Communications in Mathematical Physics最新文献

Quantum moduli algebras (mathcal {L}_{g,n}^{textrm{inv}}(H)) were introduced by Alekseev–Grosse–Schomerus and Buffenoir–Roche in the context of quantization of character varieties of surfaces and exist for any quasitriangular Hopf algebra H. In this paper we construct representations of (mathcal {L}_{g,n}^{textrm{inv}}(H)) on cohomology spaces (textrm{Ext}_H^m(X,M)) for all (m ge 0), where X is any H-module and M is any (mathcal {L}_{g,n}(H))-module endowed with a compatible H-module structure. As a corollary and under suitable assumptions on H, we obtain projective representations of mapping class groups of surfaces on such Ext spaces. This recovers the projective representations obtained in Lentner et al. (Springer Briefs in Mathematical Physics, 2023) from Lyubashenko theory, when the category (mathcal {C} = Htext {-mod}) is used in their construction. Other topological applications are matrix-valued invariants of knots in thickened surfaces and representations of skein algebras on Ext spaces.

We introduce the concept of gauged Lagrangian 1-forms, extending the notion of Lagrangian 1-forms to the setting of gauge theories. This general formalism is applied to a natural geometric Lagrangian 1-form on the cotangent bundle of the space of holomorphic structures on a smooth principal G-bundle (mathcal {P}) over a compact Riemann surface C of arbitrary genus g, with or without marked points, in order to gauge the symmetry group of smooth bundle automorphisms of (mathcal {P}). The resulting construction yields a multiform version of the 3d mixed BF action with so-called type A and B defects, providing a variational formulation of Hitchin’s completely integrable system over C. By passing to holomorphic local trivialisations and going partially on-shell, we obtain a unifying action for a hierarchy of Lax equations describing the Hitchin system in terms of meromorphic Lax matrices. The cases of genus 0 and 1 with marked points are treated in greater detail, producing explicit Lagrangian 1-forms for the rational Gaudin hierarchy and the elliptic Gaudin hierarchy, respectively, with the elliptic spin Calogero–Moser hierarchy arising as a special subcase.

For a class of wreath-like product groups with property (T), we describe explicitly all the embeddings between their von Neumann algebras. This allows us to provide a continuum of ICC groups with property (T) whose von Neumann algebras are pairwise non (stably) embeddable. We also give a construction of groups in this class only having inner injective homomorphisms. As an application, we obtain examples of group von Neumann algebras which admit only inner endomorphisms.

Motivated by the stellar wind ejected from the upper atmosphere (Corona) of a star, we explore a boundary problem of the two-species nonlinear relativistic Vlasov-Poisson systems in the 3D half space in the presence of a constant vertical magnetic field and strong background gravity. We allow species to have different mass and charge (as proton and electron, for example). As the main result, we construct stationary solutions and establish their nonlinear dynamical asymptotic stability in time and space.

We study thermodynamic formalism of dynamical systems with non-uniform structure. Precisely, we obtain the uniqueness of equilibrium states for a family of non-uniformly expansive flows by generalizing Climenhaga-Thompson’s orbit decomposition criteria. In particular, such family includes entropy expansive flows. Meanwhile, the essential part of the decomposition is allowed to satisfy an even weaker version of specification, namely controlled specification, thus also extends the corresponding results in Pavlov, R. (On controlled specification and uniqueness of the equilibrium state in expansive systems. Nonlinearity 32(7), 2441–2466 (2019)). Two applications of our abstract theorems are explored. Firstly, we introduce a notion of regularity condition called weak Walters condition, and study the uniqueness of measure of maximal entropy for a suspension flow with roof function satisfying such condition. Secondly, we investigate topologically transitive frame flows on rank one manifolds of nonpositive curvature, which is a group extension of nonuniformly hyperbolic flows. Under a bunched curvature condition and running a Gauss-Bonnet type of argument, we show the uniqueness of equilibrium states with respect to certain potentials.

