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The symmetries of the gravitational scattering are intimately tied to the symmetries which preserve asymptotic flatness at null infinity. In Penrose's definition of asymptotic flatness, a central role is played by the notion of asymptotic simplicity and the ensuing peeling behavior which dictates the decay rate of the Weyl tensor. However, there is now accumulating evidence that in a generic gravitational scattering the peeling property is broken, so that the spacetime is not asymptotically-flat in the usual sense. These obstructions to peeling can be traced back to the existence of universal radiative low frequency observables called ''tails to the displacement memory''. The universality of these tail modes is the statement of the classical logarithmic soft graviton theorem of Sahoo, Saha and Sen. Four-dimensional gravitation scattering therefore exhibits a rich infrared interplay between tail to the memory, loss of peeling, and universal logarithmic soft theorems. In this paper we study the solution space and the asymptotic symmetries for logarithmically-asymptotically-flat spacetimes. These are defined by a polyhomogeneous expansion of the Bondi metric which gives rise to a loss of peeling, and represent the classical arena which can accommodate a generic gravitational scattering containing tails to the memory. We show that while the codimension-two generalized BMS charges are sensitive to the loss of peeling at $\mathcal{I}^+$, the flux is insensitive to the fate of peeling. Due to the tail to the memory, the soft superrotation flux contains a logarithmic divergence whose coefficient is the quantity which is conserved in the scattering by virtue of the logarithmic soft theorem. In our analysis we also exhibit new logarithmic evolution equations and flux-balance laws, whose presence suggests the existence of an infinite tower of subleading logarithmic soft graviton theorems., Comment: 40 pages

We study the subleading structure of asymptotically-flat spacetimes and its relationship to the $w_{1+\infty}$ loop algebra of higher spin charges. We do so using both the Bondi-Sachs and the Newman-Penrose formalism, via a dictionary built from a preferred choice of tetrad. This enables us to access properties of the so-called higher Bondi aspects, such as their evolution equations, their transformation laws under asymptotic symmetries, and their relationship to the Newman-Penrose and the higher spin charges. By studying the recursive Einstein evolution equations defining these higher spin charges, we derive the general form of their transformation behavior under BMSW symmetries. This leads to an immediate proof that the spin 0,1 and spin $s$ brackets reproduce upon linearization the structure expected from the $w_{1+\infty}$ algebra. We then define renormalized higher spin charges which are conserved in the radiative vacuum at quadratic order, and show that they satisfy for all spins the $w_{1+\infty}$ algebra at linear order in the radiative data., Comment: 39+14 pages

In the companion paper [SciPost Phys. 13, 108 (2022), arXiv:2205.11401 [hep-th]] we have studied the solution space at null infinity for gravity in the partial Bondi gauge. This partial gauge enables to recover as particular cases and among other choices the Bondi-Sachs and Newman-Unti gauges, and to approach the question of the most general boundary conditions and asymptotic charges in gravity. Here we compute and study the asymptotic charges and their algebra in this partial Bondi gauge, by focusing on the flat case with a varying boundary metric $\delta q_{AB}\neq0$. In addition to the super-translations, super-rotations, and Weyl transformations, we find two extra asymptotic symmetries associated with non-vanishing charges labelled by free functions in the solution space. These new symmetries arise from a weaker definition of the radial coordinate and switch on traces in the transverse metric. We also exhibit complete gauge fixing conditions in which these extra asymptotic symmetries and charges survive. As a byproduct of this calculation we obtain the charges in Newman-Unti gauge, in which one of these extra asymptotic charges is already non-vanishing. We also apply the formula for the charges in the partial Bondi gauge to the computation of the charges for the Kerr spacetime in Bondi coordinates., Comment: 36 pages

