Double field theory is an approach for massless modes of string theory, unifying and geometrizing all gauge invariance in manifest $\mathbf{O}(D,D)$ covariant manner. In this approach, we derive off-shell conserved Noether current and corresponding Noether potential associated with unified gauge invariance. We add Wald-type counter two-form to the Noether potential and define conserved global charges as surface integral. We check our $\mathbf{O}(D,D)$ covariant formula against various string backgrounds, both geometric and non-geometric. In all cases we examined, we find perfect agreements with previous results. Our formula facilitates to evaluate momenta along not only ordinary spacetime directions but also dual spacetime directions on equal footing. From this, we confirm recent assertion that null wave in doubled spacetime is the same as macroscopic fundamental string in ordinary spacetime.
We construct a manifestly covariant differential Noether charge for theories with Chern-Simons terms in higher dimensional spacetimes. This is in contrast to Tachikawa's extension of the standard Lee-Iyer-Wald formalism which results in a non-covariant differential Noether charge for Chern-Simons terms. On a bifurcation surface, our differential Noether charge integrates to the Wald-like entropy formula proposed by Tachikawa in arXiv:hep-th/0611141.
Covariant conservation laws, local invariance and Noether's second theorem
Universidade dos Açores（亚速尔群岛大学）；Faculdade de Ciências, Universidade de Lisboa（里斯本大学理学院）；Instituto de Astrofísica e Ciências do Espaço（天体物理学与空间科学研究所）；Johns Hopkins University（约翰斯·霍普金斯大学）；Centro de Física Teórica e Computacional（理论与计算物理中心）
The covariant Noether charge formalism (also known as the covariant phase method) of Wald and collaborators, including its cohomological extension, is a manifestly covariant Hamiltonian formalism that, in principle, allows one to define and compute the energy, angular momenta, and chemical potentials of generic solutions of gravitational theories. However, it has been observed that for some supergravity solutions the variation of the Noether charge is not (at least manifestably) integrable, and as a result it is unclear whether there are well-defined thermodynamic charges for these solutions. …
A construction of conservation laws and conserved quantities for perturbations in arbitrary metric theories of gravity is developed. In an arbitrary field theory, with the use of incorporating an auxiliary metric into the initial Lagrangian covariantized Noether identities are carried out. Identically conserved currents with corresponding superpotentials are united into a family. Such a generalized formalism of the covariantized identities gives a natural basis for constructing conserved quantities for perturbations. A new family of conserved currents and correspondent superpotentials for pert…
The problem of finding a formulation of Noether's theorem in noncommutative geometry is very important in order to obtain conserved currents and charges for particles in noncommutative spacetimes. In this paper, we formulate Noether's theorem for translations of kappa-Minkowski noncommutative spacetime on the basis of the 5-dimensional kappa-Poincare' covariant differential calculus. We focus our analysis on the simple case of free scalar theory. We obtain five conserved Noether currents, which give rise to five energy-momentum charges. By applying our result to plane waves it follows that the…
Relying on known results of the Noether theory of symmetries extended to constrained systems, it is shown that there exists an obstruction that prevents certain tangent-space diffeomorphisms to be projectable to phase-space, for generally covariant theories. This main result throws new light on the old fact that the algebra of gauge generators in the phase space of General Relativity, or other generally covariant theories, only closes as a soft algebra and not a a Lie algebra. The deep relationship between these two issues is clarified. In particular, we see that the second one may be understo…
In this paper, we make a close comparison of a covariant definition of an energy/entropy in general relativity, recently proposed by a collaboration including the present authors, with existing definitions of energies such as the one from the pseudo-tensor and the quasi-local energy. We show that existing definitions of energies in general relativity are conserved charges from the Noether's 2nd theorem for the general coordinate transformation, whose conservations are merely identities implied by the local symmetry and always hold without using equations of motion. Thus none of existing defini…
We consider the open superstring action in the AdS$_4 \times \mathbf{CP}^3$ background and investigate the suitable boundary conditions for the open superstring describing the 1/2-BPS D-branes by imposing the $κ$-symmetry of the action. This results in the classification of 1/2-BPS D-branes from covariant open superstring. It is shown that the 1/2-BPS D-brane configurations are restricted considerably by the Kähler structure on $\mathbf{CP}^3$. We just consider D-branes without worldvolume fluxes.
