Let S be a family of n× n matrices over a field, and let S^ℓ 𝔽 be the linear span of all products of length ℓ of the matrices in S. Assume that, for some ℓ , the set S^ℓ 𝔽 is the full n× n matrix algebra. We show that, in this case, the set S^k 𝔽 is the full n× n matrix algebra, for any integer k⩾ n^2+2n-3 .
We develop a general construction of stationary stochastic processes with determinantal space-time correlations. Starting from a stochastically positive KMS system on the CAR algebra, we construct stochastic processes on point configuration space with determinantal point processes as stationary distributions. For quasi-free KMS states, we establish an Eynard–Mehta-type determinantal formula that gives the space-time correlations explicitly. This yields an operator-algebraic framework for constructing and analyzing such processes. Our framework extends and unifies classical discrete orthogonal polynomial ensembles by providing their finite-temperature versions. We further show that as the inverse temperature tends to infinity, these finite-temperature correlations converge to their zero-temperature limits.
We study a certain type of multiple commutation relations of the quantum affine algebra $$U_q(\widehat{\mathfrak {gl}}_N)$$ U q ( gl ^ N ) . We show that all the coefficients in the multiple commutation relations between the L -operator elements are given in terms of the trigonometric weight functions for the vector representation, independent of the representation of the L -operator. For rank one case, our proof also gives a conceptual understanding why the coefficients can also be expressed using the Izergin–Korepin determinants. As a related result, by specializing expressions for the universal nested Bethe vector by Pakuliak–Ragoucy–Slavnov, we also find a construction of the Gelfand–Tsetlin basis for the vector representation using different L -operator elements from the constructions by Nazarov–Tarasov or Molev. We also present corresponding results for the Yangian $$Y_h(\mathfrak {gl}_N)$$ Y h ( gl N ) .
In this paper, we consider the Klein-Gordon equations with cubic nonlinearity in three spatial dimensions, which are Hamiltonian perturbations of the linear one with potential. It is assumed that the corresponding linear Schrodinger operator admits an arbitrary number of possibly degenerate eigenvalues. By analyzing the resonance mechanisms between multiple discrete and continuous spectral modes, we determine the precise rate of energy transfer and radiation damping. Compared to [5], our results provide the first quantitative answer to the general multiple bound state problem proposed by Soffer and Weinstein in their seminal work [52]. Our proof leverages a pseudo-one-dimensional cancellation structure within each eigenspace, a renormalized damping mechanism, and an enhanced damping effect arising from interactions among discrete modes. Additionally, the analysis incorporates a refined Birkhoff normal form transformation and an extended version of Fermi’s Golden Rule, building on the foundational work of Bambusi and Cuccagna [5].
We introduce a new essential numerical range concept for Schrödinger operators with complex potentials and demonstrate that this so-called Žislin essential numerical range enjoys significant practical advantages in the analysis of spectral approximation for non-sectorial operators via domain truncation. As an application, dissipative barrier Stark operators are considered.
Motivated by the theory of holographic quantum error correction in the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, together with the kink transform conjecture on the bulk AdS description of boundary cocycle flow, we characterize (approximate) complementary recovery in terms of (approximate) intertwining of bulk and boundary cocycle derivatives. Using the geometric modular structure in vacuum AdS, we establish an operator algebraic subregion-subregion duality of boundary causal diamonds and bulk causal wedges for Klein–Gordon fields in the universal cover of AdS. Our results suggest that, from an algebraic perspective, the kink transform is bulk cocycle flow, which, in this setting, induces the bulk geometry via geometric modular action and the corresponding notion of time. As a by-product, we find that if the von Neumann algebra of a boundary CFT subregion is a type III_1 factor with an ergodic vacuum, then the von Neumann algebra of the corresponding dual bulk subregion is either ℂ1 (with a one-dimensional Hilbert space) or a type III_1 factor.
