Communications in Mathematical Physics最新文献

In this paper, we study the regularity of the Lyapunov exponent for quasiperiodic Schrödinger cocycles with (C^2) cos-type potentials, large coupling constants, and a fixed Diophantine frequency. We obtain the absolute continuity of the Lyapunov exponent. Moreover, we prove the Lyapunov exponent is (frac{1}{2})-Hölder continuous. Furthermore, for any given (rin (frac{1}{2}, 1)), we can find some energy in the spectrum where the local regularity of the Lyapunov exponent is between ((r-epsilon ))-Hölder continuity and ((r+epsilon ))-Hölder continuity.

We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani,

and establish local existence and uniqueness of smooth solutions in the scale of Sobolev spaces with exponent decreasing with time. Such a decrease of the Sobolev exponent is necessary, as we have shown in the companion paper (Chae et al. in Illposedness via degenerate dispersion for generalized surface quasi-geostrophic equations with singular velocities, arXiv:2308.02120) that the problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence of the Sobolev exponent can be removed when there is a dissipation term strictly stronger than log. These results improve wellposedness statements by Chae et al. (Comm Pure Appl Math 65(8):1037–1066, 2012). This well-posedness result can be applied to describe the long-time dynamics of the (delta )-SQG equations, defined by

for all sufficiently small (delta >0) depending on the size of the initial data. For the same range of (delta ), we establish global well-posedness of smooth solutions to the dissipative SQG equations.

A recent development in our understanding of the theory of quantum fields is the fact that familiar gauge theories in spacetime dimensions greater than two can have non-invertible symmetries generated by topological defects. The hallmark of these non-invertible symmetries is that the fusion rule deviates from the usual group-like structure, and in particular the fusion coefficients take values in topological field theories (TFTs) rather than in mere numbers. In this paper we begin an exploration of the associativity structure of non-invertible symmetries in higher dimensions. The first layer of associativity is captured by F-symbols, which we find to assume values in TFTs that have one dimension lower than that of the defect. We undertake an explicit analysis of the F-symbols for the non-invertible chiral symmetry that is preserved by the massless QED and explore their physical implications. In particular, we show the F-symbol TFTs can be detected by probing the correlators of topological defects with ’t Hooft lines. Furthermore, we derive the Ward–Takahashi identity that arises from the chiral symmetry on a large class of four-dimensional manifolds with non-trivial topologies directly from the topological data of the symmetry defects, without referring to a Lagrangian formulation of the theory.

For any generalized Frobenius manifold with non-flat unity, we construct a bihamiltonian integrable hierarchy of hydrodynamic type which is an analogue of the Principal Hierarchy of a Frobenius manifold. We show that such an integrable hierarchy, which we also call the Principal Hierarchy, possesses Virasoro symmetries and a tau structure, and the Virasoro symmetries can be lifted to symmetries of the tau-cover of the integrable hierarchy. We derive the loop equation from the condition of linearization of actions of the Virasoro symmetries on the tau function, and construct the topological deformation of the Principal Hierarchy of a semisimple generalized Frobenius manifold with non-flat unity. We also give two examples of generalized Frobenius manifolds with non-flat unity and show that they are closely related to the well-known integrable hierarchies: the Volterra hierarchy, the q-deformed KdV hierarchy and the Ablowitz–Ladik hierarchy.

A quantum cellular automaton (QCA) or a causal unitary is by definition an automorphism of a local operator algebra, by which local operators are mapped to nearby local operators. Quantum circuits of small depth, local Hamiltonian evolutions for short time, and translations (shifts) are examples. A Clifford QCA is one that maps any Pauli operator to a finite tensor product of Pauli operators. Here, we obtain a complete table of groups ({mathfrak {C}}({textsf{d}},p)) of translation invariant Clifford QCA in any spatial dimension ({textsf{d}}ge 0) modulo Clifford quantum circuits and shifts over prime p-dimensional qudits, where the circuits and shifts are allowed to obey only coarser translation invariance. The group ({mathfrak {C}}({textsf{d}},p)) is nonzero only for ({textsf{d}}= 2k+3) if (p=2) and ({textsf{d}}= 4k+3) if p is odd where (k ge 0) is any integer, in which case ({mathfrak {C}}({textsf{d}},p) cong {widetilde{mathfrak {W}}}({mathbb {F}}_p)), the classical Witt group of nonsingular quadratic forms over the finite field ({mathbb {F}}_p). It is well known that ({widetilde{mathfrak {W}}}({mathbb {F}}_2) cong {mathbb {Z}}/2{mathbb {Z}}), ({widetilde{mathfrak {W}}}({mathbb {F}}_p) cong {mathbb {Z}}/4{mathbb {Z}}) if (p = 3 bmod 4), and ({widetilde{mathfrak {W}}}({mathbb {F}}_p)cong {mathbb {Z}}/2{mathbb {Z}}oplus {mathbb {Z}}/2{mathbb {Z}}) if (p = 1 bmod 4). The classification is achieved by a dimensional descent, which is a reduction of Laurent extension theorems for algebraic L-groups of surgery theory in topology.