Arbitrary diffeomorphically invariant metric-torsion theories of gravity are considered. It is assumed that Lagrangians of such theories contain derivatives of field variables (tensor densities of arbitrary ranks and weights) up to a second order only. The generalized Klein-Noether methods for constructing manifestly covariant identities and conserved quantities are developed. Manifestly covariant expressions are constructed without including auxiliary structures like a background metric. In the Riemann-Cartan space, the following \emph{manifestly generally covariant results} are presented: (a…
We derive the Noether identities and the conservation laws for general gravitational models with arbitrarily interacting matter and gravitational fields. These conservation laws are used for the construction of the covariant equations of motion for test bodies with minimal and nonminimal coupling.
We carry out a parallel study of the covariant phase space and the conservation laws of local symmetries in two-dimensional dilaton gravity. Our analysis is based on the fact that the Lagrangian can be brought to a form that vanishes on-shell giving rise to a well-defined covariant potential for the symplectic current. We explicitly compute the symplectic structure and its potential and show that the requirement to be finite and independent of the Cauchy surface restricts the asymptotic symmetries.
We construct a duality manifest gravitational theory for the special linear group, ${\mathbf{SL}(N)}$ with $N{\neq 4}$. The spacetime is formally extended, to have the dimension $\textstyle{\frac{1}{2}} N(N-1)$, yet is `gauged'. Consequently the theory is subject to a section condition. We introduce a semi-covariant derivative and a semi-covariant `Riemann' curvature, both of which can be completely covariantized after symmetrizing or contracting the ${\mathbf{SL}(N)}$ vector indices properly. Fully covariant scalar and `Ricci' curvatures then constitute the action and the `Einstein' equation …
In the first order formalism of gravity theories, there are some theories which are not Lorentz-diffeomorphism covariant. In the framework of such theories we cannot apply the method of conserved charge calculation used in Lorentz-diffeomorphism covariant theories. In this paper we firstly introduce the total variation of a quantity due to an infinitesimal Lorentz-diffeomorphism transformation. Secondly, in order to obtain the conserved charges of Lorentz-diffeomorphism non-covariant theories, we extend the Tachikawa method \cite{3}. This extension includes not only Lorentz gauge transformatio…
It has been recently shown that there is a particular combination of conformal invariants in six dimensions which accepts a generic Einstein space as a solution. The Lagrangian of this Conformal Gravity theory -- originally found by Lu, Pang and Pope (LPP) -- can be conveniently rewritten in terms of products and covariant derivatives of the Weyl tensor. This allows one to derive the corresponding Noether prepotential and Noether-Wald charges in a compact form. Based on this expression, we calculate the Noether-Wald charges of six-dimensional Critical Gravity at the bicritical point, which is …
We construct a double field theory coupled to the fields present in Vasiliev's equations. Employing the "semi-covariant" differential geometry, we spell a functional in which each term is completely covariant with respect to $\mathbf{O}(4,4)$ T-duality, doubled diffeomorphisms, $\mathbf{Spin}(1,3)$ local Lorentz symmetry and, separately, $\mathbf{HS}(4)$ higher spin gauge symmetry. We identify a minimal set of BPS-like conditions whose solutions automatically satisfy the full Euler-Lagrange equations. As such a solution, we derive a linear dilaton vacuum. With extra algebraic constraints furth…
A manifestly Lorentz and diffeomorphism invariant form for the abelian gauge field action with local duality symmetry of Schwarz and Sen is given. Some of the underlying symmetries of the covariant action are further considered. The Noether conserved charge under continuous local duality rotations is found. The covariant couplings with gravity and the axidilaton field are discussed.
Root bundles appear prominently in studies of vector-like spectra of 4d F-theory compactifications. Of particular importance to phenomenology are the Quadrillion F-theory Standard Models (F-theory QSMs). In this work, we analyze a superset of the physical root bundles whose cohomologies encode the vector-like spectra for the matter representations $(\mathbf{3}, \mathbf{2})_{1/6}$, $(\mathbf{\overline{3}}, \mathbf{1})_{-2/3}$ and $(\mathbf{1}, \mathbf{1})_{1}$. For the family $B_3( Δ_4^\circ )$ consisting of $\mathcal{O}(10^{11})$ F-theory QSM geometries, we argue that more than $99.995\%$ of t…
Upon treating the whole closed-string massless NS-NS sector as stringy graviton fields, Double Field Theory may evolve into `Stringy Gravity'. In terms of an $\mathbf{O}(D,D)$ covariant differential geometry beyond Riemann, we present the definitions of the off-shell conserved stringy Einstein curvature tensor and the on-shell conserved stringy Energy-Momentum tensor. Equating them, all the equations of motion of the massless sector are unified into a single expression, $G_{AB}{=8πG} T_{AB}$, carrying $\mathbf{O}(D,D)$ vector indices, which we dub `the Einstein Double Field Equations'.