We investigate the formation of singularities in a baby Skyrme type energy model, which describes magnetic solitons in two-dimensional ferromagnetic systems. In presence of a diverging anisotropy term, which enforces a preferred background state of the magnetization, we establish a weak compactness of its topological charge density, which converges to an atomic measure with quantized weights. We characterize the (Gamma )-limit of the energies as the total variation of this measure. In the case of lattice type energies, we first need to carefully define a notion of discrete topological charge for (mathbb {S}^2)-valued maps. We then prove a corresponding compactness and (Gamma )-convergence result, thereby bridging the discrete and continuum theories.

We prove the essential self-adjointness of the d’Alembertian (square _g), allowing a larger class of spacetimes than previously considered, including those that arise from perturbing Minkowski spacetime by gravitational radiation. We emphasize the fact, proven by Taira in related settings, that all tempered distributions u satisfying (square _g u = lambda u +f) for (lambda in mathbb {C}backslash mathbb {R}) and f Schwartz are Schwartz. The proof is fully microlocal and relatively quick given the “de,sc-” machinery recently developed by the third author.

The geometric description of incompressible hydrodynamics, as geodesic motion on the infinite-dimensional group of volume-preserving diffeomorphisms, enables notions of curvature in the study of fluids in order to study stability. Formulas for Ricci curvature are often simpler than those for sectional curvature, which typically takes both signs, but the drawback is that Ricci curvature is rarely well-defined in infinite-dimensional spaces. Here we suggest a definition of Ricci curvature in the case of two-dimensional hydrodynamics, based on the finite-dimensional Zeitlin models arising in quantization theory, which gives a natural tool for renormalization. We provide formulae for the finite-dimensional approximations and give strong numerical evidence that these converge in the infinite-dimensional limit, based in part on four new conjectured identities for Wigner 6j symbols. The suggested limiting expression for (average) Ricci curvature is surprisingly simple and demonstrates an average instability for high-frequency modes which helps explain long-term numerical observations of spherical hydrodynamics due to mixing.

In this paper, we investigate the Langevin dynamics of various lattice formulations of the Yang–Mills–Higgs model, with an inverse Yang–Mills coupling (beta ) and a Higgs parameter (kappa ). The Higgs component is either a bounded field taking values in a compact target space, or an unbounded field taking values in a vector space in which case the model also has a Higgs mass parameter m. We study the regime where ((beta ,kappa )) are small in the first case or ((beta ,kappa /m)) are small in the second case. We prove the exponential ergodicity of the dynamics on the whole lattice via functional inequalities. We establish exponential decay of correlations for a broad class of observables, namely, the infinite volume measure exhibits a strictly positive mass gap. Moreover, when the target space of the Higgs field is compact, appropriately rescaled observables exhibit factorized correlations in the large N limit . These extend the earlier results (Shen et al. in Comm Math Phys 400(2):805–851, 2023) on pure lattice Yang–Mills to the case with a coupled Higgs field. Unlike pure lattice Yang–Mills where the field is always bounded, in the case where the coupled Higgs component is unbounded, the control of its behavior is much harder and requires new techniques. Our approach involves a disintegration argument and a delicate analysis of correlations to effectively control the unbounded Higgs component.

We revisit the classification, and give explicit realisations, of unitary irreducible representations of the BMS group. As compared to McCarthy’s seminal work, we make use of a unique, Lorentz-invariant, decomposition of supermomenta into a hard and a soft piece, that we introduce and properly define, to investigate the extent to which generic representations depart from usual Poincaré particles and highlight their relations to gravitational infrared physics. We insist on making wavefunctions as explicit as possible. Similarly, we explain how branching to a Poincaré subgroup works in practice: this is physically relevant because this amounts to reading off the field content of a given BMS state in terms of a choice of gravity vacuum. In particular, we emphasise how different gravity vacua differ in their interpretation of the same BMS state, here again providing concrete examples as well as the general procedure. Finally, we demonstrate on an example that generic BMS particles are flexible enough to encode memory, as opposed to usual Poincaré particles.