We present an action for minimal massive gravity (MMG) in three dimensions in terms of a dreibein and an independent spin connection. Furthermore, the construction provides an action principle for an infinite family of so-called third-way consistent generalizations of the three-dimensional Einstein field equations, including exotic massive gravity and new higher-order generalizations. It allows to systematically construct the matter couplings for these models, including the couplings to fermions, depending on the spin connection. In particular, we construct different supersymmetric extensions of MMG, and derive their second order fermionic field equations. This establishes a new class of three-dimensional supergravity theories and we discuss their limit to topological massive supergravity. Finally, we identify the landscape of (A)dS vacua of the supersymmetric models. We analyze the spectrum and the unitarity properties of these vacua. We recover the known AdS vacua of MMG which are bulk and boundary unitary., Comment: 41 pages, 2 figures

We study the null asymptotic structure of Einstein-Maxwell theory in three-dimensional (3D) spacetimes. Although devoid of bulk gravitational degrees of freedom, the system admits a massless photon and can therefore accommodate electromagnetic radiation. We derive fall-off conditions for the Maxwell field that contain both Coulombic and radiative modes with non-vanishing news. The latter produces non-integrability and fluxes in the asymptotic surface charges, and gives rise to a non-trivial 3D Bondi mass loss formula. The resulting solution space is thus analogous to a dimensional reduction of 4D pure gravity, with the role of gravitational radiation played by its electromagnetic cousin. We use this simplified setup to investigate choices of charge brackets in detail, and compute in particular the recently introduced Koszul bracket. When the latter is applied to Wald-Zoupas charges, which are conserved in the absence of news, it leads to the field-dependent central extension found earlier in [arXiv:1503.00856]. We also consider (Anti-)de Sitter asymptotics to further exhibit the analogy between this model and 4D gravity with leaky boundary conditions., Comment: 50 pages, 1 figure. v2: added references and a paragraph in the conclusion on supersymmetric extensions

Reduced general relativity for four-dimensional spherically-symmetric stationary space-times, more simply called the black hole mini-superspace, was shown in previous work to admit a symmetry under the three-dimensional Poincar\'e group ISO(2,1). Such a non-semi-simple symmetry group usually signals that the system is a special case of a more general model admitting a semi-simple Lie group symmetry. We explore here possible modifications of the Hamiltonian constraint of the mini-superspace. We identify in particular a continuous deformation of the dynamics that lifts the degeneracy of the Poincar\'e group and leads to a SO(3,1) or SO(2,2) symmetry. This deformation is not related to the cosmological constant. We show that the deformed dynamics can be represented as the superposition of two non-interacting homogeneous FRW cosmologies, with flat slices filled with perfect fluid. The resulting modified black hole metrics are found to be non-singular., Comment: 12 pages, v2: extended discussion with added comparison with existing modified black hole ansatz, accepted for publication in Phys Rev D

By virtue of the Noether theorems, the vast gauge redundancy of general relativity provides us with a rich algebra of boundary charges that generate physical symmetries. These charges are located at codimension-2 entangling surfaces called corners. The presence of non-trivial corner symmetries associated with any entangling cut provides stringent constraints on the theory's mathematical structure and a guide through quantization. This report reviews new and recent results for non-perturbative quantum gravity, which are natural consequences of this structure. First, we establish that the corner symmetry derived from the gauge principle encodes quantum entanglement across internal boundaries. We also explain how the quantum representation of the corner symmetry algebra provides us with a notion of quantum geometry. We then focus our discussion on the first-order formulation of gravity and show how many results obtained in the continuum connect naturally with previous results in loop quantum gravity. In particular, we show that it is possible to get, purely from quantization and without discretization, an area operator with discrete spectrum, which is covariant under local Lorentz symmetry. We emphasize that while loop gravity correctly captures some of the gravitational quantum numbers, it does not capture all of them, which points towards important directions for future developments. Finally, we discuss the understanding of the gravitational dynamics along null surfaces as a conservation of symmetry charges associated with a Carrollian fluid., Comment: 29 pages. Revised version taking into account comments by the referees. This is a preprint of a chapter to appear in the "Handbook of Quantum Gravity", edited by Cosimo Bambi, Leonardo Modesto and Ilya Shapiro, 2023, Springer

