Symplectic Potential Research Articles

Centerless BMS charge algebra

We show that when the Wald-Zoupas prescription is implemented, the resulting charges realize the Bondi-Van der Burg–Metzner-Sachs (BMS) symmetry algebra without any 2-cocycle nor central extension, at any cut of future null infinity. We refine the covariance prescription for application to the charge aspects, and introduce a new aspect for Geroch’s supermomentum with better covariance properties. For the extended BMS symmetry with singular conformal Killing vectors we find that a Wald-Zoupas symplectic potential exists, if one is willing to modify the symplectic structure by a corner term. The resulting algebra of Noether currents between two arbitrary cuts is centerless. The charge algebra at a given cut has a residual field-dependent 2-cocycle, but time-independent and nonradiative. More precisely, superrotation fluxes act covariantly, but superrotation charges act covariantly only on global translations. The take home message is that in any situation where 2-cocycles appears in the literature, covariance has likely been lost in the charge prescription, and that the criterium of covariance is a powerful one to reduce ambiguities in the charges, and can be used also for ambiguities in the charge aspects. Published by the American Physical Society 2024

Renormalization of conformal infinity as a stretched horizon

Abstract In this paper, we provide a comprehensive study of asymptotically flat spacetime in even dimensions d ⩾ 4 . We analyze the most general boundary condition and asymptotic symmetry compatible with Penrose’s definition of asymptotic null infinity I through conformal compactification. Following Penrose’s prescription and using a minimal version of the Bondi–Sachs gauge, we show that I is naturally equipped with a Carrollian stress tensor whose radial derivative defines the asymptotic Weyl tensor. This analysis describes asymptotic infinity as a stretched horizon in the conformally compactified spacetime. We establish that charge aspects conservation can be written as Carrollian Bianchi identities for the asymptotic Weyl tensor. We then provide a covariant renormalization for the asymptotic symplectic potential, which results in a finite symplectic flux and asymptotic charges. The renormalization scheme works even in the presence of logarithmic anomalies.

A highly accurate and efficient Genocchi‐based spectral technique applied to singular fractional order boundary value problems

This article focuses on an efficient and highly accurate approximate solver for a class of generalized singular boundary value problems (SBVPs) having nonlinearity and with two‐term fractional derivatives. The involved fractional derivative operators are given in the form of Liouville–Caputo. The developed algorithm for solving the generalized SBVPs consists of two main stages. The first stage is devoted to an iterative quasilinearization method (QLM) to conquer the (strong) nonlinearity of the governing SBVPs. Secondly, we employ the generalized Genocchi polynomials (GGPs) to treat the resulting sequence of linearized SBVPs numerically. An upper error estimate for the Genocchi series solution in the norm is obtained via a rigorous error analysis. The main benefit of the presented QLM‐GGPs procedure is that the required number of iteration in the first stage is within a few steps, and an accurate polynomial solution is obtained through computer implementations in the second stage. Three widely applicable test cases are investigated to observe the efficacy as well as the high‐order accuracy of the QLM‐GGPs algorithm. The comparable accuracy and robustness of the presented algorithm are validated by doing comparisons with the results of some well‐established available computational methods. It is apparently shown that the QLM‐GGPs algorithm provides a promising tool to solve strongly nonlinear SBVPs with two‐term fractional derivatives accurately and efficiently.

Global Gevrey solvability of a complex of differential operators associated with an involutive system

We consider a complex of differential operators associated with an involutive system of vector fields on the torus. We present a complete characterization of the global Gevrey solvability in terms of Diophantine conditions involving exponential Liouville forms.

Chebyshev potentials, Fubini–Study metrics, and geometry of the space of Kähler metrics

AbstractThe Chebyshev potential of a Hermitian metric on an ample line bundle over a projective variety, introduced by Witt Nyström, is a convex function defined on the Okounkov body. It is a generalization of the symplectic potential of a torus‐invariant Kähler potential on a toric variety, introduced by Guillemin, that is a convex function on the Delzant polytope. A folklore conjecture asserts that a curve of Chebyshev potentials associated to a subgeodesic in the space of positively curved Hermitian metrics is linear in the time variable if and only if the subgeodesic is a geodesic in the Mabuchi metric. This is classically true in the special toric setting, and in general Witt Nyström established the sufficiency. The main obstacle in the conjecture is that it is difficult to compute Chebyshev potentials, that are currently only known on the Riemann sphere and toric varieties. The goal of this article is to disprove this conjecture. To that end we characterize the geodesics consisting of Fubini–Study metrics for which the conjecture is true on the hyperplane bundle of the projective space. The proof involves explicitly solving the Monge–Ampère equation describing geodesics on the subspace of Fubini–Study metrics and computing their Chebyshev potentials.

