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In the first part of the paper, we propose a different viewpoint on the theory of higher Airy structures (or Airy ideals), which may shed light on its origin. We define Airy ideals in the ħ -adic completion of the Rees Weyl algebra and show that Airy ideals are defined exactly such that they are always related to the canonical left ideal generated by derivatives by automorphisms of the Rees Weyl algebra of a simple type, which we call transvections. The standard existence and uniqueness result in the theory of Airy structures then follow immediately. In the second part of the paper, we construct Airy ideals generated by the nonnegative modes of the strong generators of the principal W -algebra of sp 2 N at level - N - 1 / 2 , following the approach developed in Borot et al. (Mem Am Math Soc, 2021). This provides an example of an Airy ideal in the Heisenberg algebra that requires realizing the zero modes as derivatives instead of variables, which leads to an interesting interpretation for the resulting partition function. [ABSTRACT FROM AUTHOR]

The goal of this paper is to make first steps towards the quantisation of integrable nonlinear sigma models using the formalism of affine Gaudin models, by approaching these theories through their conformal limits. We focus mostly on the example of the Klimčík model, which is a two-parameter deformation of the principal chiral model on a Lie group G. We show that the UV fixed point of this theory is described classically by two decoupled chiral affine Gaudin models, encoding its left- and right-moving degrees of freedom, and give a detailed analysis of the chiral and integrable structures of these models. Their quantisation is then explored within the framework of Feigin and Frenkel. We study the quantum local integrals of motion using the formalism of quantised affine Gaudin models and show agreement of the first two integrals with known results in the literature for G = SU (2) . Evidence is given for the existence of a monodromy matrix satisfying the Yang–Baxter algebra for this model, thus paving the way for the quantisation of the non-local integrals of motion. We conclude with various perspectives, including on generalisations of this programme to a larger class of integrable sigma models and applications of the ODE/IQFT correspondence to the description of their quantum spectrum. [ABSTRACT FROM AUTHOR]

In the TQFT formalism of Moore–Tachikawa for describing Higgs branches of theories of class S , the space associated to the unpunctured sphere in type g is the universal centraliser Z G , where g = L i e (G) . In more physical terms, this space arises as the Coulomb branch of pure N = 4 gauge theory in three dimensions with gauge group G ˇ , the Langlands dual. In the analogous formalism for describing chiral algebras of class S , the vertex algebra associated to the sphere has been dubbed the chiral universal centraliser. In this paper, we construct an open, symplectic embedding from a cover of the Kostant–Toda lattice of type g to the universal centraliser of G—extending a classic result of Kostant. Using this embedding and some observations on the Poisson algebraic structure of Z G , we propose a free field realisation of the chiral universal centraliser for any simple group G. We exploit this realisation to develop free field realisations of chiral algebras of class S of type a 1 for theories of genus zero with n = 1 , ... , 6 punctures. These realisations make generalised S-duality completely manifest, and the generalisation to n ⩾ 7 punctures is conceptually clear, though technically burdensome. [ABSTRACT FROM AUTHOR]

We continue the study of fuzzy geometries inside Connes' spectral formalism and their relation to multimatrix models. In this companion paper to Pérez-Sánchez (Ann Henri Poincaré 22:3095–3148, 2021, arXiv:2007.10914), we propose a gauge theory setting based on noncommutative geometry, which—just as the traditional formulation in terms of almost-commutative manifolds—has the ability to also accommodate a Higgs field. However, in contrast to 'almost-commutative manifolds', the present framework, which we call gauge matrix spectral triples, employs only finite-dimensional algebras. In a path-integral quantization approach to the Spectral Action, this allows to state Yang–Mills–Higgs theory (on four-dimensional Euclidean fuzzy space) as an explicit random multimatrix model obtained here, whose matrix fields exactly mirror those of the Yang–Mills–Higgs theory on a smooth manifold. [ABSTRACT FROM AUTHOR]