $G\times G$ and $G\times G/H$ models and construct a new class of integrable $G\times G/H$ models with abelian $H$. We then present a new and different link between integrability and the RG flow in the context of $\sigma$-models with 'local couplings' depending explicitly on 2d time. Such models are naturally obtained in the light-cone gauge in string theory, pointing to the possibility of a large, new class of solvable string models.