We present a Sugawara-type construction for boundary charges in 4d BF theory and in a general family of related TQFTs. Starting from the underlying current Lie algebra of boundary symmetries, this gives rise to well-defined quadratic charges forming an algebra of vector fields. In the case of 3d BF theory (i.e. 3d gravity), it was shown in [PRD 106 (2022), arXiv:2012.05263 [hep-th]] that this construction leads to a two-dimensional family of diffeomorphism charges which satisfy a certain modular duality. Here we show that adapting this construction to 4d BF theory first requires to split the underlying gauge algebra. Surprisingly, the space of well-defined quadratic generators can then be shown to be once again two-dimensional. In the case of tangential vector fields, this canonically endows 4d BF theory with a $\mathrm{diff}(S^2)\times\mathrm{diff}(S^2)$ or $\mathrm{diff}(S^2)\ltimes\mathrm{vect}(S^2)_\mathrm{ab}$ algebra of boundary symmetries depending on the gauge algebra. The prospect is to then understand how this can be reduced to a gravitational symmetry algebra by imposing Pleba\'nski simplicity constraints.

Minimal massive gravity in three dimensions propagates a single massive spin-2 mode around an AdS vacuum. It is distinguished by allowing for vacua with positive central charges of the asymptotic conformal algebra and a bulk graviton of positive energy. We present a new action for the model (and its higher order extensions) in terms of a dreibein and an independent spin connection. From this, we construct its supersymmetric extension. Surprisingly, all vacua complying with bulk and boundary unitarity appear to break supersymmetry spontaneously. In contrast, all supersymmetric vacua have a negative central charge whenever the bulk graviton has positive energy., Comment: 6 pages, 1 figure v2: matches published version

We present a detailed analysis of gravity in a partial Bondi gauge, where only the three conditions $g_{rr}=0=g_{rA}$ are fixed. We relax in particular the so-called determinant condition on the transverse metric, which is only assumed to admit a polyhomogeneous radial expansion. This is sufficient in order to build the solution space, which here includes a cosmological constant, time-dependent sources in the boundary metric, logarithmic branches, and an extra trace mode at subleading order in the transverse metric. The evolution equations are studied using the Newman-Penrose formalism in terms of covariant functionals identified from the Weyl scalars, and we build the explicit dictionary between this formalism and the tensorial Einstein equations. This provides in particular a new derivation of the (A)dS mass loss formula. We then study the holographic renormalisation of the symplectic potential, and the transformation laws under residual asymptotic symmetries. The advantage of the partial Bondi gauge is that it allows to contrast and treat in a unified manner the Bondi-Sachs and Newman-Unti gauges, which can each be reached upon imposing a further specific gauge condition. The differential determinant condition leads to the $\Lambda$-BMSW gauge, while a differential condition on $g_{ur}$ leads to a generalized Newman-Unti gauge. This latter gives access to a new asymptotic symmetry which acts on the asymptotic shear and further extends the $\Lambda$-BMSW group by an extra abelian radial translation. This generalizes results which we have recently obtained in three dimensions., Comment: 80 pages

We investigate the phase space symmetries and conserved charges of homogeneous gravitational minisuperspaces. These (0+1)-dimensional reductions of general relativity are defined by spacetime metrics in which the dynamical variables depend only on a time coordinate, and are formulated as mechanical systems with a non-trivial field space metric (or supermetric) and effective potential. We show how to extract conserved charges for those minisuperspaces from the homothetic Killing vectors of the field space metric. In the case of two-dimensional field spaces, we exhibit a universal 8-dimensional symmetry algebra given by the semi-direct sum of $\mathfrak{sl}(2,\mathbb{R})\oplus\mathbb{R}$ with the two-dimensional Heisenberg algebra $\mathfrak{h}_2\simeq\mathbb{R}^4$. We apply this to the systematic study of the Bianchi models for homogeneous cosmology. This extends previous results on the $\mathfrak{sl}(2,\mathbb{R})$ algebra for Friedmann-Lemaitre-Robertson-Walker cosmology, and the Poincar\'e symmetry for Kantowski-Sachs metrics describing the black hole interior. The presence of this rich symmetry structure already in minisuperspace models opens new doors towards quantization and the study of solution generating mechanisms., Comment: 40 pages

