Communications in Mathematical Physics最新文献

We develop a computer-assisted symbolic method to show that a linearized Boussinesq flow in self-similar coordinates gives rise to an invertible operator.

We give a simplified proof of the quantum null energy condition (QNEC). Our proof is based on an explicit formula for the shape derivative of the relative entropy, with respect to an entangling cut. It allows bypassing the analytic continuation arguments of a previous proof by Ceyhan and Faulkner and can be used e.g., for defining entropy current fluctuations.

The disordered ferromagnet is a disordered version of the ferromagnetic Ising model in which the coupling constants are non-negative quenched random. A ground configuration is an infinite-volume configuration whose energy cannot be reduced by finite modifications. It is a long-standing challenge to ascertain whether the disordered ferromagnet on the (mathbb {Z}^D) lattice admits non-constant ground configurations. We answer this affirmatively in dimensions (Dge 4), when the coupling constants are sampled independently from a sufficiently concentrated distribution. The obtained ground configurations are further shown to be translation-covariant with respect to (mathbb {Z}^{D-1}) translations of the disorder. Our result is proved by showing that the finite-volume interface formed by Dobrushin boundary conditions is localized, and converges to an infinite-volume interface. This may be expressed in purely combinatorial terms, as a result on the fluctuations of certain minimal cutsets in the lattice (mathbb {Z}^D) endowed with independent edge capacities.

Q-map spaces form an important class of quaternionic Kähler manifolds of negative scalar curvature. Their one-loop deformations are always inhomogeneous and have been used to construct cohomogeneity one quaternionic Kähler manifolds as deformations of homogeneous spaces. Here we study the group of isometries in the deformed case. Our main result is the statement that it always contains a semidirect product of a group of affine transformations of (mathbb {R}^{n-1}) with a Heisenberg group of dimension (2n+1) for a q-map space of dimension 4n. The affine group and its action on the normal Heisenberg factor in the semidirect product depend on the cubic affine hypersurface which encodes the q-map space.

We consider the joint system of equivariant quantum differential equations (qDE) and qKZ difference equations for the Grassmannian G(k, n), which parametrizes k-dimensional subspaces of ({{mathbb {C}}}^n). First, we establish a connection between this joint system for G(k, n) and the corresponding system for the projective space ({{mathbb {P}}}^{n-1}). Specifically, we show that, under suitable Satake identifications of the equivariant cohomologies of G(k, n) and ({{mathbb {P}}}^{n-1}), the joint system for G(k, n) is gauge equivalent to a differential-difference system on the k-th exterior power of the cohomology of ({{mathbb {P}}}^{n-1}). Secondly, we demonstrate that the Б-theorem for Grassmannians, as stated in Cotti and Varchenko (Equivariant quantum differential equation and qKZ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-Theorem, Integrability, quantization, and geometry: dedicated to the memory of Boris Dubrovin, 1950, 2019, 2021), Tarasov and Varchenko (J Geom Phys 184:104711, 2023), is compatible with the Satake identification. This implies that the Б-theorem for ({{mathbb {P}}}^{n-1}) extends to G(k, n) through the Satake identification. As a consequence, we derive determinantal formulas and new integral representations for multi-dimensional hypergeometric solutions of the joint qDE and qKZ system for G(k, n). Finally, we analyze the Stokes phenomenon for the joint system of qDE and qKZ equations associated with G(k, n). We prove that the Stokes bases of solutions correspond to explicit K-theoretical classes of full exceptional collections in the derived category of equivariant coherent sheaves on G(k, n). Furthermore, we show that the Stokes matrices equal the Gram matrices of the equivariant Euler–Poincaré–Grothendieck pairing with respect to these exceptional K-theoretical bases.

We study the complexity of local Hamiltonians in which the terms pairwise commute. Commuting local Hamiltonians (CLHs) provide a way to study the role of non-commutativity in the complexity of quantum systems and touch on many fundamental aspects of quantum computing and many-body systems, such as the quantum PCP conjecture and the area law. Much of the recent research has focused on the physically motivated 2D case, where particles are located on vertices of a 2D grid and each term acts non-trivially only on the particles on a single square (or plaquette) in the lattice. In particular, Schuch showed that the CLH problem on 2D with qubits is in NP. Resolving the complexity of the 2D CLH problem with higher dimensional particles has been elusive. We prove two results for the CLH problem in 2D: We give a non-constructive proof that the CLH problem in 2D with qutrits is in (textbf{NP}). As far as we know, this is the first result for the commuting local Hamiltonian problem on 2D beyond qubits. Our key lemma works for general qudits and might give new insights for tackling the general case. We consider the factorized case, also studied by Bravyi and Vyalyi, where each term is a tensor product of single-particle Hermitian operators. We show that a factorized CLH in 2D, even on particles of arbitrary finite dimension, is equivalent to a direct sum of qubit stabilizer Hamiltonians. This implies that the factorized 2D CLH problem is in (textbf{NP}).

“Positive geometries” are a class of semi-algebraic domains which admit a unique “canonical form”: a logarithmic form whose residues match the boundary structure of the domain. The study of such geometries is motivated by recent progress in particle physics, where the corresponding canonical forms are interpreted as the integrands of scattering amplitudes. We recast these concepts in the language of mixed Hodge theory, and identify “genus zero pairs” of complex algebraic varieties as a natural and general framework for the study of positive geometries and their canonical forms. In this framework, we prove some basic properties of canonical forms which have previously been proved or conjectured in the literature. We give many examples and study in detail the case of arrangements of hyperplanes and convex polytopes.

We extend the notion of topological T-duality from oriented sphere bundles to transgressive fibrations, a more general type fibration characterised by the abundance of transgressive elements. Examples of transgressive fibrations include principal (textrm{U}(n))-bundles therefore our notion of T-duality belongs to the realm of non-Abelian T-duality. We prove that transgressive T-duals have isomorphic twisted cohomology. We then introduce Clifford–Courant algebroids, show that one can assign such an algebroid to a transgressive fibration and that transgressive T-duals have isomorphic Clifford–Courant algebroids. We provide several examples illustrating different properties of T-dual spaces.

We study mesoscopic fluctuations of orthogonal polynomial ensembles on the unit circle. We show that asymptotics of such fluctuations are stable under decaying perturbations of the recurrence coefficients, where the appropriate decay rate depends on the scale considered. By directly proving Gaussian limits for certain constant coefficient ensembles, we obtain mesoscopic scale Gaussian limits for a large class of orthogonal polynomial ensembles on the unit circle. As a corollary we prove mesoscopic central limit theorems (for all mesoscopic scales) for the (beta =2) circular Jacobi ensembles with real parameter (delta >-1/2).

Spreadable, exchangeable, and rotatable states on the CAR algebra are shown to be the same.