Duality Mapping Research Articles

The geometry of large charge multi-traces in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM

We construct a one-parameter family of half-BPS solutions of type IIB supergravity using a consistent truncation to gauged five-dimensional supergravity. For small values of the parameter, the solution reduces to the linear perturbation of AdS5 × S5 dual to the chiral primary operator in the stress-tensor multiplet, and we give evidence that the geometry is regular and asymptotes AdS in a normalisable way for arbitrarily large values of the parameter. We conjecture that the solution is the gravity dual of a “heavy” multi-trace operator in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SU(N) Super Yang-Mills made by p copies of the stress-tensor chiral primary operator, with p of order N2 in the large N limit. We perform some holographic checks supporting this duality map.

Quantum Duality Maps, Skein Algebras and their Ensemble Compatibility

We generalize the quantum duality map IA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {I}_{\\mathcal {A}}$$\\end{document} of Allegretti–Kim (Adv Math 306:1164–1208, 2017) for punctured closed surfaces to general marked surfaces. When the marked surface has no interior marked points, we investigate its compatibility with the quantum duality map IX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {I}_{\\mathcal {X}}$$\\end{document} on the dual side based on the quantum bracelets basis (Mandel and Qin in arXiv:2301.11101; Thurston in Proc Natl Acad Sci USA 111(27):9725–9732, 2014). Our construction factors through reduced stated skein algebras, based on the quantum trace maps (Lê in Quantum Topol 9(3):591–632, 2018) together with an appropriate way of skein lifting of integral A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}-laminations. We also give skein theoretic proofs for some expected properties of Laurent expressions, and positivity of structure constants for marked disks and a marked annulus.

Non-supersymmetric duality cascade of QCD(BF) via semiclassics on ℝ2× T2 with the baryon-’t Hooft flux

We study the phase diagrams of the bifundamental QCD (QCD(BF)) of different ranks, which is the 4d SU(N1) × SU(N2) gauge theory coupled with a bifundamental Dirac fermion. After discussing the anomaly constraints on possible vacuum structures, we apply a novel semiclassical approach on ℝ2× T2 with the baryon-’t Hooft flux to obtain the concrete dynamics. The 2d effective theory is derived by the dilute gas approximation of center vortices, and it serves as the basis for determining the phase diagram of the model under the assumption of adiabatic continuity. As an application, we justify the non-supersymmetric duality cascade between different QCD(BF), which has been conjectured in the large-N argument. Combined with the semiclassics and the large-N1,2 limit, we construct the explicit duality map from the parent theory, SU(N1) × SU(N2) QCD(BF), to the daughter theory, SU(N1) × SU(N2− N1) QCD(BF), including the correspondence of the coupling constants. We numerically examine the validity of the duality also for finite N1,2 within our semiclassics, finding a remarkable agreement of the phase diagrams between the parent and daughter sides.

Solutions for the Nonlinear Mixed Variational Inequality Problem in the System

Our paper proposes a system of nonlinear mixed variational inequality problems (SNMVIPs) on Banach spaces. Under suitable assumptions, using the K-Fan fixed point theorem and Minty techniques, we demonstrate that the solution set to the SNMVIP is nonempty, weakly compact, and unique. Additionally, we suggest a stability result for the SNMVIPs by perturbing the duality mappings. Furthermore, we present an optimal control problem that is governed by the SNMVIPs and show that it can be solved.

