Letters in Mathematical Physics最新文献

We investigate the long-time asymptotics of the solution to the Cauchy problem for the nonlocal PT-symmetric derivative nonlinear Schrödinger equation (nPT-DNLS)

where (q_{0}(x)) belongs to the Schwartz class (mathcal {S}(R)) with the assumption (Vert q_0(x)Vert _{L^1(mathbb {R})}<0.817).Beginning with the Lax pair, we define the corresponding Jost functions and scattering data, then formulate the Riemann–Hilbert problem related to the solution. Due to nonlocal effects, we introduce two distinct reflection coefficients (r_1(lambda )) and (r_2(lambda )) to address the broken space-inversion symmetry. Then we adapt the Deift–Zhou nonlinear steepest-descent method to analyze the long-time asymptotics for the solution of the nPT-DNLS equation. Note that our main results are presented in two cases, corresponding to the distinct positions of the stationary phase points.In contrast with the local derivative nonlinear Schrödinger equation, we have some different results on the decay rate of the leading asymptotic term for the nPT-DNLS equation.

In this note, we establish mass–capacity inequalities for asymptotically flat manifolds whose boundary capacity potential satisfies an overdetermined boundary condition—so-called critical area-normalized capacitors. As a consequence, we derive uniqueness results for the Schwarzschild metric and sharpen uniqueness theorems for spin asymptotically flat static vacuum spacetimes with a connected photon surface, as well as for spin asymptotically flat static manifolds satisfying an overdetermined Robin-type boundary condition.

We consider the centre of the affine vertex algebra at the critical level associated with the orthosymplectic Lie superalgebra. It is well-known that the centre is a commutative superalgebra, and we construct a family of its elements in an explicit form. In particular, this gives a new proof of the formulas for the central elements for the orthogonal and symplectic Lie algebras. Our arguments rely on the properties of a new extended Brauer-type algebra.

Employing a recent technology of tree surgery, we prove a “deletion–constriction” formula for products of rooted spanning-trees on weighted directed graphs that generalizes deletion–contraction on undirected graphs. The formula implies that, letting (tau _texttt{x}^varnothing ), (tau _texttt{x}^+), and (tau _texttt{x}^-) be the rooted spanning-tree polynomials obtained, respectively, by removing both directed edges between two vertices, or by forcing the tree to pass through either edge, the vectors ((tau _texttt{x}^varnothing , tau _texttt{x}^+, tau _texttt{x}^-)) are coplanar for all roots (texttt{x}). We deploy the result to give an alternative derivation of a recently found mutual linearity of stationary currents of Markov chains. We generalize deletion–constriction and current linearity for two pairs of edges and conjecture that similar results may hold for arbitrary subsets of edges.

We prove that a wide class of random quantum channels with few Kraus operators, sampled as random matrices with some sparsity and moment assumptions, typically exhibit a large spectral gap and are therefore optimal quantum expanders. In particular, our result provides a recipe to construct random quantum expanders from their classical (random or deterministic) counterparts. This considerably enlarges the list of known constructions of optimal quantum expanders, which was previously limited to few examples. Our proofs rely on recent progress in the study of the operator norm of random matrices with dependence and non-homogeneity, which we expect to have further applications in several areas of quantum information.

Refined BPS indices give rise to a quantum Riemann–Hilbert problem that is inherently related to a non-commutative deformation of moduli spaces arising in gauge and string theory compactifications. We reformulate this problem in terms of a non-commutative deformation of a TBA-like equation and obtain its formal solution as an expansion in refined indices. As an application of this construction, we derive a generating function of solutions of the TBA equation in the unrefined case.

We explore the relation of the super Macdonald polynomials and the BPS state counting on the blow-up of (mathbb {P}^2), which is mathematically described by framed stable perverse coherent sheaves. Fixed points of the torus action on the moduli space of BPS states are labeled by super partitions. From the equivariant character of the tangent space at the fixed points we can define the Nekrasov factor for a pair of super partitions, which is used for the localization computation of the partition function. The Nekrasov factor also allows us to compute matrix elements of the action of the quantum toroidal algebra of type (mathfrak {gl}_{1|1}) on the K group of the moduli space. We confirm that these matrix elements are consistent with the Pieri rule of the super Macdonald polynomials.

Drinfeld classified Poisson homogeneous spaces of a Poisson Lie group in terms of Dirac structures of the Lie bialgebra. In this paper, we study homogeneous spaces of a 2-group and develop Drinfeld theorem in the Poisson 2-group context.

Recently, a set of q-series invariants, labeled by (operatorname {Spin}^c) structures, for weakly negative definite plumbed 3-manifolds called the (widehat{Z}_a) invariants were discovered by Gukov, Pei, Putrov and Vafa. The leading rational power of the (widehat{Z}_a) invariants are invariants themselves denoted by (Delta _a). In this paper, we further analyze the structure of these (Delta _a) invariants. We review some of the foundations of the (Delta _a) invariants and analyze their structure for a subclass of integer homology spheres. In particular, we provide a complete description of the (Delta _0) invariants for Brieskorn spheres. Along the way we show that the (Delta _a) invariants are not homology cobordism invariants, thereby answering an open question in the literature.

We discuss the construction and classification of the irreducible modules and intertwining operators between the irreducible modules of a rational Lorentzian lattice vertex operator algebra (LLVOA) based on an even, self-dual Lorentzian lattice (Lambda subset mathbb {R}^{m,n}) of signature (m, n). We also discuss various equivalent characterizations of rationality of the modular invariant LLCFT and recover (and slightly generalize) some old results of Wendland. Finally, we describe the standard construction of modular tensor category (MTC) associated with rational LLCFTs. We explicitly construct the modular data and braiding and fusing matrices for the MTC. As a concrete example, we show that the LLCFT based on a certain even, self-dual Lorentzian lattice of signature (m, n), with m even, realizes the (D(m bmod 8)) level 1 Kac–Moody MTC.