Universal 1-loop divergences for integrable sigma models

Integrable string sigma models on AdS3 backgrounds with 16 supersymmetries have the distinguishing feature that their superisometry group is a direct product. As a result the deformation theory of these models is particularly rich since the two supergroups in the product can be deformed independently. We construct bi-η and bi-λ deformations of two classes of ℤ4 permutation supercoset sigma models, which describe sectors of the Green-Schwarz and pure-spinor string worldsheet theories on type II AdS3 backgrounds with pure R-R flux. We discuss an important limit of these models when one supergroup is undeformed. The associated deformed supergravity background should preserve 8 supersymmetries and is expected to have better properties than the full bi-deformation. As a step towards investigating the quantum properties of these models, we study the two-loop RG flow of the bosonic truncation of the bi-λ deformation.

Bi-η and bi-λ deformations of ℤ4 permutation supercosets

Hoare

Levine

Seibold

2023

An elliptic integrable deformation of the Principal Chiral Model

Lacroix,

Wallberg

2024

We introduce a new elliptic integrable σ-model in the form of a two-parameter deformation of the Principal Chiral Model on the group SLℝ(N), generalising a construction of Cherednik for N = 2 (up to reality conditions). We exhibit the Lax connection and $$ \mathcal{R} $$ R -matrix of this theory, which depend meromorphically on a spectral parameter valued in the torus. Furthermore, we explain the origin of this model from an equivariant semi-holomorphic 4-dimensional Chern-Simons theory on the torus. This approach opens the way for the construction of a large class of elliptic integrable σ-models, with the deformed Principal Chiral Model as the simplest example.

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

Lacroix,

Wallberg

2024

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 σ-models. These models depend on the data of a Riemann surface C (here of genus 0 or 1) and of a meromorphic 1-form ω 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 ω. These conjectures were put forward in [1] and [2] and were proven in a variety of cases. After extending the proposal of [1] 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 $${{\text{SL}}}_{\mathbb{R}}\left(N\right)$$.