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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]

A presentation of the centralizer of the threefold tensor product of the spin s representation of the quantum group U q (sl 2) is provided. It is expressed as a quotient of the Askey–Wilson braid algebra. This newly defined algebra combines the Askey–Wilson relations with the braid group relations, on three strands, together with a characteristic equation of degree 2 s + 1 for the braid generators. Explicit bases are given for the centralizer. [ABSTRACT FROM AUTHOR]

We show that four-dimensional superconformal algebras admit an infinite-dimensional derived enhancement after performing a holomorphic twist. The type of higher symmetry algebras we find is closely related to algebras studied by Faonte–Hennion–Kapranov, Hennion–Kapranov, and the second author with Gwilliam in the context of holomorphic QFT. We show that these algebras are related to the two-dimensional chiral algebras extracted from four-dimensional superconformal theories by Beem and collaborators; further deforming by a superconformal element induces the Koszul resolution of a plane in C 2 ≅ R 4 . The central charges at the level of chiral algebras arise from central extensions of the higher symmetry algebras. [ABSTRACT FROM AUTHOR]

We consider the dynamics t ↦ τ t of an infinite quantum lattice system that is generated by a local interaction. If the interaction decomposes into a finite number of terms that are themselves local interactions, we show that τ t can be efficiently approximated by a product of n automorphisms, each of them being an alternating product generated by the individual terms. For any integer m, we construct a product formula (in the spirit of Trotter) such that the approximation error scales as n - m . Our bounds hold in norm, pointwise for algebra elements that are sufficiently well approximated by finite volume observables. [ABSTRACT FROM AUTHOR]

We generalize the concepts of weak quantum logarithmic Sobolev inequality (LSI) and weak hypercontractivity (HC), introduced in the quantum setting by Olkiewicz and Zegarlinski, to the case of non-primitive quantum Markov semigroups (QMS). The originality of this work resides in that this new notion of hypercontractivity is given in terms of the so-called amalgamated L p norms introduced recently by Junge and Parcet in the context of operator spaces theory. We make three main contributions. The first one is a version of Gross' integration lemma: we prove that (weak) HC implies (weak) LSI. Surprisingly, the converse implication differs from the primitive case as we show that LSI implies HC but with a weak constant equal to the cardinal of the center of the decoherence-free algebra. Building on the first implication, our second contribution is the fact that strong LSI and therefore strong HC do not hold for non-trivially primitive QMS. This implies that the amalgamated L p norms are not uniformly convex for 1 ≤ p ≤ 2 . As a third contribution, we derive universal bounds on the (weak) logarithmic Sobolev constants for a QMS on a finite dimensional Hilbert space, using a similar method as Diaconis and Saloff-Coste in the case of classical primitive Markov chains, and Temme, Pastawski and Kastoryano in the case of primitive QMS. This leads to new bounds on the decoherence rates of decohering QMS. Additionally, we apply our results to the study of the tensorization of HC in non-commutative spaces in terms of the completely bounded norms (CB norms) recently introduced by Beigi and King for unital and trace preserving QMS. We generalize their results to the case of a general primitive QMS and provide estimates on the (weak) constants. [ABSTRACT FROM AUTHOR]

We discuss a set of heterotic and type II string theory compactifications to 1 + 1 dimensions that are characterized by factorized internal worldsheet CFTs of the form V 1 ⊗ V ¯ 2 , where V 1 , V 2 are self-dual (super) vertex operator algebras. In the cases with spacetime supersymmetry, we show that the BPS states form a module for a Borcherds–Kac–Moody (BKM) (super)algebra, and we prove that for each model the BKM (super)algebra is a symmetry of genus zero BPS string amplitudes. We compute the supersymmetric indices of these models using both Hamiltonian and path integral formalisms. The path integrals are manifestly automorphic forms closely related to the Borcherds–Weyl–Kac denominator. Along the way, we comment on various subtleties inherent to these low-dimensional string compactifications. [ABSTRACT FROM AUTHOR]

The higher rank Racah algebra R(n) introduced in De Bie et al. (J Phys A Math Theor 51(2):025203, 2017. arXiv:1610.02638) is recalled. A quotient of this algebra by central elements, which we call the special Racah algebra sR(n), is then introduced. Using the results from classical invariant theory, this sR(n) algebra is shown to be isomorphic to the centralizer Z n (s l 2) of the diagonal embedding of U (s l 2) in U (s l 2) ⊗ n . This leads to a first and novel presentation of the centralizer Z n (s l 2) in terms of generators and defining relations. An explicit formula of its Hilbert–Poincaré series is also obtained and studied. The extension of the results to the study of the special Askey–Wilson algebra and its higher rank generalizations is discussed. [ABSTRACT FROM AUTHOR]

Given an (irreducible) Möbius covariant net A , we prove a Bisognano–Wichmann theorem for its categorical extension E d associated with the braided C ∗ -tensor category Rep d (A) of dualizable (more precisely, "dualized") Möbius covariant A -modules. As a closely related result, we prove a (modified) Bisognano–Wichmann theorem for any (possibly) non-local extension of A obtained by a C ∗ -Frobenius algebra Q in Rep d (A) . As an application, we discuss the relation between the domains of modular operators and the preclosedness of certain unbounded operators in E d . [ABSTRACT FROM AUTHOR]

The centralizer of the image of the diagonal embedding of U q (sl 2) in the tensor product of three irreducible representations is examined in a Schur–Weyl duality spirit. The aim is to offer a description in terms of generators and relations. A conjecture in this respect is offered with the centralizers presented as quotients of the Askey–Wilson algebra. Support for the conjecture is provided by an examination of the representations of the quotients. The conjecture is also shown to be true in a number of cases thereby exhibiting in particular the Temperley–Lieb, Birman–Murakami–Wenzl and one-boundary Temperley–Lieb algebras as quotients of the Askey–Wilson algebra. [ABSTRACT FROM AUTHOR]

We propose an abstract framework describing energy-renormalized Hamiltonians in terms of local algebras. Within the framework, we examine the positivity improvingness of the semigroup generated by the renormalized Hamiltonian. As examples, we discuss the renormalized Nelson Hamiltonian and the renormalized Nelson Hamiltonian at fixed total momentum. The characteristic features of our approach are as follows: (i) in contrast with the probabilistic approach in the Schrödinger representation, our method works well in the Fock representation and (ii) the method covers the massless case. [ABSTRACT FROM AUTHOR]