I will introduce a first-order formalism for two-dimensional sigma models with the Kähler target space. I will explain how to compute the metric beta function in this approach using the conformal perturbation methods. Comparing the answer with the standard geometric background field methods we observe certain anomalies, which we later resolve with supersymmetric completion. Based on 2312.01885 and 2307.04665.

The behavior of dynamical correlation functions in one-dimensional quantum systems at zero temperature is now very well understood in terms of linear and non-linear Luttinger models. The "microscopic" justification of these models consists in exactly accounting for the soft-mode excitations around the vacuum state and at most a few high-energy excitations. At finite temperature, or more generically for finite entropy states, this direct approach is not strictly applicable due to the different structure of soft excitations. To address these issues we study the asymptotic behavior of correlation functions in one-dimensional free fermion models. On the one hand, we obtain exact answers in terms of Fredholm determinants. On the other hand, based on "microscopic" numerical resummations, we develop a phenomenological approach that provides results depending only on the state-dependent dressing of the scattering phase. Our main example will be the sine-kernel and correlation functions in XY model. -------------------- Part of the London Integrability Journal Club. New participants please register using the form at integrability-london.weebly.com.