Covariant Phase Space Research Articles

Canonical aspects of pregeometric vector-based first order gauge theory

A recently proposed pregeometric auxiliary vector mediated gauge theory is studied in its canonical domain, by performing the Legendre transform on a curved background and by considering its covariant phase space, with further application to duality. The constraints become differential equations, but the Dirac-Bergmann algorithm appears consistent with electromagnetic degrees of freedom, metric background permitting. Solving the consistency conditions provides a preferred direction in an intermediary form of spontaneous symmetry breaking. In parallel, the covariant phase space defines the symplectic structure, and establishes the conserved currents and quantum phenomenology with the generated background. The formalism immediately allows one to study the parent path integrals of dual theories, with quartic Proca to Kalb-Ramond inequivalence and Maxwell-Chern-Simons path integral consistency as practical applications, while the differential geometry gives a global description of the issue of Yang-Mills duality rotations. Properties of degenerate metric geometry are discussed throughout, and the viability of inherent backgrounds leads into the fundamental question of background independence in all physical theories. Published by the American Physical Society 2024

BRST covariant phase space and holographic Ward identities

This paper develops an enlarged BRST framework to treat the large gauge transformations of a given quantum field theory. It determines the associated infinitely many Noether charges stemming from a gauge fixed and BRST invariant Lagrangian, a result that cannot be obtained from Noether’s second theorem. The geometrical significance of this result is highlighted by the construction of a trigraded BRST covariant phase space, allowing a BRST invariant gauge fixing procedure. This provides an appropriate framework for determining the conserved BRST Noether current of the global BRST symmetry and the associated global Noether charges. The latter are found to be equivalent with the usual classical corner charges of large gauge transformations. It allows one to prove the gauge independence of their physical effects at the perturbative quantum level. In particular, the underlying BRST fundamental canonical relation provides the same graded symplectic brackets as in the classical covariant phase space. A unified Lagrangian Ward identity for small and large gauge transformations is built. It consistently decouples into a bulk part for small gauge transformations, which is the standard BRST-BV quantum master equation, and a boundary part for large gauge transformations. The boundary part provides a perturbation theory origin for the invariance of the Hamiltonian physical -matrix under asymptotic symmetries. Holographic anomalies for the boundary Ward identity are studied and found to be solutions of a codimension one Wess-Zumino consistency condition. Such solutions are studied in the context of extended BMS symmetry. Their existence clarifies the status of the 1-loop correction to the subleading soft graviton theorem.

Linearized second law for higher curvature gravity and nonminimally coupled vector fields

Expanding the work of Wall [], we show that black holes obey a second law for linear perturbations to bifurcate Killing horizons, in any covariant higher curvature gravity coupled to scalar and vector fields. The vector fields do not need to be gauged, and (like the scalars) can have arbitrary nonminimal couplings to the metric. The increasing entropy has a natural expression in covariant phase space language, which makes it manifestly invariant under Jacobson-Kang-Myers ambiguities. An explicit entropy formula is given for f(Riemann) gravity coupled to vectors, where at most one derivative acts on each vector. Besides the previously known curvature terms, there are three extra terms involving differentiating the Lagrangian by the symmetric vector derivative (which therefore vanish for gauge fields). Published by the American Physical Society 2024

Covariant phase space formalism for fluctuating boundaries

We reconsider formulating D dimensional gauge theories, with the focus on the case of gravity theories, in spacetimes with boundaries. We extend covariant phase space formalism to the cases in which boundaries are allowed to fluctuate. We analyze the symplectic form, the freedoms (ambiguities), and its conservation for this case. We show that boundary fluctuations render all the surface charges integrable. We study the algebra of charges and its central extensions, charge conservation, and fluxes. We briefly comment on memory effects and questions regarding semiclassical aspects of black holes in the fluctuating boundary setup.