In the present paper, we study the asymptotics of the Fredholm determinant D(x, s) of the finite-temperature deformation of the sine kernel, which represents the probability that there are no particles in the interval ((-x/pi ,x/pi )) in the bulk scaling limit of the finite-temperature fermion system. The variable s in D(x, s) is related to the temperature. This determinant also corresponds to the finite-temperature correlation function of the one-dimensional Bose gas. We derive the asymptotics of D(x, s) in several different regimes in the (x, s)-plane. A third-order phase transition is observed in the asymptotic expansions as both x and s tend to positive infinity at certain related speed. The phase transition is then shown to be described by an integral involving the Hastings–McLeod solution of the second Painlevé equation.

We provide an explicit expression for the modular hamiltonian of the von Neumann algebras associated to the unit double cone for the (fermionic) quantum field theories of the 2-component Weyl (helicity 1/2) field, and of the 4-component massless Dirac and Majorana fields. To this end, we represent the one particle spaces of these theories in terms of solutions of the corresponding wave equations, and obtain the action of the modular group on them. As an application, we compute the relative entropy between the vacuum of the massless Majorana field and one particle states associated to waves with Cauchy data localized in the spatial unit ball.

The double copy relationship between Yang–Mills theory and general relativity can be stated in terms of a field theory Kawai–Lewellen–Tye (KLT) momentum kernel, which maps two colour-ordered gluon amplitudes to a graviton amplitude at tree-level. These amplitudes can also be written in compact, helicity-graded representations on twistor space which include the famous Parke–Taylor and Hodges formulae in the maximal helicity violating sector. However, a double copy formulation of these helicity-graded formulae has proved elusive. In this paper, we use graph-theoretic methods to obtain an explicit double copy representation of the tree-level, helicity graded S-matrix of general relativity in terms of a KLT-like integral kernel in twistor space. This integral kernel glues together two colour-ordered integrands for tree-level gluon scattering on twistor space to produce tree-level graviton amplitudes, and admits a chiral splitting into positive and negative helicity degrees of freedom. Furthermore, the kernel can be inverted to obtain a new formula for the tree-level S-matrix of biadjoint scalar theory, which we verify using recursion relations. We also derive extensions of this integral kernel to graviton scattering in anti-de Sitter space and self-dual radiative spacetimes, commenting on their potential double copy interpretations.

We construct highest weight vectors of (widehat{mathfrak {sl}_2}_{,k+1} oplus textsf{Vir}) in tensor products of highest weight modules of (widehat{mathfrak {sl}_2}_{,k}) and (widehat{mathfrak {sl}_2}_{,1}), and thus for generic weights we find the decomposition of the tensor product into irreducibles of (widehat{mathfrak {sl}_2}_{k+1} oplus textsf{Vir}). The construction uses Wakimoto representations of (widehat{mathfrak {sl}_2}_{,k}), but the obtained vectors can be mapped back to Verma modules. Singularities of this mapping are cancelled by a renormalization. A detailed study of “degenerations” of Wakimoto modules allowed to find the renormalization factor explicitly. The obtained result is a significant step forward in a proof of equivalence of certain two-dimensional CFT models.

In this article we study r-neutralized local entropy and derive some entropy formulas. For an ergodic hyperbolic measure of a smooth system, we show that the r-neutralized local entropy equals the Brin-Katok local entropy plus r times the pointwise dimension of the measure. We further establish the existence of ergodic measures that maximize the r-neutralized entropy for certain hyperbolic systems. Moreover, we construct a uniformly hyperbolic system, for which such measures are not unique. Finally, we present some rigidity results related to these ergodic measures.