In the context of the infrared triangle there have been recent discussions on the existence and the role of dual charges. We present a new viewpoint on dual magnetic charges in $p$-form theories, and argue that they can be inherited from the charges of a first order formulation as a topological BF theory with potential. This happens because, depending on the spacetime dimension and on the form degree, the so-called translational gauge symmetries of BF theory become reducible and therefore admit zero-modes. Although such zero-modes lead to trivial symmetries of the $p$-form theory, they are associated with non-trivial charges. These turn out to be precisely the dual magnetic charges. The centrally-extended current algebra of electric and magnetic charges in the $p$-form theory then descends naturally from that of BF theory. This is an effort towards finding an existence criterion for dual charges., Comment: 10 pages, 1 figure

The spacetime in the interior of a black hole can be described by an homogeneous line element, for which the Einstein--Hilbert action reduces to a one-dimensional mechanical model. We have shown in [SciPost Phys. 10, 022 (2021), [2010.07059]] that this model exhibits a symmetry under the $(2+1)$-dimensional Poincar\'e group. Here we explain how this can be understood as a broken infinite-dimensional BMS$_3$ symmetry. This is done by reinterpreting the action for the model as a geometric action for BMS$_3$, where the configuration space variables are elements of the algebra $\mathfrak{bms}_3$ and the equations of motion transform as coadjoint vectors. The Poincar\'e subgroup then arises as the stabilizer of the vacuum orbit. This symmetry breaking is analogous to what happens with the Schwarzian action in AdS$_2$ JT gravity, although in the present case there is no direct interpretation in terms of boundary symmetries. This observation, together with the fact that other lower-dimensional gravitational models (such as the BTZ black hole) possess the same broken BMS$_3$ symmetries, provides yet another illustration of the ubiquitous role played by this group., Comment: 32 pages

In the companion paper [Phys. Rev. D 103 (2021) 10, [2101.02951]] we have derived the short-ranged potentials for the Teukolsky equations for massless spins $(0,1/2,1,2)$ in general spherically-symmetric and static metrics. Here we apply these results to numerically compute the Hawking radiation spectra of such particles emitted by black holes (BHs) in three different ansatz: charged BHs, higher-dimensional BHs, and polymerized BHs arising from models of quantum gravity. In order to ensure the robustness of our numerical procedure, we show that it agrees with newly derived analytic formulas for the cross-sections in the high and low energy limits. We show how the short-ranged potentials and precise Hawking radiation rates can be used inside the code $\texttt{BlackHawk}$ to predict future primordial BH evaporation signals for a very wide class of BH solutions, including the promising regular BH solutions derived from loop quantum gravity. In particular, we derive the first Hawking radiation constraints on polymerized BHs from AMEGO. We prove that the mass window $10^{16}-10^{18}\,$g for all dark matter into primordial BHs can be reopened with high values of the polymerization parameter, which encodes the typical scale and strength of quantum gravity corrections., Comment: 22 pages, 8 figures, second in a series of 2 articles

We introduce a new gauge and solution space for three-dimensional gravity. As its name Bondi-Weyl suggests, it leads to non-trivial Weyl charges, and uses Bondi-like coordinates to allow for an arbitrary cosmological constant and therefore spacetimes which are asymptotically locally (A)dS or flat. We explain how integrability requires a choice of integrable slicing and also the introduction of a corner term. After discussing the holographic renormalization of the action and of the symplectic potential, we show that the charges are finite, symplectic and integrable, yet not conserved. We find four towers of charges forming an algebroid given by $\mathfrak{vir}\oplus\mathfrak{vir}\oplus\text{Heisenberg}$ with three central extensions, where the base space is parametrized by the retarded time. These four charges generate diffeomorphisms of the boundary cylinder, Weyl rescalings of the boundary metric, and radial translations. We perform this study both in metric and triad variables, and use the triad to explain the covariant origin of the corner terms needed for renormalization and integrability., Comment: 36 pages