General gravitational charges on null hypersurfaces

We perform a detailed study of the covariance properties of the symplectic potential of general relativity on a null hypersurface, and of the different polarizations that can be used to study conservative as well as leaky boundary conditions. This allows us to identify a one-parameter family of covariant symplectic potentials. We compute the charges and fluxes for the most general phase space with arbitrary variations. We study five symmetry groups that arise when different restrictions on the variations are included. Requiring stationarity as in the original Wald-Zoupas prescription selects a unique member of the family of symplectic potentials, the one of Chandrasekaran, Flanagan and Prabhu. The associated charges are all conserved on non-expanding horizons, but not on flat spacetime. We show that it is possible to require a weaker notion of stationarity which selects another symplectic potential, again in a unique way, and whose charges are conserved on both non-expanding horizons and flat light-cones. Furthermore, the flux of future-pointing diffeomorphisms at leading-order around an outgoing flat light-cone is positive and reproduces a tidal heating plus a memory term. We also study the conformal conservative boundary conditions suggested by the alternative polarization and identify under which conditions they define a non-ambiguous variational principle. Our results have applications for dynamical notions of entropy, and are useful to clarify the interplay between different boundary conditions, charge prescriptions, and symmetry groups that can be associated with a null boundary.

Phase space renormalization and finite BMS charges in six dimensions

We perform a complete and systematic analysis of the solution space of six-dimensional Einstein gravity. We show that a particular subclass of solutions — those that are analytic near I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{I} $$\\end{document}+ — admit a non-trivial action of the generalised Bondi-Metzner-van der Burg-Sachs (GBMS) group which contains infinite-dimensional supertranslations and superrotations. The latter consists of all smooth volume-preserving Diff×Weyl transformations of the celestial S4. Using the covariant phase space formalism and a new technique which we develop in this paper (phase space renormalization), we are able to renormalize the symplectic potential using counterterms which are local and covariant. The Hamiltonian charges corresponding to GBMS diffeomorphisms are non-integrable. We show that the integrable part of these charges faithfully represent the GBMS algebra and in doing so, settle a long-standing open question regarding the existence of infinite-dimensional asymptotic symmetries in higher even dimensional non-linear gravity. Finally, we show that the semi-classical Ward identities for supertranslations and superrotations are precisely the leading and subleading soft-graviton theorems respectively.

The cell-centered Finite-Volume self-consistent approach for heterostructures: 1D electron gas at the Si–SiO2 interface

Achieving self-consistent convergence with the conventional effective-mass approach at ultra-low temperatures (below 4.2 K) is a challenging task, which mostly lies in the discontinuities in material properties (e.g. effective-mass, electron affinity, dielectric constant). In this article, we develop a novel self-consistent approach based on cell-centered finite-volume discretization of the Sturm–Liouville form of the effective-mass Schrödinger equation and generalized Poisson’s equation (FV-SP). We apply this approach to simulate the one-dimensional electron gas formed at the Si–SiO2 interface via a top gate. We find excellent self-consistent convergence from high to extremely low (as low as 50 mK) temperatures. We further examine the solidity of FV-SP method by changing external variables such as the electrochemical potential and the accumulative top gate voltage. Our approach allows for counting electron–electron interactions. Our results demonstrate that FV-SP approach is a powerful tool to solve effective-mass Hamiltonians.

Sign-changing bubble tower solutions for sinh-Poisson type equations on pierced domains

For asymmetric sinh-Poisson type problems with Dirichlet boundary condition arising as a mean field equation of equilibrium turbulence vortices with variable intensities of interest in hydrodynamic turbulence, we address the existence of sign-changing bubble tower solutions on a pierced domain Ωϵ:=Ω∖B(ξ,ϵ)‾, where Ω is a smooth bounded domain in R2 and B(ξ,ϵ) is a ball centered at ξ∈Ω with radius ϵ>0. Precisely, given a small parameter ρ>0 and any integer m≥2, there exist a radius ϵ=ϵ(ρ)>0 small enough such that each sinh-Poisson type equation, either in Liouville form or mean field form, has a solution uρ with an asymptotic profile as a sign-changing tower of m singular Liouville bubbles centered at the same ξ and with ϵ(ρ)→0+ as ρ approaches to zero.