Dynamical edge modes and entanglement in Maxwell theory

Previous work on black hole partition functions and entanglement entropy suggests the existence of “edge” degrees of freedom living on the (stretched) horizon. We identify a local and “shrinkable” boundary condition on the stretched horizon that gives rise to such degrees of freedom. They can be interpreted as the Goldstone bosons of gauge transformations supported on the boundary, with the electric field component normal to the boundary as their symplectic conjugate. Applying the covariant phase space formalism for manifolds with boundary, we show that both the symplectic form and Hamiltonian exhibit a bulk-edge split. We then show that the thermal edge partition function is that of a codimension-two ghost compact scalar living on the horizon. In the context of a de Sitter static patch, this agrees with the edge partition functions found by Anninos et al. in arbitrary dimensions. It also yields a 4D entanglement entropy consistent with the conformal anomaly. Generalizing to Proca theory, we find that the prescription of Donnelly and Wall reproduces existing results for its edge partition function, while its classical phase space does not exhibit a bulk-edge split.

The quasilocal energy and thermodynamic first law in accelerating AdS black holes

We scrutinize the conserved energy of an accelerating AdS black hole by employing the off-shell quasilocal formalism, which amalgamates the ADT formalism with the covariant phase space approach. In the presence of conical singularities in the accelerating black hole, the energy expression is articulated through the surface term derived from our formalism. The essence of our analysis of the quasilocal energy resides in the surface contributions coming from the conical singularities as well as the conventional radial boundary. Consequently, the resultant conserved quasilocal energy naturally conforms the thermodynamic first law for the black hole without necessitating any augmentation of thermodynamic variables.

On the charge algebra of causal diamonds in three dimensional gravity

Covariant phase space methods are applied to the analysis of a causal diamond in 2+1-dimensional pure Einstein gravity. It is found that the reduced phase space is parametrized by a family of charges with a dual geometrical interpretation: they are geometric observables on the corner of the diamond, and they generate diffeomorphisms. The Poisson brackets among them close into an algebra. Knowledge of the corner charges therefore permits reconstruction of the diamond geometry, which realizes a form of local holography. The results are contrasted with the literature, and the path to a quantum description of spacetime geometry is discussed.

The dressing field method for diffeomorphisms: a relational framework

The dressing field method (DFM) is a tool to reduce gauge symmetries. Here we extend it to cover the case of diffeomorphisms. The resulting framework is a systematic scheme to produce Diff(M) -invariant objects, which has a natural relational interpretation. Its precise formulation relies on a clear understanding of the bundle geometry of field space. By detailing it, among other things we stress the geometric nature of field-independent and field-dependent diffeomorphisms, and highlight that the heuristic ‘extended bracket’ for field-dependent vector fields often featuring in the covariant phase space literature can be understood as arising from the Frölicher-Nijenhuis bracket. Furthermore, by articulating this bundle geometry with the covariant phase space approach, we give a streamlined account of the elementary objects of the (pre)symplectic structure of a Diff(M) -theory: Noether charges and their bracket, as induced by the standard prescription for the presymplectic potential and 2-form. We give conceptually transparent expressions allowing to read the integrability conditions and the circumstances under which the bracket of charge is Lie, and the resulting Poisson algebras of charges are central extensions of the Lie algebras of field-independent ( diff(M) ) and field-dependent vector fields.We show that, applying the DFM, one obtains a Diff(M) -invariant and manifestly relational formulation of a general relativistic field theory. Relying on results just mentioned, we easily derive the ‘dressed’ (relational) presymplectic structure of the theory. This reproduces or extends results from the gravitational edge modes and gravitational dressings literature. In addition to simplified technical derivations, the conceptual clarity of the framework supplies several insights and allows us to dispel misconceptions.

Covariant phase space analysis of Lanczos-Lovelock gravity with boundaries

This work introduces a novel prescription for the expression of the thermodynamic potentials associated with the couplings of a Lanczos-Lovelock theory. These potentials emerge in theories with multiple couplings, where the ratio between them provide intrinsic length scales that break scale invariance. Our prescription, derived from the covariant phase space formalism, differs from previous approaches by enabling the construction of finite potentials without reference to any background. To do so, we consistently work with finite-size systems with Dirichlet boundary conditions and rigorously take into account boundary and corner terms: including these terms is found to be crucial for relaxing the integrability conditions for phase space quantities that were required in previous works. We apply this prescription to the first law of (extended) thermodynamics for stationary black holes, and derive a version of the Smarr formula that holds for static black holes with arbitrary asymptotic behaviour.