In the context of the dynamics and stability of black holes in modified theories of gravity, we derive the Teukolsky equations for massless fields of all spins in general spherically-symmetric and static metrics. We then compute the short-ranged potentials associated with the radial dynamics of spin 1 and spin 1/2 fields, thereby completing the existing literature on spin 0 and 2. These potentials are crucial for the computation of Hawking radiation and quasi-normal modes emitted by black holes. In addition to the Schwarzschild metric, we apply these results and give the explicit formulas for the radial potentials in the case of charged (Reissner--Nordstr\"om) black holes, higher-dimensional black holes, and polymerized black holes arising from loop quantum gravity. These results are in particular relevant and applicable to a large class of regular black hole metrics. The phenomenological applications of these formulas will be the subject of a companion paper., Comment: 19 pages, 3 figures, first in a series of 2 articles

We study the finite distance boundary symmetry current algebra of the most general first order theory of 3d gravity. We show that the space of quadratic generators contains diffeomorphisms but also a notion of dual diffeomorphisms, which together form either a double Witt or centreless BMS$_3$ algebra. The relationship with the usual asymptotic symmetry algebra relies on a duality between the null and angular directions, which is possible thanks to the existence of the dual diffeomorphisms., Comment: 5 pages

We show that the phase space of three-dimensional gravity contains two layers of dualities: between diffeomorphisms and a notion of "dual diffeomorphisms" on the one hand, and between first order curvature and torsion on the other hand. This is most elegantly revealed and understood when studying the most general Lorentz-invariant first order theory in connection and triad variables, described by the so-called Mielke-Baekler Lagrangian. By analyzing the quasi-local symmetries of this theory in the covariant phase space formalism, we show that in each sector of the torsion/curvature duality there exists a well-defined notion of dual diffeomorphism, which furthermore follows uniquely from the Sugawara construction. Together with the usual diffeomorphisms, these duals form at finite distance, without any boundary conditions, and for any sign of the cosmological constant, a centreless double Virasoro algebra which in the flat case reduces to the BMS$_3$ algebra. These algebras can then be centrally-extended via the twisted Sugawara construction. This shows that the celebrated results about asymptotic symmetry algebras are actually generic features of three-dimensional gravity at any finite distance. They are however only revealed when working in first order connection and triad variables, and a priori inaccessible from Chern-Simons theory. As a bonus, we study the second order equations of motion of the Mielke-Baekler model, as well as the on-shell Lagrangian. This reveals the duality between Riemannian metric and teleparallel gravity, and a new candidate theory for three-dimensional massive gravity which we call teleparallel topologically massive gravity., Comment: 41+25 pages; v2: appendix D on Kosmann derivative and symplectic symmetries added, references added

We reveal an $\mathfrak{iso}(2,1)$ Poincar\'e algebra of conserved charges associated with the dynamics of the interior of black holes. The action of these Noether charges integrates to a symmetry of the gravitational system under the Poincar\'e group ISO$(2,1)$, which allows to describe the evolution of the geometry inside the black hole in terms of geodesics and horocycles of AdS${}_2$. At the Lagrangian level, this symmetry corresponds to M\"obius transformations of the proper time together with translations. Remarkably, this is a physical symmetry changing the state of the system, which also naturally forms a subgroup of the much larger $\textrm{BMS}_{3}=\textrm{Diff}(S^1)\ltimes\textrm{Vect}(S^1)$ group, where $S^1$ is the compactified time axis. It is intriguing to discover this structure for the black hole interior, and this hints at a fundamental role of BMS symmetry for black hole physics. The existence of this symmetry provides a powerful criterion to discriminate between different regularization and quantization schemes. Following loop quantum cosmology, we identify a regularized set of variables and Hamiltonian for the black hole interior, which allows to resolve the singularity in a black-to-white hole transition while preserving the Poincar\'e symmetry on phase space. This unravels new aspects of symmetry for black holes, and opens the way towards a rigorous group quantization of the interior., Comment: 46 pages, 6 figures

