Annales Henri Poincaré最新文献

The Aharonov–Casher theorem is a result on the number of the so-called zero modes of a system described by the magnetic Pauli operator in (mathbb {R}^2). In this paper we address the same question for the Dirac operator on a flat two-dimensional manifold with boundary and Atiyah–Patodi–Singer boundary condition. More concretely we are interested in the plane and a disc with a finite number of circular holes cut out. We consider a smooth compactly supported magnetic field on the manifold and an arbitrary magnetic field inside the holes.

The extended algebra of the free electromagnetic fields, including infrared-singular fields, and the almost radial gauge, both introduced earlier, are postulated for the construction of the quantum electrodynamics in a Hilbert space (no indefinite metric). Both the Dirac and electromagnetic fields are constructed up to the first order (based on the incoming fields) as operators in the Hilbert space and shown to have physically well-interpretable asymptotic behavior in far past and spacelike separations. The Dirac field tends in far past to the free incoming field, carrying its own Coulomb field, but with no ‘soft photon dressing.’ The spacelike asymptotic limit of the electromagnetic field yields a conserved operator field, which is a sum of contributions of the incoming Coulomb field, and of the low-energy limit of the incoming free electromagnetic field. This should agree with the operator field similarly constructed with the use of outgoing fields, which then relates these past and future characteristics. Higher orders are expected not to change this picture, but their construction needs a treatment of the UV question, which has not been undertaken and remains a problem for further investigation.

Classical dynamical r-matrices arise naturally in the combinatorial description of the phase space of Chern–Simons theories, either through the inclusion of dynamical sources or through a gauge fixing procedure involving two punctures. Here we consider classical dynamical r-matrices for the family of Lie algebras which arise in the Chern–Simons formulation of 3d gravity, for any value of the cosmological constant. We derive differential equations for classical dynamical r-matrices in this case and show that they can be viewed as generalized complexifications, in a sense which we define, of the equations governing dynamical r-matrices for (mathfrak {su}(2)) and (mathfrak {sl}(2,{mathbb {R}})). We obtain explicit families of solutions and relate them, via Weierstrass factorization, to solutions found by Feher, Gabor, Marshall, Palla and Pusztai in the context of chiral WZWN models.

In this note, we consider the asymmetric nearest neighbor ferromagnetic Ising model on the ((d+s))-dimensional unit cubic lattice ({mathbb {Z}}^{d+s}), at inverse temperature (beta =1) and with coupling constants (J_s>0) and (J_d>0) for edges of ({mathbb {Z}}^s) and ({mathbb {Z}}^d), respectively. We obtain a lower bound for the critical curve in the phase diagram of ((J_s,J_d)). In particular, as (J_d) approaches its critical value from below, our result is directly related to the so-called dimensional crossover phenomenon.

This paper extends the Bakry-Émery criterion relating the Ricci curvature and logarithmic Sobolev inequalities to the noncommutative setting. We obtain easily computable complete modified logarithmic Sobolev inequalities of graph Laplacians and Lindblad operators of the corresponding graph Hörmander systems. We develop the anti-transference principle stating that the matrix-valued modified logarithmic Sobolev inequalities of sub-Laplacian operators on a compact Lie group are equivalent to such inequalities of a family of the transferred Lindblad operators with a uniform lower bound.

Given a globally hyperbolic spacetime (M={mathbb {R}}times Sigma ) of dimension four and regularity (C^tau ), we estimate the Sobolev wavefront set of the causal propagator (K_G) of the Klein–Gordon operator. In the smooth case, the propagator satisfies (WF'(K_G)=C), where (Csubset T^*(Mtimes M)) consists of those points ((tilde{x},tilde{xi },tilde{y},tilde{eta })) such that (tilde{xi },tilde{eta }) are cotangent to a null geodesic (gamma ) at (tilde{x}) resp. (tilde{y}) and parallel transports of each other along (gamma ). We show that for (tau >2),

for every ({epsilon }>0). Furthermore, in regularity (C^{tau +2}) with (tau >2),

holds for (0<epsilon <tau +frac{1}{2}). In the ultrastatic case with (Sigma ) compact, we show (WF'^{-frac{3}{2}+tau -epsilon }(K_G)subset C) for (epsilon >0) and (tau >2) and (WF'^{-frac{3}{2}+tau -epsilon }(K_G)= C) for (tau >3) and (epsilon <tau -3). Moreover, we show that the global regularity of the propagator (K_G) is (H^{-frac{1}{2}-epsilon }_{loc}(Mtimes M)) as in the smooth case.

A discrete version of the conformal field theory of symplectic fermions is introduced and discussed. Specifically, discrete symplectic fermions are realised as holomorphic observables in the double-dimer model. Using techniques of discrete complex analysis, the space of local fields of discrete symplectic fermions on the square lattice is proven to carry a representation of the Virasoro algebra with central charge (-2).

We consider operators acting in (L^2({mathbb {R}}^d)) with (dge 3) that locally behave as a magnetic Schrödinger operator. For the magnetic Schrödinger operators, we suppose the magnetic potentials are smooth and the electric potential is five times differentiable and the fifth derivatives are Hölder continuous. Under these assumptions, we establish sharp spectral asymptotics for localised counting functions and Riesz means.

It is well established that the spectral analysis of canonically quantized four-dimensional Seiberg–Witten curves can be systematically studied via the Nekrasov–Shatashvili functions. In this paper, we explore another aspect of the relation between ({mathcal {N}}=2) supersymmetric gauge theories in four dimensions and operator theory. Specifically, we study an example of an integral operator associated with Painlevé equations and whose spectral traces are related to correlation functions of the 2d Ising model. This operator does not correspond to a canonically quantized Seiberg–Witten curve, but its kernel can nevertheless be interpreted as the density matrix of an ideal Fermi gas. Adopting the approach of Tracy and Widom, we provide an explicit expression for its eigenfunctions via an ({{,mathrm{O(2)},}}) matrix model. We then show that these eigenfunctions are computed by surface defects in ({{,mathrm{SU(2)},}}) super Yang–Mills in the self-dual phase of the (Omega )-background. Our result also yields a strong coupling expression for such defects which resums the instanton expansion. Even though we focus on one concrete example, we expect these results to hold for a larger class of operators arising in the context of isomonodromic deformation equations.