Asymptotic structure of higher dimensional Yang-Mills theory

Using the covariant phase space formalism, we construct the phase space for non-Abelian gauge theories in (d+2)-dimensional Minkowski spacetime for any d ≥ 2, including the edge modes that symplectically pair to the low energy degrees of freedom of the gauge field. Despite the fact that the symplectic form in odd and even-dimensional spacetimes appear ostensibly different, we demonstrate that both cases can be treated in a unified manner by utilizing the shadow transform. Upon quantization, we recover the algebra of the vacuum sector of the Hilbert space and derive a Ward identity that implies the leading soft gluon theorem in (d+2)-dimensional spacetime.

Heisenberg soft hair on Robinson-Trautman spacetimes

We study 4 dimensional (4d) gravitational waves (GWs) with compact wavefronts, generalizing Robinson-Trautman (RT) solutions in Einstein gravity with an arbitrary cosmological constant. We construct the most general solution of the GWs in the presence of a causal, timelike, or null boundary when the usual tensor modes are turned off. Our solution space besides the shape and topology of the wavefront which is a generic compact, smooth, and orientable 2d surface Σ, is specified by a vector over Σ satisfying the conformal Killing equation and two scalars that are arbitrary functions over the causal boundary, the boundary modes (soft hair). We work out the symplectic form over the solution space using covariant phase space formalism and analyze the boundary symmetries and charges. The algebra of surface charges is a Heisenberg algebra. Only the overall size of the compact wavefront and not the details of its shape appears in the boundary symplectic form and is canonical conjugate to the overall mass of the GW. Hence, the information about the shape of the wavefront can’t be probed by the boundary observer. We construct a boundary energy-momentum tensor and a boundary current, whose conservation yields the RT equation for both asymptotically AdS and flat spacetimes. The latter provides a hydrodynamic description for our RT solutions.

Notes on thermodynamics of Schwarzschild-like bumblebee black hole

In this work, we revisited the thermodynamics of Schwarzschild-like bumblebee black hole using Iyer–Wald covariant phase space formalism. With a non zero vacuum expectation value, the bumblebee field is responsible for spontaneously breaking the Lorentz symmetry. As the bumblebee field is non-minimally coupled to gravity, we showed that the thermodynamic variables will be different from the counterpart in Einstein gravity. Especially, by using Iyer–Wald formalism, we found that the black hole entropy also differs from the result obtained from Wald entropy formula. Like Horndeski gravity, this mismatch is due to the divergence of bumblebee one-form field at the horizon. After figuring out the thermodynamics, we also briefly discussed the evaporation behavior of Schwarzschild like bumblebee black hole. We found that although bumblebee field has no influence on the critical impact factor, it can influence the black hole evaporation time.

Gravitational radiations of Kerr black hole from warped symmetries

The warped conformal symmetries have been found in the covariant phase space of a set of non-trivial diffeomorphism near the Kerr black hole horizon. In this paper, we consider the retarded Green’s function and the absorption probability for the scalar and higher spin perturbations on a generic non-extreme Kerr black hole background, and perform their holographic calculations in terms of the warped CFT. This provide further evidence on the conjecture that the warped CFT is relevant to the microstate description of the Kerr black hole.

The action of geometric entropy in topologically massive gravity

Due to the presence of a gravitational anomaly in topologically massive gravity (TMG), the geometric entropy is no longer simply the Hubeny-Rangamani-Takayanagi (HRT) area; instead, it is given by the HRT area plus an anomalous contribution. We study the action of this geometric entropy on the covariant phase space of classical solutions for TMG with matter fields whose action is algebraic in the metric. The result agrees precisely with the action of HRT area operators in Einstein-Hilbert gravity given in [8], i.e., it is a boundary-condition-preserving kink transformation. Furthermore, we show our result to be consistent with direct computations of semiclassical commutators of geometric entropies in pure TMG spacetimes asymptotic to planar AdS, as computed in [21].