BPS states in type II string theory compactified on a Calabi-Yau threefold can typically be decomposed as moduli-dependent bound states of absolutely stable constituents, with a hierarchical structure labeled by attractor flow trees. This decomposition is best understood from the scattering diagram, an arrangement of real codimension-one loci (or rays) in the space of stability conditions where BPS states of given electromagnetic charge and fixed phase of the central charge exist. The consistency of the diagram when rays intersect determines all BPS indices in terms of the 'attractor indices' carried by the initial rays. In this work we study the scattering diagram for a non-compact toric CY threefold known as local F0, namely the total space of the canonical bundle over P1 & times;P1. We first construct the scattering diagram for the quiver, valid near the orbifold point, and for the large volume slice, valid when both P1's have large (and nearly equal) area. We then combine the insights gained from these simple limits to construct the scattering diagram along the physical slice of Pi-stability conditions, which carries an action of a Z4 extension of the modular group Gamma 0(4). We sketch a proof of the split attractor flow tree conjecture in this example, albeit for a restricted range of the central charge phase. Most arguments are similar to our early study of local P2 (Bousseau et al. in Commun. Math. Phys. 405(4):108, 2024.arXiv:2210.10712), but complicated by the occurrence of an extra mass parameter and ramification points on the Pi-stability slice.
This paper is a summary of the work by the author (The global stability of the Minkowski space-time in higher dimensions, Energy estimates for the Einstein-Yang-Mills fields and applications, Exterior stability of the (1+3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+3)$$\end{document}-dimensional Minkowski space-time solution to the Einstein-Yang-Mills equations), where we study the Einstein-Yang-Mills system in both the Lorenz and harmonic gauges, where the Yang-Mills fields are valued in any arbitrary Lie algebra G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {G}}$$\end{document} , associated with any compact Lie group G . In the paper by the author (The global stability of the Minkowski space-time in higher dimensions), we first showed that in the Lorenz gauge and in wave coordinates, we can recast the problem as equivalent to studying solutions of the Einstein-Yang-Mills equations that solve a covariant system of tensorial nonlinear hyperbolic partial differential equations. In another papers by the author (Energy estimates for the Einstein-Yang-Mills fields and applications, Exterior stability of the (1+3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(1+3)}$$\end{document}-dimensional Minkowski space-time solution to the Einstein-Yang-Mills equations), we exhibited that this system does not satisfy neither the null condition nor the weak-null condition of Lindblad-Rodnianski, rather a new weak-null condition with a different nonlinearity that has new complications that are not present neither for the Einstein vacuum equations nor for the Einstein-Maxwell system. Based on a separate energy estimate for each component, as well as based on a new separate estimate for the commutator term for the higher order energy for the tangential components, established by the author [Ghanem, S.: arXiv:2310.08611; arXiv:2310.08196], we sum up here the long detailed proof by the author (Exterior stability of the (1+3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(1+3)}$$\end{document}-dimensional Minkowski space-time solution to the Einstein-Yang-Mills equations), of the exterior stability of the Minkowski space-time, R1+3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}<^>{1+3}$$\end{document} , governed by the fully coupled Einstein-Yang-Mills system in the Lorenz gauge, valued in any arbitrary Lie algebra G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {G}}$$\end{document}, without any assumption of spherical symmetry. This provides a first detailed proof of the exterior stability of Minkowski governed by the fully nonlinear Einstein-Yang-Mills equations in the Lorenz gauge, by using a null-frame decomposition that was first used by H. Lindblad and I. Rodnianski in their celebrated seminal work for the case of the Einstein vacuum equations. We note, however, that to the best of our knowledge, the L infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>\infty $$\end{document}-estimate of Lindblad-Rodnianski does not work for the Einstein-Yang-Mills system in the Lorenz gauge. We replace their L infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>\infty $$\end{document}-estimate with our new separate energy estimates for the higher-order energy norm of the tangential components that allow us to treat the new type of nonlinearity that arises from the Yang-Mills fields in the Lorenz gauge.
We study the S=12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S=\frac{1}{2}$$\end{document} quantum spin system on the d-dimensional hypercubic lattice with d >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 2$$\end{document} with uniform nearest-neighbor interaction of the XY or XYZ type and arbitrary uniform magnetic field. By extending the method recently developed for quantum spin chains, we prove that the model possesses no local conserved quantities except for the trivial ones, such as the Hamiltonian. This result strongly suggests that the model is nonintegrable. We note that our result applies to the XX model without a magnetic field, which is one of the easiest solvable models in one dimension.