The effective geometry and the gravitational coupling of nonabelian gauge and scalar fields on generic NC branes in Yang-Mills matrix models is determined. Covariant field equations are derived from the basic matrix equations of motions, known as Yang-Mills algebra. Remarkably, the equations of motion for the Poisson structure and for the nonabelian gauge fields follow from a matrix Noether theorem, and are therefore protected from quantum corrections. This provides a transparent derivation and generalization of the effective action governing the SU(n) gauge fields obtained in [1], including t…
In this short review we compare the rigid Noether charges to topological gauge charges. One important extension is that one should consider each boundary component of spacetime independently. The argument that relates bulk charges to surface terms can be adapted to the perfect fluid situation where one can recognise the helicity and enstrophies as Noether charges. More generally a forcing procedure that increases for instance any Noether charge is demonstrated. In the gauge theory situation, the key idea can be summarized by one sentence: ``go to infinity and stay there''. A new variational fo…
It is shown that the nature of physical time requires the extended phase-space in mechanics to have a bundle structure with time as the 1-dimensional base manifold and the phase space as the fiber. This bundle picture of the extended phase space is then applied to fields in a covariant, `directly' Hamiltonian formalism. Variational principle in the covariant phase-space is discussed, Noether currents calculated for symmetry fields and a new bracket analogous to the Peierls bracket is defined.
We investigate the covariant formulation of Chern-Simons theories in a general odd dimension which can be obtained by introducing a vacuum connection field as a reference. Field equations, Noether currents and superpotentials are computed so that results are easily compared with the well-known results in dimension 3. Finally we use this covariant formulation of Chern-Simons theories to investigate their relation with topological BF theories.
We give an explicit form of the PST-type SL(2;R)-covariant super D3-brane action for the flat Minkowski background. To this end, we follow the prescription developed by Hatsuda and Kamimura. As an application of the action, we obtain the supercharge of the action by using the standard Noether's method and calculate the Poisson bracket algebra of the supercharge. The central charge of the supersymmetry algebra is given in a manifestly SL(2;R)-covariant way.
We revisit the gauge-covariant canonical formalism by separating explicitly physical and gauge degrees of freedom. We show in particular that the gauge-invariant linear and angular momentum operators proposed by Chen et al. can consistently be derived from the standard procedure based on the Noether's theorem. Finally, we demonstrate that this approach is essentially equivalent to the gauge-invariant canonical formalism based on the concept of Dirac variables. Because of many similarities with the background field method, the formalism developed here should also be relevant to general relativi…
We develop an approach based on the Noether method to construct nilpotent BRST charges and BRST-invariant actions. We apply this approach first to the holomorphic part of the flat-space covariant superstring, and we find that the ghosts b, c_z which we introduced by hand in our earlier work, are needed to fix gauge symmetries of the ghost action. Then we apply this technique to the superparticle and determine its cohomology. Finally, we extend our results to the combined left- and right-moving sectors of the superstring.
Two techniques for computing black hole entropy in generally covariant gravity theories including arbitrary higher derivative interactions are studied. The techniques are Wald's Noether charge approach introduced recently, and a field redefinition method developed in this paper. Wald's results are extended by establishing that his local geometric expression for the black hole entropy gives the same result when evaluated on an arbitrary cross-section of a Killing horizon (rather than just the bifurcation surface). Further, we show that his expression for the entropy is not affected by ambiguiti…
It has recently been shown that, in the vicinity of their event horizons, black holes exhibit an infinite-dimensional symmetry. This symmetry captures relevant physical information about the black hole, and in particular about its thermodynamics. Here, we show that the same holds in a cosmological setup. More precisely, we show that around the de Sitter cosmological horizon, an infinite set of diffeomorphisms preserving a sensible set of boundary conditions emerges. These boundary conditions are similar to those considered by Price and Thorne in the context of the membrane paradigm, and permit…
In this paper we describe 4-dimensional gravity coupled to scalar and Maxwell fields by the Einstein-Katz action, that is, the covariant version of the "Gamma-Gamma $-$ Gamma-Gamma" part of the Hilbert action supplemented by the divergence of a generalized "Katz vector". We consider static solutions of Einstein's equations, parametrized by some integration constants, which describe an ensemble of asymptotically AdS black holes. Instead of the usual Dirichlet boundary conditions, which aim at singling out a specific solution within the ensemble, we impose that the variation of the action vanish…
