Communications in Mathematical Physics最新文献

One of the features of Baxter’s Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang–Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang–Baxter equation) associated to these representations. In the case of quantum affine (mathfrak {sl}_2) and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.

Motivated by an extension to Gallavotti–Cohen Chaotic Hypothesis, we study local and global existence of a gradient flow of the Sinai–Ruelle–Bowen entropy functional in the space of transitive Anosov maps. For the space of expanding maps on the unit circle, we equip it with a Hilbert manifold structure using a Sobolev norm in the tangent space of the manifold. Under the additional measure-preserving assumption and a slightly modified metric, we show that the gradient flow exists globally and every trajectory of the flow converges to a unique limiting map where the SRB entropy attains the maximal value. In a simple case, we obtain an explicit formula for the flow’s ordinary differential equation representation. This gradient flow has close connection to a nonlinear partial differential equation, a gradient-dependent diffusion equation.

We study Jánossy densities of a randomly thinned Airy kernel determinantal point process. We prove that they can be expressed in terms of solutions to the Stark and cylindrical Korteweg–de Vries equations; these solutions are Darboux tranformations of the simpler ones related to the gap probability of the same thinned Airy point process. Moreover, we prove that the associated wave functions satisfy a variation of Amir–Corwin–Quastel’s integro-differential Painlevé II equation. Finally, we derive tail asymptotics for the relevant solutions to the cylindrical Korteweg–de Vries equation and show that they decompose asymptotically into a superposition of simpler solutions.

We define a new gauge independent quasi-local mass and energy, and show its relation to the Brown–York Hamilton–Jacobi analysis. A quasi-local proof of the positivity, based on spacetime harmonic functions, is given for admissible closed spacelike 2-surfaces which enclose an initial data set satisfying the dominant energy condition. Like the Wang-Yau mass, the new definition relies on isometric embeddings into Minkowski space, although our notion of admissibility is different from that of Wang and Yau. Rigidity is also established, in that vanishing energy implies that the 2-surface arises from an embedding into Minkowski space, and conversely the mass vanishes for any such surface. Furthermore, we show convergence to the ADM mass at spatial infinity, and provide the equation associated with optimal isometric embedding.

We start from the remark that in wave turbulence theory, exemplified by the cubic two-dimensional Schrödinger equation (NLS) on the real plane, the regularity of the resonant manifold is linked with dispersive properties of the equation and thus with scattering phenomena. In contrast with classical analysis starting with a dynamics on a large periodic box, we propose to study NLS set on the real plane using the dispersive effects, by considering the time evolution operator in various time scales for deterministic and random initial data. By considering periodic functions embedded in the whole space by gaussian truncation, this allows explicit calculations and we identify two different regimes where the operators converges towards the kinetic operator but with different form of convergence.

The spectrum of BPS states in type IIA string theory compactified on a Calabi–Yau threefold famously jumps across codimension-one walls in complexified Kähler moduli space, leading to an intricate chamber structure. The Split Attractor Flow Conjecture posits that the BPS index (Omega _z(gamma )) for given charge (gamma ) and moduli z can be reconstructed from the attractor indices (Omega _star (gamma _i)) counting BPS states of charge (gamma _i) in their respective attractor chamber, by summing over a finite set of decorated rooted flow trees known as attractor flow trees. If correct, this provides a classification (or dendroscopy) of the BPS spectrum into different topologies of nested BPS bound states, each having a simple chamber structure. Here we investigate this conjecture for the simplest, albeit non-compact, Calabi–Yau threefold, namely the canonical bundle over (mathbb {P}^2). Since the Kähler moduli space has complex dimension one and the attractor flow preserves the argument of the central charge, attractor flow trees coincide with scattering sequences of rays in a two-dimensional slice of the scattering diagram ({mathcal {D}}_psi ) in the space of stability conditions on the derived category of compactly supported coherent sheaves on (K_{mathbb {P}^2}). We combine previous results on the scattering diagram of (K_{mathbb {P}^2}) in the large volume slice with an analysis of the scattering diagram for the three-node quiver valid in the vicinity of the orbifold point (mathbb {C}^3/mathbb {Z}_3), and prove that the Split Attractor Flow Conjecture holds true on the physical slice of (Pi )-stability conditions. In particular, while there is an infinite set of initial rays related by the group (Gamma _1(3)) of auto-equivalences, only a finite number of possible decompositions (gamma =sum _i gamma _i) contribute to the index (Omega _z(gamma )) for any (gamma ) and z, with constituents (gamma _i) related by spectral flow to the fractional branes at the orbifold point. We further explain the absence of jumps in the index between the orbifold and large volume points for normalized torsion free sheaves, and uncover new ‘fake walls’ across which the dendroscopic structure changes but the total index remains constant.