Only recently the concept of achronal localization has been developed as the adequate frame for the description of the localizability of a relativistic quantum mechanical system. Here covariant achronal localizations are gained out of covariant conserved currents computing their flux passing through achronal surfaces. This general method is applied to the probability density currents with causal kernel regarding the massive scalar boson. As (covariant) achronal localizations correspond one-to-one to (covariant) representations of the causal logic, thus, apparently for the first time, a covariant representation of the causal logic for an elementary relativistic quantum mechanical system has been achieved. Similarly a covariant family of representations of the causal logic is derived from the stress-energy tensor of the massive scalar boson. The construction of an achronal localization from a conserved current relies on a version of the divergence theorem for open sets with almost Lipschitz boundary. This result is stated and proved in this work.
We derive exact, convergent representations of multiloop sunset Feynman integrals in two dimensions for arbitrary mass configurations and all loop orders valid for large Euclidean momentum. The integrals are expressed as sums of symmetric polynomials in logarithmic mass ratios, normalized by the external momentum squared, with coefficients determined by analytic series expansions. For the equal-mass case, we establish a dimension-lowering relation expressing the L loop sunset integrals in D+2 as the one in D dimensions acted on by a differential operator of order L-1 . These representations are free of complicated transcendental functions, making them well-suited to both formal analysis and high-precision numerical evaluation. The two-dimensional results serve as boundary conditions for dimension-shifting relations, enabling systematic reconstruction of four-dimensional sunset integrals via analytic continuation to D = 4 - 2ϵ .
We consider correlation functions of topologically twisted, 𝒩=2 supersymmetric Yang–Mills theory with gauge group SU(2) and N_f≤ 3 massive hypermultiplets in the fundamental representation. For a smooth, oriented, closed four-manifold X with b_2^+>1 , the correlation functions are expressed in terms of a finite set of universal functions. The mass dependence of these functions encodes intersection numbers of the moduli space of instantons. We determine closed expressions for the universal functions by combining techniques of the Seiberg–Witten geometry, the u-plane integral, and the blowup formula. If X is specialised to a complex algebraic surface S, the correlation functions can be identified with generating functions of Segre invariants for moduli spaces of sheaves on S. With appropriate identifications of parameters, we establish agreement of our results with results by Göttsche and Kool for these generating functions.
We study dilute Bose gases in the thermodynamic limit interacting via two-body and three-body interaction potentials. We prove that the leading order of the thermodynamic ground state energy is entirely characterized by both the scattering length of the two-body potential and the scattering energy of the three-body potential. The corresponding result for two-body interactions was proven in seminal papers of Dyson (1957) [1] and Lieb–Yngvason (1998) [2], and the result for three-body interactions was proven much more recently by Nam–Ricaud–Triay (2022) [3]. The present result resolves a conjecture of Nam–Ricaud–Triay (2022) [4].
We study an Aganagic–Vafa brane supported on a special Lagrangian submanifold ℒ in a non-compact toric Calabi–Yau threefold 𝒳 . From the perspective of geometric engineering, the Aganagic–Vafa branes give rise to a special class of half-BPS codimension-two defects in 5d 𝒩=1 supersymmetric field theories in the presence of Ω -background. We propose that the defect partition functions give generating functions of refined, nonnegative, integral open BPS invariants of the pair (𝒳,ℒ) , across different Kähler moduli chambers they are expanded in. In the Nekrasov–Shatashvili limit, the partition function provides a partially resummed solution to a q-difference equation that quantizes the mirror curve of 𝒳 in an unambiguous fashion, in a polarization determined by the discrete labels of the Aganagic–Vafa brane. We demonstrate our method at examples of ℂ^3 , resolved conifold, resolved A_1 -singularity, local F_0 , and local F_1 .
We introduce a family of algebras 𝒜_M,N , M,N∈ℤ , as an extension of a pair of commuting quantum toroidal 𝔤𝔩_1 subalgebras ℰ_1,ℰ̌_1 , wherein the parameters are tuned in a specific way according to M, N. In the case M=± 1 , algebra 𝒜_± 1,N is a shifted quantum toroidal 𝔤𝔩_2 algebra introduced in Feigin et al (Affinization of shifted quantum affine gl2. arXiv:2511.12178). Conjecturally there is a coproduct homomorphism 𝒜_M,N_1+N_2→𝒜_M,N_1⊗̂𝒜_M,N_2 to a completed tensor product, whose restriction to the subalgebras ℰ_1,ℰ̌_1 coincides with the standard Drinfeld coproduct. We give examples of 𝒜_M,N modules constructed on certain direct sums of tensor products of Fock modules of ℰ_1⊗ℰ̌_1 .