In Einstein's gravitational theory, the spacetime is Riemannian, that is, it has vanishing torsion and vanishing nonmetricity (covariant derivative of the metric). In the gauging of the general affine group ${A}(4,R)$ and of its subgroup ${GL}(4,R)$ in four dimensions, energy--momentum and hypermomentum currents of matter are canonically coupled to the one--form basis and to the connection of a metric--affine spacetime with nonvanishing torsion and nonmetricity, respectively. Fermionic matter can be described in this framework by half--integer representations of the $\overline{SL}(4,R)$ coveri…
We consider systems of higher spin gauge fields that are described by a free field Lagrangian and one interaction of arbitrary order $N$ that is local and satisfies abelian gauge invariance. Such "solitary" interactions are derived from Noether potentials solving the Noether equations. They are constructed using a free conformal field theory carried by the same flat space as the higher spin fields. In this field theory we consider $N$-loop functions of conserved, conformally covariant currents, they are UV divergent. The residue of the first order pole in the dimensional regularisation approac…
Motivated by the study of physical models associated with General Relativity, we review some finite-dimensional, geometric and covariant formulations that allow us to characterize in a simple manner the symmetries for classical field theory by implementing an appropriate fibre-bundle structure, either at the Lagrangian, the multisymplectic or the polysymplectic levels. In particular, we are able to formulate Noether's theorems by means of the covariant momentum maps and to systematically introduce a covariant Poisson-Hamiltonian framework. Also, by focusing on the space plus time decomposition…
Noether's 2nd theorem applied to a total system states that a global symmetry which is a part of local symmetries does not provide a physically meaningful conserved charge but it instead leads to off-shell constraints as a form of conserved currents. In this paper, we propose a general method to derive a matter conserved current associated with a special global symmetry in the presence of local symmetries. While currents derived from local symmetries of a matter sector with a covariant background gauge field are not conserved in general, we show that the current associated with a special type …
We show that Belinfante construction of an improved energy-momentum tensor can be carried over to curved backgrounds, in analogy to the case of flat spacetime. The results hold irrespective of the background being dynamical or a fixed, non-backreacting one. It turns out that the analogous would-be canonical energy-momentum tensor is not covariantly conserved in general, but its Belinfante "improvement" is. We relate this last tensor with the Hilbert tensor obtained by functionally derivating the Lagrangian with respect to the metric. When the background in non-dynamical, we discuss some issues…
We prove the renormalizability to all orders of a refined Gribov-Zwanziger type action in linear covariant gauges in four-dimensional Euclidean space. In this model, the Gribov copies are taken into account by requiring that the Faddeev-Popov operator is positive definite with respect to the transverse component of the gauge field, a procedure which turns out to be analogous to the restriction to the Gribov region in the Landau gauge. The model studied here can be regarded as the first approximation of a more general nonperturbative BRST invariant formulation of the refined Gribov-Zwanziger ac…
We try to give hereafter an answer to some open questions about the definition of conserved quantities in Chern-Simons theory, with particular reference to Chern-Simons AdS_3 Gravity. Our attention is focused on the problem of global covariance and gauge invariance of the variation of Noether charges. A theory which satisfies the principle of covariance on each step of its construction is developed, starting from a gauge invariant Chern-Simons Lagrangian and using a recipe developed in gr-qc/0110104 and gr-qc/0107074 to calculate the variation of conserved quantities. The problem to give a mat…
We investigate the covariant Hamiltonian symplectic structure of General Relativity for spatially bounded regions of spacetime with a fixed time-flow vector. For existence of a well-defined Hamiltonian variational principle taking into account a spatial boundary, it is necessary to modify the standard Arnowitt-Deser-Misner Hamiltonian by adding a boundary term whose form depends on the spatial boundary conditions for the gravitational field. The most general mathematically allowed boundary conditions and corresponding boundary terms are shown to be determined by solving a certain equation obta…
In the framework of classical field theory, we first review the Noether theory of symmetries, with simple rederivations of its essential results, with special emphasis given to the Noether identities for gauge theories. Will this baggage on board, we next discuss in detail, for Poincaré invariant theories in flat spacetime, the differences between the Belinfante energy-momentum tensor and a family of Hilbert energy-momentum tensors. All these tensors coincide on shell but they split their duties in the following sense: Belinfante's tensor is the one to use in order to obtain the generators of …
