Communications in Mathematical Physics最新文献

We prove that every unitary invertible quantum field theory satisfies a generalization of the famous spin statistics theorem. To formulate this extension, we define a higher spin action of the stable orthogonal group O on appropriate spacetime manifolds, which extends both the reflection involution and spin flip. On the algebraic side, we define a higher statistics action of O on the universal target for invertible field theories, (Imathbb {Z}), which extends both complex conjugation and fermion parity ((-1)^F). We prove that every unitary invertible quantum field theory intertwines these actions.

We give a rigorous derivation of the Hartree equation for the many-body dynamics of pseudo-relativistic Fermi systems at high density (varrho gg 1), on arbitrarily large domains, at zero temperature. With respect to previous works, we show that the many-body evolution can be approximated by the Hartree dynamics locally, proving convergence of the expectation of observables that are supported in regions with fixed volume, independent of (varrho ). The result applies to initial data describing fermionic systems at equilibrium confined in arbitrarily large domains, under the assumption that a suitable local Weyl-type estimate holds true. The proof relies on the approximation of the initial data through positive temperature quasi-free states, that satisfy strong local semiclassical bounds, which play a key role in controlling the growth of the local excitations of the quasi-free state along the many-body dynamics.

In the theory of renormalization for classical dynamical systems, e.g. unimodal maps and critical circle maps, topological conjugacy classes are stable manifolds of renormalization and unstable manifolds of renormalization are full families of minimal dimension. On the other hand, physically more realistic systems may exhibit renormalization phenomena which are surprisingly different when compared with the classical theory. In phase space one observes the coexistence phenomenon, i.e. even for bounded combinatorial type there are systems whose attractor has bounded geometry but which are topologically conjugate to systems whose attractor has degenerate geometry. In parameter space there is dimensional discrepancy at the renormalization fixed point, i.e. the unstable manifold of the renormalization fixed point contains a strong unstable manifold which is a full family of minimal dimension but the whole unstable manifold has a strictly larger dimension.

We show that the 1D Hou–Luo model on the real line admits exact self-similar finite-time blowup solutions with smooth self-similar profiles. The existence of these profiles is established via a fixed-point method that is purely analytic. We also prove that the profiles satisfy some monotonicity and convexity properties that were unknown before, and we give rigorous estimates on the algebraic decay rates of the profiles in the far field. Our result supplements the previous computer-assisted proof of self-similar finite-time blowup for the Hou–Luo model with finer characterizations of the profiles.

We investigate the Navier–Stokes–Cattaneo–Christov (NSC) system in (mathbb {R}^d) ((dge 3)), a model of heat-conductive compressible flows serving as a finite speed of propagation approximation of the Navier–Stokes–Fourier (NSF) system. Due to the presence of Oldroyd’s upper-convected derivatives, the system (NSC) exhibits a lack of hyperbolicity which makes it challenging to establish its well-posedness, especially in multi-dimensional contexts. In this paper, within a critical regularity functional framework, we prove the global-in-time well-posedness of (NSC) for initial data that are small perturbations of constant equilibria, uniformly with respect to the approximation parameter (varepsilon >0). Then, building upon this result, we obtain the sharp large-time asymptotic behaviour of (NSC) and, for all time (t>0), we derive quantitative error estimates between the solutions of (NSC) and (NSF). To the best of our knowledge, our work provides the first strong convergence result for this relaxation procedure in the three-dimensional setting and for ill-prepared data. The (NSC) system is partially dissipative and incorporates both partial diffusion and partial damping mechanisms. To address these aspects and ensure the large-time stability of the solutions, we construct localized-in-frequency perturbed energy functionals based on the hypocoercivity theory. More precisely, our analysis relies on partitioning the frequency space into three distinct regimes: low, medium and high frequencies. Within each frequency regime, we introduce effective unknowns and Lyapunov functionals, revealing the spectrally expected dissipative structures.

We investigate the hydrostatic approximation for inviscid stratified fluids, described by the two-dimensional Euler–Boussinesq equations in a periodic channel. Through a perturbative analysis of the hydrostatic homogeneous setting, we exhibit a stratified steady state violating the Miles-Howard criterion and generating a growing mode, both for the linearized hydrostatic and non-hydrostatic equations. By leveraging long-wave nonlinear instability for the original Euler–Boussinesq system, we demonstrate the breakdown of the hydrostatic limit around such unstable profiles. Finally, we establish the generic nonlinear ill-posedness of the limiting hydrostatic system in Sobolev spaces.

We define F-topological recursion (F-TR) as a non-symmetric version of topological recursion, which associates a vector potential to some initial data. We describe the symmetries of the initial data for F-TR and show that, at the level of the vector potential, they include the F-Givental (non-linear) symmetries studied by Arsie, Buryak, Lorenzoni, and Rossi within the framework of F-manifolds. Additionally, we propose a spectral curve formulation of F-topological recursion. This allows us to extend the correspondence between semisimple cohomological field theories (CohFTs) and topological recursion, as established by Dunin-Barkowski, Orantin, Shadrin, and Spitz, to the F-world. In the absence of a full reconstruction theorem à la Teleman for F-CohFTs, this demonstrates that F-TR holds for the ancestor vector potential of a given F-CohFT if and only if it holds for some F-CohFT in its F-Givental orbit. We turn this into a useful statement by showing that the correlation functions of F-topological field theories (F-CohFTs of cohomological degree 0) are governed by F-TR. We apply these results to the extended 2-spin F-CohFT. Furthermore, we exhibit a large set of linear symmetries of F-CohFTs, which do not commute with the F-Givental action.

We prove that there exists a diffusion process whose invariant measure is the three-dimensional polymer measure (nu _lambda ) for all (lambda >0). We follow in part a previous incomplete unpublished work of the first named author with M. Röckner and X. Y. Zhou (Stochastic quantization of the three-dimensional polymer measure, 1996). For the construction of (nu _lambda ) we rely on previous work by J. Westwater, E. Bolthausen and X.Y. Zhou. Using (nu _lambda ), the diffusion is constructed by means of the theory of Dirichlet forms on infinite-dimensional state spaces. The closability of the appropriate pre-Dirichlet form which is of gradient type is proven, by using a general closability result by the first named author and Röckner (Probab Theory Related Fields 83(3):405–434, 1989). This result does not require an integration by parts formula (which does not even hold for the two-dimensional polymer measure (nu _lambda )) but requires the quasi-invariance of (nu _lambda ) along a basis of vectors in the classical Cameron-Martin space such that the Radon-Nikodym derivatives have versions which form a continuous process.

We study microscopic observables of the Discrete Gaussian model (i.e., the Gaussian free field restricted to take integer values) at high temperature using the renormalisation group method. In particular, we show the central limit theorem for the two-point function of the Discrete Gaussian model by computing the asymptotic of the moment generating function (big langle e^{{mathcalligra {z}}(sigma (0) - sigma (y))} big rangle _{beta , {mathbb {Z}}^2}^{operatorname {DG}}) for ({mathcalligra {z}}in {mathbb {C}}) sufficiently small. The method we use has direct connection with the multi-scale polymer expansion used in Bauerschmidt et al. (Ann Probab 52(4):1253–1359, 2024, Ann Probab 52(4):1360–1398, 2024), where it was used to study the scaling limit of the Discrete Gaussian model. The method also applies to multi-point functions and lattice models of sine-Gordon type studied in Fröhlich and Spencer (Commun Math Phys 81(4): 527–602, 1981).