Naturality of $\mathrm{SL}_{3}$ quantum trace maps for surfaces

Fock–Goncharov’s moduli spaces \mathscr{X}_{\mathrm{PGL}_3,\mathfrak{S}} of framed \mathrm{PGL}_{3} -local systems on punctured surfaces \mathfrak{S} provide prominent examples of cluster \mathscr{X} -varieties and higher Teichmüller spaces. In a previous paper of the author (2022), building on the works of others, the so-called \mathrm{SL}_{3} quantum trace map is constructed for each triangulable punctured surface \mathfrak{S} and an ideal triangulation \Delta of \mathfrak{S} , as a homomorphism from the stated \mathrm{SL}_{3} -skein algebra of the surface to a quantum torus algebra that deforms the ring of Laurent polynomials in the cube-roots of the cluster coordinate variables for the cluster \mathscr{X} -chart for \mathscr{X}_{\mathrm{PGL}_3,\mathfrak{S}} associated to \Delta . We develop quantum mutation maps between special subalgebras of the cube-root quantum torus algebras for different triangulations and show that the \mathrm{SL}_{3} quantum trace maps are natural, in the sense that they are compatible under these quantum mutation maps. As an application, the quantum \mathrm{SL}_{3} - \mathrm{PGL}_{3} duality map constructed in the previous paper is shown to be independent of the choice of an ideal triangulation.

Normalized duality mappings and projections in Bochner spaces

Normalized duality mappings and projections in Bochner spaces

Tensionless strings on AdS3 orbifolds

The bound state of one NS5 brane (wrapped on a \U0001d54b4) and N NS1-branes has two dual descriptions: its low-energy dynamics is described by the symmetric orbifold of \U0001d54b4, while the near horizon geometry is captured by string theory on AdS3 × S3 × \U0001d54b4 with one unit of NS flux. The latter theory is exactly solvable in the hybrid formalism, and this allows one to prove the equivalence of the two descriptions. In this paper we extend this duality to ℤk orbifolds of this AdS3 × S3 background. In particular, we show that the corresponding worldsheet spectrum reproduces exactly the perturbative excitations on top of a certain non-perturbative state in the dual symmetric orbifold theory. Since the AdS/CFT duality map is exact for these models, we obtain an interesting picture of how the duality relates boundary and bulk descriptions.

A dual mapping associated to a closed convex set and some subdifferential properties

In this paper, we establish some properties of the multivalued mapping (x, d) ⇒ DC (x; d) that associates to every element x of a linear normed space X the set of linear continuous functionals of norm d ≥ 0 and which separates the closed ball B (x; d) from a closed convex set C ⊂ X. Using this mapping we give links with other important concepts in convex analysis (ε-approximation element, ε-subdifferential of distance function, duality mapping, polar cone). Thus, we establish a dual characterization of ε-approximation elements with respect to a nonvoid closed convex set as a generalization of a known result of Garkavi. Also, we give some properties of univocity and monotonicity of mapping DC. Mathematics Subject Classification (2010): 32A70, 41A65, 46B20, 46N10. Received 18 May 2023; Accepted 27 November 2023

Dynamical localization transition in the non-Hermitian lattice gauge theory

Local constraint in the lattice gauge theory provides an exotic mechanism that facilitates the disorder-free localization. However, the understanding of nonequilibrium dynamics in the non-Hermitian lattice gauge model remains limited. Here, we investigate the quench dynamics in a system of spinless fermions with nonreciprocal hopping in the Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb{Z}}}_{2}$$\\end{document} gauge field. By employing a duality mapping, we systematically explore the non-Hermitian skin effect, localization-delocalization transition, and real-complex transition. Through the identification of diverse scaling behaviors of quantum mutual information for fermions and spins, we propose that the non-Hermitian quantum disentangled liquids exist both in the localized and delocalized phases, the former originates from the Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb{Z}}}_{2}$$\\end{document} gauge field and the latter arises from the non-Hermitian skin effect. Furthermore, we demonstrate that the nonreciprocal dissipation causes the flow of quantum information. Our results provide valuable insights into the nonequilibrium dynamics in the gauge field, and may be experimentally validated using quantum simulators.

Multi-label learning with incomplete labels via dual manifold mappings

Multi-label learning with incomplete labels via dual manifold mappings

Evolution equations with nonlocal multivalued Cauchy problems

We consider evolution equations in Banach spaces. Their linear parts generate a strongly continuous C0-semigroup of contractions. The nonlinear term is a Carathéodory function. When the semigroup is not compact the nonlinearity has an additional restriction, involving the Hausdorff measure of noncompactness. We provide solutions satisfying nonlocal, multivalued Cauchy conditions. Our approach involves a suitable degree argument. The duality mapping is used for guaranteeing the lack of fixed points of the associated homotopic fields along the boundary of their domain. We apply our results for the investigation of transport and diffusion equations for which we provide the existence of nonlocal solutions.