It is well known that in general theories of gravity with the diffeomorphism symmetry, the black hole entropy is a Noether charge. But what will happen if the symmetry is explicitly broken? By investigating the covariant first law of black hole mechanics with background fields, we show that the Noether entropy is still applicable due to the local nature of the black hole entropy. Moreover, motivated by the proposal that the cosmological constant behaves as a thermodynamic variable, we allow the non-dynamical background fields to be varied. To illustrate this general formalism, we study a gener…
This is the first paper of a five part work in which we study the Lagrangian and Hamiltonian structure of classical field theories with constraints. Our goal is to explore some of the connections between initial value constraints and gauge transformations in such theories (either relativistic or not). To do this, in the course of these four papers, we develop and use a number of tools from symplectic and multisymplectic geometry. Of central importance in our analysis is the notion of the ``energy-momentum map'' associated to the gauge group of a given classical field theory. We hope to demonst…
Eleven-dimensional supergravity can be formulated in superspaces locally of the form $\mathbf X\times Y$ where $\mathbf X$ is 4D $N=1$ conformal superspace and $Y$ is an arbitrary 7-manifold admitting a $G_2$-structure. The eleven-dimensional 3-form and the stable 3-form on $Y$ define the lowest component of a gauge superfield on $\mathbf X \times Y$ that is chiral as a superfield on $\mathbf X$. This chiral field is part of a tensor hierarchy giving rise to a superspace Chern-Simons action and its real field strength defines a lifting of the Hitchin functional on $Y$ to the $G_2$ superspace $…
We formulate the variational problem for AdS gravity with Dirichlet boundary conditions and demonstrate that the covariant counterterms are necessary to make the variational problem well-posed. The holographic charges associated with asymptotic symmetries are then rederived via Noether's theorem and `covariant phase space' techniques. This allows us to prove the first law of black hole mechanics for general asymptotically locally AdS black hole spacetimes. We illustrate our discussion by computing the conserved charges and verifying the first law for the four dimensional Kerr-Newman-AdS and th…
An overlap between the general relativist and particle physicist views of Einstein gravity is uncovered. Noether's 1918 paper developed Hilbert's and Klein's reflections on the conservation laws. Energy-momentum is just a term proportional to the field equations and a 'curl' term with identically zero divergence. Noether proved a \emph{converse} "Hilbertian assertion": such "improper" conservation laws imply a generally covariant action. Later and independently, particle physicists derived the nonlinear Einstein equations assuming the absence of negative-energy degrees of freedom ("ghosts") fo…
Soft cutoffs in the covariant phase space of dynamical reference frames
Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University（扬州大学物理科学与技术学院引力与宇宙学中心）；School of Physics and Astronomy, China West Normal University（西华师范大学物理与天文学院）
We consider gravity in four dimensions in the vielbein formulation, where the fundamental variables are a tetrad $e$ and a SO(3,1) connection $ω$. We start with the most general action principle compatible with diffeomorphism invariance which includes, besides the standard Palatini term, other terms that either do not change the equations of motion, or are topological in nature. For our analysis we employ the covariant Hamiltonian formalism where the phase space $Γ$ is given by solutions to the equations of motion. We consider spacetimes that include a boundary at infinity, satisfying asymptot…
We investigate generators of local gauge transformations in the covariant canonical formalism (CCF) for matter fields, gauge fields and the second order formalism of gravity. The CCF treats space and time on an equal footing regarding the differential forms as the basic variables. The conjugate forms $π_A$ are defined as derivatives of the Lagrangian $d$-form $L(ψ^A, dψ^A)$ with respect to $dψ^A$, namely $π_A := \partial L/\partial dψ^A$, where $ψ^A $ are $p$-form dynamical fields. The form-canonical equations are derived from the form-Legendre transformation of the Lagrangian form $H:=dψ^A \w…
In string theory the closed-string massless NS-NS sector forms a multiplet of $\mathbf{O}(D,D)$ symmetry. This suggests a specific modification to General Relativity in which the entire NS-NS sector is promoted to stringy graviton fields. Imposing off-shell $\mathbf{O}(D,D)$ symmetry fixes the correct couplings to other matter fields and the Einstein field equations are enriched to comprise $D^{2}+1$ components, dubbed recently as the Einstein Double Field Equations. Here we explore the cosmological implications of this framework. We derive the most general homogeneous and isotropic ansatzes f…
Starting with the generalized field equation of dyons and gravito-dyons, we study the theory of octonion variables to the SU (2) non-Abelian gauge formalism. We demonstrate the resemblance of octonion covariant derivative with the gauge covariant derivative of generalized fields of dyons.Expressing the generalized four-potential, current and fields of gravito-dyons in terms of split octonion variables, the U (1) abelian and SU (2) non-Abelian gauge structure of dyons and gravito-dyons are described. It is emphasized that in general the generalized four-current is not conserved but only the Noe…
