Communications in Mathematical Physics最新文献

We show that the Calabi–Yau metrics with isolated conical singularities of Hein-Sun (Publ Math de l’IHÉS, 126(1):73–130, 2017) admit polyhomogeneous expansions near their singularities. Moreover, we show that, under certain generic assumptions, natural families of smooth Calabi-Yau metrics on crepant resolutions and on polarized smoothings of conical Calabi–Yau manifolds degenerating to the initial conical Calabi-Yau metric admit polyhomogeneous expansions where the singularities are forming. The construction proceeds by performing weighted Melrose-type blow-ups and then gluing conical and scaled asymptotically conical Calabi-Yau metrics on the fibers, close to the blow-up’s front face without compromising polyhomogeneity. This yields a polyhomogeneous family of Kähler metrics that are approximately Calabi-Yau. Solving formally a complex Monge-Ampère equation, we obtain a polyhomogeneous family of Kähler metrics with Ricci potential converging rapidly to zero as the family is degenerating. We can then conclude that the corresponding family of degenerating Calabi-Yau metrics is polyhomogeneous by using a fixed point argument.

Dualities play a central role in the study of quantum spin chains, providing insight into the structure of quantum phase diagrams and phase transitions. In this work, we study categorical dualities, which are defined as bounded-spread isomorphisms between algebras of symmetry-respecting local operators on a spin chain. We consider generalized global symmetries that correspond to unitary fusion categories, which are represented by matrix-product operator algebras. A fundamental question about dualities is whether they can be extended to quantum cellular automata on the larger algebra generated by all local operators in the the unit matrix-product operator sector. For on-site representations of Hopf algebra symmetries, this larger algebra is the usual tensor product quasi-local algebra. We present a solution to the extension problem using the machinery of Doplicher–Haag–Roberts bimodules. Our solution provides a crisp categorical criterion for when an extension of a duality exists. We show that the set of possible extensions form a torsor over the invertible objects in the relevant symmetry category. As a corollary, we obtain a classification result concerning dualities in the group case.

We show that semi-arithmetic surfaces of arithmetic dimension two which admit a modular embedding have exponential growth of mean multiplicities in their length spectrum. Prior to this work large mean multiplicities were rigorously confirmed only for the length spectra of arithmetic surfaces. We also discuss the relation of the degeneracies in the length spectrum and quantization of the Hamiltonian mechanical system on the surface.

In this paper, we provide a continuum model for the fluctuations of the symmetric simple exclusion process about its hydrodynamic limit. The model is based on an approximating sequence of stochastic PDEs with nonlinear, conservative noise. In the small-noise limit, we show that the fluctuations of the solutions are to first-order the same as the fluctuations of the particle system. Furthermore, the SPDEs correctly simulate the rare events in the particle process. We prove that the solutions satisfy a zero-noise large deviations principle with rate function equal to that describing the deviations of the symmetric simple exclusion process.

For Schrödinger operators (H_V=-Delta _g+V) with critically singular potentials V on compact manifolds, we prove sharp estimates for the restriction of eigenfunctions to submanifolds. Our method refines the perturbative argument by Blair et al. (J Geom Anal 31(7):6624–6661, 2021) and enables us to deal with submanifolds of all codimensions. As applications, we obtain improved estimates on negatively curved manifolds and flat tori. In particular, we extend the uniform (L^2) restriction estimates on flat tori by Bourgain and Rudnick (Geom Funct Anal 22(4):878–937, 2012) to singular potentials.

In mirror symmetry, after the work by J. Walcher, the number of holomorphic disks with boundary on the real quintic lagrangian in a general quintic threefold is related to the periods of the mirror quintic family with boundary on two homologous rational curves, known as Deligne conics. Following the ideas of H. Movasati, we construct a quasi-affine space parametrizing such objects enhanced with a frame for the relative de Rham cohomology with boundary at the curves compatible with the mixed Hodge structure. We also compute a modular vector field attached to such a parametrization.

For any complex number b and nonzero complex number (lambda ), we construct a class of (N=1) Neveu-Schwarz algebra modules (mathcal {L}(P,V,lambda ,b)) from module P over the Weyl superalgebra and restricted module V over the positive-part subalgebra of the (N=1) Neveu-Schwarz algebra. The necessary and sufficient conditions for (mathcal {L}(P,V,lambda ,b)) to be irreducible are obtained. If such a module (mathcal {L}(P,V,lambda ,b)) is not irreducible, we also construct its submodules concretely. Then we determine the necessary and sufficient conditions for two such Neveu-Schwarz Virasoro superalgebra modules to be isomorphic.

We examine the solution of the Benjamin–Ono Cauchy problem for rational initial data in three types of double-scaling limits in which the dispersion tends to zero while simultaneously the independent variables either approach a point on one of the two branches of the caustic curve of the inviscid Burgers equation, or approach the critical point where the branches meet. The results reveal universal limiting profiles in each case that are independent of details of the initial data. We compare the results obtained with corresponding results for the Korteweg-de Vries equation found by Claeys–Grava in three papers (Claeys and Grava in Commun Math Phys 286:979–1009, 2009, Commun Pure Appl Math 63:203–232, 2010, SIAM J Math Anal 42:2132–2154, 2010). Our method is to analyze contour integrals appearing in an explicit representation of the solution of the Cauchy problem, in various limits involving coalescing saddle points.

We study the volume of rigid loop–O(n) quadrangulations with a boundary of length 2p in the non-generic critical regime, for all (nin (0,2]). We prove that, as the half-perimeter p goes to infinity, the volume scales in distribution to an explicit random variable. This limiting random variable is described in terms of the multiplicative cascades of Chen et al. (Ann Inst Henri Poincaré D 7(4):535–584, 2020), or alternatively (in the dilute case) as the law of the area of a unit-boundary (gamma )–quantum disc, as determined by Ang and Gwynne (Ann Inst Henri Poincaré D 57(1): 1–53, 2021), for suitable (gamma ). Our arguments go through a classification of the map into several regions, where we rule out the contribution of bad regions to be left with a tractable portion of the map. One key observable for this classification is a Markov chain which explores the nested loops around a size-biased vertex pick in the map, making explicit the spinal structure of the discrete multiplicative cascade. We stress that our techniques enable us to include the boundary case (n=2), that we define rigorously, and where the nested cascade structure is that of a critical branching random walk. In that case the scaling limit is given by the limit of the derivative martingale and is inverse-exponentially distributed, which answers a conjecture of Aïdékon and Da Silva (Probab Theory Relat Fields 183(1):125–166, 2022).

Understanding the statistics of collisions among locally confined gas particles poses a major challenge. In this work we investigate (mathbb {Z}^d)-map lattices coupled by collision with simplified local dynamics that offer significant insights for the above challenging problem. We obtain a first order approximation for the first collision rate at a site ({textbf{p}}^*in mathbb {Z}^d) and we prove a distributional convergence for the first collision time to an exponential, with sharp error term. Moreover, we prove that the number of collisions at site ({textbf{p}}^*) converge in distribution to a compound Poisson distributed random variable. Key to our analysis in this infinite dimensional setting is the use of transfer operators associated with the decoupled map lattice at site ({textbf{p}}^*).