Density of a thin film billiard reflection pseudogroup in a Hamiltonian symplectomorphism pseudogroup

Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to the canonical symplectic form. We consider an arbitrary C∞-smooth hypersurface γ ⊂ ℝn+1 that is either a global strictly convex closed hypersurface, or a germ of a hypersurface. We deal with the pseudogroup generated by compositional ratios of reflections from γ and of reflections from its small deformations. In the case when γ is a global convex hypersurface, we show that the C∞-closure of the latter pseudogroup contains the pseudogroup of Hamiltonian diffeomorphisms between domains in the phase cylinder: the space of oriented lines intersecting γ transversally. We prove an analogous local result in the case when γ is a germ. The derivatives of the above compositional differences in the deformation parameter are Hamiltonian vector fields calculated by Ron Perline. To prove the main results, we find the Lie algebra generated by them and prove its C∞-density in the Lie algebra of Hamiltonian vector fields. We also prove analogues of the above results for hypersurfaces in Riemannian manifolds.

On the uniqueness of supersymmetric AdS(5) black holes with toric symmetry

We consider the classification of supersymmetric AdS5 black hole solutions to minimal gauged supergravity that admit a torus symmetry. This problem reduces to finding a class of toric Kähler metrics on the base space, which in symplectic coordinates are determined by a symplectic potential. We derive the general form of the symplectic potential near any component of the horizon or axis of symmetry, which determines its singular part for any black hole solution in this class, including possible new solutions such as black lenses and multi-black holes. We find that the most general known black hole solution in this context, found by Chong, Cvetic, Lü and Pope (CCLP), is described by a remarkably simple symplectic potential. We prove that any supersymmetric and toric solution that is timelike outside a smooth horizon, with a Kähler base metric of Calabi type, must be the CCLP black hole solution or its near-horizon geometry.

Solutions of Initial Value Problems with Non-Singular, Caputo Type and Riemann-Liouville Type, Integro-Differential Operators

Motivated by the recent interest in generalized fractional order operators and their applications, we consider some classes of integro-differential initial value problems based on derivatives of the Riemann–Liouville and Caputo form, but with non-singular kernels. We show that, in general, the solutions to these initial value problems possess discontinuities at the origin. We also show how these initial value problems can be re-formulated to provide solutions that are continuous at the origin but this imposes further constraints on the system. Consideration of the intrinsic discontinuities, or constraints, in these initial value problems is important if they are to be employed in mathematical modelling applications.

Holographic entanglement entropy in $$T{\bar{T}}$$-deformed CFTs

In this paper, we study the holographic entanglement entropy computation of the ultraviolet, integrable deformation of the $$2-$$ dimensional conformal field theory ( $$T{\bar{T}}$$ -deformed conformal field theory) that would be dual to some massive deformations of 3D gravity in asymptotically $$AdS_{3}$$ spacetimes. We compute the correction due to the deformation up to the leading order of the deformation parameter in higher curvature 3D gravities such as new massive gravity, general minimal massive gravity, and exotic general massive gravity. We also use the evaluation of the symplectic potential to obtain the entanglement entropy for deformed theories. In each case, we find agreement between the results.

Goldman form, flat connections and stable vector bundles

We consider the moduli space $\mathscr{N}$ of stable vector bundles of degree 0 over a compact Riemann surface and the affine bundle $\mathscr{A}\to \mathscr{N}$ of flat connections. Following the similarity between the Teichmüller spaces and the moduli of bundles, we introduce the analogue of the quasi-Fuchsian projective connections – local holomorphic sections of $\mathscr{A}$ – that allow to pull back the Liouville symplectic form on $T\*\mathscr{N}$ to $\mathscr{A}$. We prove that the pullback of the Goldman form to $\mathscr{A}$ by the Riemann–Hilbert correspondence coincides with the pullback of the Liouville form. We also include a simple proof, in the spirit of Riemann bilinear relations, of the classic result – the pullback of Goldman symplectic form to $\mathscr{N}$ by the Narasimhan–Seshadri connection is the natural symplectic form on $\mathscr{N}$, introduced by Narasimhan and Atiyah & Bott.

A general framework for gravitational charges and holographic renormalization

We develop a general framework for constructing charges associated with diffeomorphisms in gravitational theories using covariant phase space techniques. This framework encompasses both localized charges associated with space–time subregions, as well as global conserved charges of the full space–time. Expressions for the charges include contributions from the boundary and corner terms in the subregion action, and are rendered unambiguous by appealing to the variational principle for the subregion, which selects a preferred form of the symplectic flux through the boundaries. The Poisson brackets of the charges on the subregion phase space are shown to reproduce the bracket of Barnich and Troessaert for open subsystems, thereby giving a novel derivation of this bracket from first principles. In the context of asymptotic boundaries, we show that the procedure of holographic renormalization can be always applied to obtain finite charges and fluxes once suitable counterterms have been found to ensure a finite action. This enables the study of larger asymptotic symmetry groups by loosening the boundary conditions imposed at infinity. We further present an algorithm for explicitly computing the counterterms that renormalize the action and symplectic potential, and, as an application of our framework, demonstrate that it reproduces known expressions for the charges of the generalized Bondi–Metzner–Sachs algebra.

