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In the past few years, the unifying frameworks of 4-dimensional Chern-Simonstheory and affine Gaudin models have allowed for the systematic construction ofa large family of integrable $\sigma$-models. These models depend on the dataof 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 theanalytic structure of their Lax connection. The main subject of this paper isthe renormalisation of these theories and in particular two conjecturesdescribing their 1-loop RG-flow in terms of the 1-form $\omega$. Theseconjectures were put forward in [2010.07879] and [2106.09781] and were provenin a variety of cases. After extending the proposal of [2010.07879] to theelliptic setup (with $C$ of genus 1), we establish the equivalence of these twoconjectures and discuss some of their applications. Moreover, we check theirveracity on an explicit example, namely an integrable elliptic deformation ofthe Principal Chiral Model on $\text{SL}_N(\mathbb{R})$.

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