Gaudin models and multipoint conformal blocks. Part II. Comb channel vertices in 3D and 4D

It was recently shown that multi-point conformal blocks in higher dimensional conformal field theory can be considered as joint eigenfunctions for a system of commuting differential operators. The latter arise as Hamiltonians of a Gaudin integrable system. In this work we address the reduced fourth order differential operators that measure the choice of 3-point tensor structures for all vertices of 3- and 4-dimensional comb channel conformal blocks. These vertices come associated with a single cross ratio. Remarkably, we identify the vertex operators as Hamiltonians of a crystallographic elliptic Calogero-Moser-Sutherland model that was discovered originally by Etingof, Felder, Ma and Veselov. Our construction is based on a further development of the embedding space formalism for mixed-symmetry tensor fields. The results thereby also apply to comb channel vertices of 5- and 6-point functions in arbitrary dimension.

Gaudin models and multipoint cnformal blocks: general theory

The construction of conformal blocks for the analysis of multipoint correlation functions with N > 4 local field insertions is an important open problem in higher dimensional conformal field theory. This is the first in a series of papers in which we address this challenge, following and extending our short announcement in [1]. According to Dolan and Osborn, conformal blocks can be determined from the set of differential eigenvalue equations that they satisfy. We construct a complete set of commuting differential operators that characterize multipoint conformal blocks for any number N of points in any dimension and for any choice of OPE channel through the relation with Gaudin integrable models we uncovered in [1]. For 5-point conformal blocks, there exist five such operators which are worked out smoothly in the dimension d.

Integrable Kondo problems

We discuss the integrability and wall-crossing properties of Kondo problems, where an 1d impurity is coupled to a 2d chiral CFT and triggers a defect RG flow. We review several new and old examples inspired by constructions in four-dimensional Chern-Simons theory and by affine Gaudin models.

New integrable coset sigma models

By using the general framework of affine Gaudin models, we construct a new class of integrable sigma models. They are defined on a coset of the direct product of N copies of a Lie group over some diagonal subgroup and they depend on 3N − 2 free parameters. For N = 1 the corresponding model coincides with the well-known symmetric space sigma model. Starting from the Hamiltonian formulation, we derive the Lagrangian for the N = 2 case and show that it admits a remarkably simple form in terms of the classical ℛ-matrix underlying the integrability of these models. We conjecture that a similar form of the Lagrangian holds for arbitrary N. Specifying our general construction to the case of SU(2) and N = 2, and eliminating one of the parameters, we find a new three-parametric integrable model with the manifold T1,1 as its target space. We further comment on the connection of our results with those existing in the literature.

4D Chern–Simons theory and affine Gaudin models

We relate two formalisms recently proposed for describing classical integrable field theories. The first (Costello and Yamazaki in Gauge Theory and Integrability, III, 2019) is based on the action of four-dimensional Chern–Simons theory introduced and studied by Costello, Witten and Yamazaki. The second (Costello and Yamazaki, in Gauge Theory and Integrability, III, 2017) makes use of classical generalised Gaudin models associated with untwisted affine Kac–Moody algebras.

Perfect Integrability and Gaudin Models

We suggest the notion of perfect integrability for quantum spin chains and conjecture that quantum spin chains are perfectly integrable. We show the perfect integrability for Gaudin models associated to simple Lie algebras of all finite types, with periodic and regular quasi-periodic boundary conditions.