Geometry of the spectral parameter and renormalisation of integrable $\sigma$-models

In the past few years, the unifying frameworks of 4-dimensional Chern-Simons theory and affine Gaudin models have allowed for the systematic construction of a large family of integrable $\sigma$-models. These models depend on the data of a Riemann surface $C$ (here of genus 0 or 1) and of a meromorphic 1-form $\omega$ on $C$, which encodes the geometry of their spectral parameter and the analytic structure of their Lax connection. The main subject of this paper is the renormalisation of these theories and in particular two conjectures describing their 1-loop RG-flow in terms of the 1-form $\omega$. These conjectures were put forward in [2010.07879] and [2106.09781] and were proven in a variety of cases. After extending the proposal of [2010.07879] to the elliptic setup (with $C$ of genus 1), we establish the equivalence of these two conjectures and discuss some of their applications. Moreover, we check their veracity on an explicit example, namely an integrable elliptic deformation of the Principal Chiral Model on $\text{SL}_N(\mathbb{R})$.