Near horizon gravitational charges

In this paper, we study the near horizon symmetry and gravitational charges in the Newman-Penrose formalism. In particular we investigate the effect from topological terms. We find that the Pontryagin term and Gauss-Bonnet term have significant influence on the near horizon charges and bring interesting novel features. We show that the gravitational charge derived from a general class of topological terms including the Pontryagin term and Gauss-Bonnet term can be obtained from the ambiguities of the symplectic potential.

Subdiffusion Equations with a Source Term and Their Extensions

We propose an alternative derivation of the subdiffusion equation with a source term. Basing on this approach, we obtain the variable-order and the distributed-order equations that reduce to the well-known Riemann–Liouville and Caputo forms of the time-fractional subdiffusion equation. We find out that the naive inclusion of sources/sinks as a separate term is correct for the equations with the Riemann–Liouville derivative. However, for the equations with the Caputo derivative, the source term must go under a fractional integral.

Sewing spacetime with Lorentzian threads: complexity and the emergence of time in quantum gravity

Holographic entanglement entropy was recently recast in terms of Riemannian flows or ‘bit threads’. We consider the Lorentzian analog to reformulate the ‘complexity=volume’ conjecture using Lorentzian flows — timelike vector fields whose minimum flux through a boundary subregion is equal to the volume of the homologous maximal bulk Cauchy slice. By the nesting of Lorentzian flows, holographic complexity is shown to obey a number of properties. Particularly, the rate of complexity is bounded below by conditional complexity, describing a multi-step optimization with intermediate and final target states. We provide multiple explicit geometric realizations of Lorentzian flows in AdS backgrounds, including their time-dependence and behavior near the singularity in a black hole interior. Conceptually, discretized flows are interpreted as Lorentzian threads or ‘gatelines’. Upon selecting a reference state, complexity thence counts the minimum number of gatelines needed to prepare a target state described by a tensor network discretizing the maximal volume slice, matching its quantum information theoretic definition. We point out that suboptimal tensor networks are important to fully characterize the state, leading us to propose a refined notion of complexity as an ensemble average. The bulk symplectic potential provides a specific ‘canonical’ thread configuration characterizing perturbations around arbitrary CFT states. Consistency of this solution requires the bulk satisfy the linearized Einstein’s equations, which are shown to be equivalent to the holographic first law of complexity, thereby advocating for a principle of ‘spacetime complexity’. Lastly, we argue Lorentzian threads provide a notion of emergent time. This article is an expanded and detailed version of [1], including several new results.

Lorentzian Threads as Gatelines and Holographic Complexity.

The continuous min flow-max cut principle is used to reformulate the "complexity=volume" conjecture using Lorentzian flows-divergenceless norm-bounded timelike vector fields whose minimum flux through a boundary subregion is equal to the volume of the homologous maximal bulk Cauchy slice. The nesting property is used to show the rate of complexity is bounded below by "conditional complexity," describing a multistep optimization with intermediate and final target states. Conceptually, discretized Lorentzian flows are interpreted in terms of threads or gatelines such that complexity is equal to the minimum number of gatelines used to prepare a conformal field theory (CFT) state by an optimal tensor network (TN) discretizing the state. We propose a refined measure of complexity, capturing the role of suboptimal TNs, as an ensemble average. The bulk symplectic potential provides a "canonical" thread configuration characterizing perturbations around arbitrary CFT states. Its consistency requires the bulk to obey linearized Einstein's equations, which are shown to be equivalent to the holographic first law of complexity, thereby advocating a notion of "spacetime complexity."

Tau-Functions and Monodromy Symplectomorphisms

We derive a new Hamiltonian formulation of Schlesinger equations in terms of the dynamical r-matrix structure. The corresponding symplectic form is shown to be the pullback, under the monodromy map, of a natural symplectic form on the extended monodromy manifold. We show that Fock–Goncharov coordinates are log-canonical for the symplectic form. Using these coordinates we define the symplectic potential on the monodromy manifold and interpret the Jimbo–Miwa–Ueno tau-function as the generating function of the monodromy map. This, in particular, solves a recent conjecture by A. Its, O. Lisovyy and A. Prokhorov.