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We study the charges and first law of thermodynamics for accelerating, non-rotating black holes with dyonic charges in AdS4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_4$$\\end{document} using the covariant phase space formalism. In order to apply the formalism to these solutions (which are asymptotically locally AdS and admit a non-smooth conformal boundary I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {I}}$$\\end{document}) we make two key improvements: (1) We relax the requirement to impose Dirichlet boundary conditions and demand merely a well-posed variational problem. (2) We keep careful track of the codimension-2 corner term induced by the holographic counterterms, a necessary requirement due to the presence of “cosmic strings” piercing I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {I}}$$\\end{document}. Using these improvements we are able to match the holographic Noether charges to the Wald Hamiltonians of the covariant phase space and derive the first law of black hole thermodynamics with the correct “thermodynamic length” terms arising from the strings. We investigate the relationship between the charges imposed by supersymmetry and show that our first law can be consistently applied to various classes of non-supersymmetric solutions for which the cross-sections of the horizon are spindles.

No logarithmic corrections to entropy in shift-symmetric Gauss-Bonnet gravity

Employing the covariant phase space formalism, we discuss black hole thermodynamics in four-dimensional scalar-tensor Einstein-Gauss-Bonnet gravity. We argue that logarithmic corrections to Wald entropy previously reported in this theory do not appear, due to the symmetry of the theory under constant shifts of the scalar field. Instead, we obtain the standard Bekenstein entropy of general relativity. Then, to satisfy the first law of black hole mechanics, the Hawking temperature must be modified. It has been proposed that such temperature modifications occur generically in scalar-tensor theories, due to different propagation speeds of gravitons and photons. We show that the temperature modifications also emerge in the Euclidean canonical ensemble approach to black hole thermodynamics. Notably, the boundary terms of the type we consider here can be considered in any scalar-tensor gravitational theories. Hence, we illustrate that adding a suitable boundary term to action may drastically affect black hole thermodynamics, changing both the entropy and the temperature.

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.

Finite charges from the bulk action

Constructing charges in the covariant phase space formalism often leads to formally divergent expressions, even when the fields satisfy physically acceptable fall-off conditions. These expressions can be rendered finite by corner ambiguities in the definition of the presymplectic potential, which in some cases may be motivated by arguments involving boundary Lagrangians. We show that the necessary corner terms are already present in the variation of the bulk action and can be extracted in a straightforward way. Once these corner terms are included in the presymplectic potential, charges derived from an associated codimension-2 form are automatically finite. We illustrate the procedure with examples in two and three dimensions, working in Bondi gauge and obtaining integrable charges. As a by-product, actions are derived for these theories that admit a well-defined variational principle when the fields satisfy boundary conditions on a timelike surface with corners. An interesting feature of our analysis is that the fields are not required to be fully on-shell.

Systematics of boundary actions in gauge theory and gravity

We undertake a general study of the boundary (or edge) modes that arise in gauge and gravitational theories defined on a space with boundary, either asymptotic or at finite distance, focusing on efficient techniques for computing the corresponding boundary action. Such actions capture all the dynamics of the system that are implied by its asymptotic symmetry group, such as correlation functions of the corresponding conserved currents. Working in the covariant phase space formalism, we develop a collection of approaches for isolating the boundary modes and their dynamics, and illustrate with various examples, notably AdS3 gravity (with and without a gravitational Chern-Simons terms) subject to assorted boundary conditions.

Integrable teleparallel gravity in the presence of boundaries

Symmetric teleparallel gravity is shown to be integrable in the presence of boundaries, given the consistent implementation of constraints in the covariant phase space formalism.Received 14 October 2022Accepted 3 January 2023DOI:https://doi.org/10.1103/PhysRevD.107.024019© 2023 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAlternative gravity theoriesGeneral relativityRelativistic aspects of cosmologyGravitation, Cosmology & Astrophysics