Seismic Resolution Enhancement Using a Cycle Generative Adversarial Neural Network with Pseudo-Well Data

High-resolution seismic processing involves the recovery of high-frequency components from seismic data with lower resolution. Traditional methods typically impose prior knowledge or predefined subsurface structures when modeling seismic high-resolution processes, and they are usually model-driven. Nowadays, there has been a growing utilization of deep learning techniques to enhance seismic resolution. These approaches involve feature learning from extensive training datasets through multi-layered neural networks and are fundamentally data-driven. However, the reliance on labeled data has consistently posed a primary challenge for deploying these methods in practical applications. To address this issue, a novel approach for seismic high-resolution reconstruction is introduced, employing a Cycle Generative Adversarial Neural Network (CycleGAN) trained on authentic pseudo-well data. The application of the CycleGAN involves creating dual mappings connecting low-resolution and high-resolution data. This enables the model to comprehend both the forward and inverse processes, ensuring the stability of the inverse process, particularly in the context of high-resolution reconstruction. More importantly, statistical distributions are extracted from well logs and used to randomly generate extensive sets of low-resolution and high-resolution training pairs. This training set captures the structural characteristics of the actual subsurface and leads to significant improvement of the proposed method. The results from experiments conducted on both synthetic and field examples validate the effectiveness of the proposed approach in significantly enhancing seismic resolution and achieving superior recovery of thin layers when compared with the conventional method and the deep-learning-based method.

Deconfined criticalities and dualities between chiral spin liquid, topological superconductor and charge density wave Chern insulator

We propose bi-critical and tri-critical theories between chiral spin liquid (CSL), topological superconductor (SC) and charge density wave (CDW) ordered Chern insulator with Chern number C=2C=2 on square, triangular and Kagome lattices. The three CDW order parameters form a manifold of S^2S2 or S^1S1 depending on whether there is easy-plane anisotropy. The skyrmion defect of the CDW order carries physical charge 2e2e and its condensation leads to a topological superconductor. The CDW-SC transitions are in the same universality classes as the celebrated deconfined quantum critical points (DQCP) between Neel order and valence bond solid order on square lattice. Both SC and CDW order can be accessed from the CSL phase through a continuous phase transition. At the CSL-SC transition, there is still CDW order fluctuations although CDW is absent in both sides. We propose three different theories for the CSL-SC transition (and CSL to easy-plane CDW transition): a U(1)U(1) theory with two bosons, a U(1)U(1) theory with two Dirac fermions, and an SU(2)SU(2) theory with two bosons. Our construction offers a derivation of the duality between these three theories as well as a promising physical realization. The SU(2)SU(2) theory offers a unified framework for a series of fixed points with explicit SO(5), O(4)SO(5),O(4) or SO(3)× O(2)SO(3)×O(2) symmetry. There is also a transparent duality transformation mapping SC order to easy-plane CDW order. The CSL-SC-CDW tri-critical points are invariant under this duality mapping and have an enlarged SO(5)SO(5) or O(4)O(4) symmetry. The DQCPs between CDW and SC inherit the enlarged symmetry, emergent anomaly, and self-duality from the tri-critical point. Our analysis unifies the well-studied DQCP between symmetry breaking phases into a larger framework where they are proximate to a topologically ordered phase. Experimentally the theory demonstrates the possibility of a rich phase diagram and criticality through closing the Mott gap of a quantum spin liquid with projective symmetry group.

Simplified REGINN-IT method in Banach spaces for nonlinear ill-posed operator equations

Abstract In 2021, Z. Fu, Y. Chen and B. Han introduced an inexact Newton regularization (REGINN-IT) using an idea involving the non-stationary iterated Tikhonov regularization scheme for solving nonlinear ill-posed operator equations. In this paper, we suggest a simplified version of the REGINN-IT scheme by using the Bregman distance, duality mapping and a suitable parameter choice strategy to produce an approximate solution. The method is comprised of inner and outer iteration steps. The outer iterates are stopped by a Morozov-type stopping rule, while the inner iterate is executed by making use of the non-stationary iterated Tikhonov scheme. We have studied convergence of the proposed method under some standard assumptions and utilizing tools from convex analysis. The novelty of the method is that it requires computation of the Fréchet derivative only at an initial guess of an exact solution and hence can be identified as more efficient compared to the method given by Z. Fu, Y. Chen and B. Han. Further, in the last section of the paper, we discuss test examples to inspect the proficiency of the method.

Asplunds, yet again

We present some partially new characterizations of Asplund spaces via Baire class one selectors. Sufficient conditions for Asplundness in terms of properties of duality mappings are also considered.

A Modified Algorithms for New Krasnoselskii's Type for Strongly Monotone and Lipschitz Mappings

Let E be a 2 uniformly smooth and convex real Banach space and let a mapping A: E → E∗ be lipschitz and strongly monotone such that A−1(0)≠∅. For an arbitrary ({x1}, {y1})∈E, we define the sequences {xn} and {yn} byyn = xn − θnJ−1(Axn), n≥1xn+1 = yn − λnJ−1(Ayn), n≥1where λn and θn are positive real number and J is the duality mapping of E. Letting (λn, θn)∈(0, 1), then xn and yn converges strongly to ρ∗, a unique solution of the equation Ax = 0. We also applied our algorithm in convex minimization and also proved the convergence of it in Lp, lp or Wm,p. At the end we proposed the algorithm of it in Lp(Ω) and its inverse Lq(Ω).

Solutions for Some Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 1

This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or Newmann problems for these important operators. This article details three ways to use surjectivity results for a special type of operator involving the duality mapping and a Nemytskii operator, three methods starting from Ekeland’s variational principle and, lastly, one with a generalized variational principle to solve or describe the above-mentioned solutions. The relevance of these operators and the possibility of their involvement in the modeling of an important class of real phenomena determined the author to group these seven procedures together, presented in detail, followed by many applications, accompanied by a general overview of specialty domains. The use of certain variational methods facilitates the complete solution of the problem via appropriate numerical methods and computational algorithms. The exposure of the sequence of theoretical results, together with their demonstration in as much detail as possible has been fulfilled as an opportunity for the complete development of these topics.

A characterization of the star duality mapping

We establish a characterization of the star duality mapping for star-shaped sets in n-dimensional Euclidean vector space.

On the Notion of Metaplectic Barbasch–Vogan Duality

Abstract In analogy with the Barbasch–Vogan duality for real reductive linear groups, we introduce a duality notion useful for the representation theory of the real metaplectic groups. This is a map on the set of nilpotent orbits in a complex symplectic Lie algebra, whose range consists of the so-called metaplectic special nilpotent orbits. We relate this duality notion with the theory of primitive ideals and extend the notion of special unipotent representations to the real metaplectic groups. We also interpret the duality map in terms of double cells of Weyl group representations.

Solution of master equations by fermionic-duality: Time-dependent charge and heat currents through an interacting quantum dot proximized by a superconductor

We analyze the time-dependent solution of master equations by exploiting fermionic duality, a dissipative symmetry applicable to a large class of open systems describing quantum transport. Whereas previous studies mostly exploited duality relations after partially solving the evolution equations, we here systematically exploit the invariance under the fermionic duality mapping from the very beginning when setting up these equations. Moreover, we extend the resulting simplifications -so far applied to the local state evolution- to non-local observables such as transport currents. We showcase the exploitation of fermionic duality for a quantum dot with strong interaction -covering both the repulsive and attractive case- proximized by contact with a large-gap superconductor which is weakly probed by charge and heat currents into a wide-band normal-metal electrode. We derive the complete time-dependent analytical solution of this problem involving non-equilibrium Cooper pair transport, Andreev bound states and strong interaction. Additionally exploiting detailed balance we show that even for this relatively complex problem the evolution towards the stationary state can be understood analytically in terms of the stationary state of the system itself via its relation to the stationary state of a dual system with inverted Coulomb interaction, superconducting pairing and applied voltages.