It has been suggested that the homogeneous black hole interior spacetime, when quantized following the techniques of loop quantum cosmology, has a resolved singularity replaced by a black-to-white hole transition. This result has however been derived so far only using effective classical evolution equations, and depends on details of the so-called polymerization scheme for the Hamiltonian constraint. Here we propose to use the unimodular formulation of general relativity to study the full quantum dynamics of this mini-superspace model. When applied to such cosmological models, unimodular gravity has the advantage of trivializing the problem of time by providing a true Hamiltonian which follows a Schr\"odinger evolution equation. By choosing variables adapted to this setup, we show how to write semi-classical states agreeing with that of the Wheeler-DeWitt theory at late times, and how in loop quantum cosmology they evolve through the would-be singularity while remaining sharply peaked. This provides a very simple setup for the study of the full quantum dynamics of these models, which can hopefully serve to tame regularization ambiguities., Comment: 39 pages, 10 figures

In the tetrad formulation of gravity, the so-called simplicity constraints play a central role. They appear in the Hamiltonian analysis of the theory, and in the Lagrangian path integral when constructing the gravity partition function from topological BF theory. We develop here a systematic analysis of the corner symplectic structure encoding the symmetry algebra of gravity, and perform a thorough analysis of the simplicity constraints. Starting from a precursor phase space with Poincar\'e and Heisenberg symmetry, we obtain the corner phase space of BF theory by imposing kinematical constraints. This amounts to fixing the Heisenberg frame with a choice of position and spin operators. The simplicity constraints then further reduce the Poincar\'e symmetry of the BF phase space to a Lorentz subalgebra. This picture provides a particle-like description of (quantum) geometry: The internal normal plays the role of the four-momentum, the Barbero-Immirzi parameter that of the mass, the flux that of a relativistic position, and the frame that of a spin harmonic oscillator. Moreover, we show that the corner area element corresponds to the Poincar\'e spin Casimir. We achieve this central result by properly splitting, in the continuum, the corner simplicity constraints into first and second class parts. We construct the complete set of Dirac observables, which includes the generators of the local $\mathfrak{sl}(2,\mathbb{C})$ subalgebra of Poincar\'e, and the components of the tangential corner metric satisfying an $\mathfrak{sl}(2,\mathbb{R})$ algebra. We then present a preliminary analysis of the covariant and continuous irreducible representations of the infinite-dimensional corner algebra. Moreover, as an alternative path to quantization, we also introduce a regularization of the corner algebra and interpret this discrete setting in terms of an extended notion of twisted geometries., Comment: 71 pages, 1 figure

In this second paper of the series we continue to spell out a new program for quantum gravity, grounded in the notion of corner symmetry algebra and its representations. Here we focus on tetrad gravity and its corner symplectic potential. We start by performing a detailed decomposition of the various geometrical quantities appearing in BF theory and tetrad gravity. This provides a new decomposition of the symplectic potential of BF theory and the simplicity constraints. We then show that the dynamical variables of the tetrad gravity corner phase space are the internal normal to the spacetime foliation, which is conjugated to the boost generator, and the corner coframe field. This allows us to derive several key results. First, we construct the corner Lorentz charges. In addition to sphere diffeomorphisms, common to all formulations of gravity, these charges add a local $\mathfrak{sl}(2,\mathbb{C})$ component to the corner symmetry algebra of tetrad gravity. Second, we also reveal that the corner metric satisfies a local $\mathfrak{sl}(2,\mathbb{R})$ algebra, whose Casimir corresponds to the corner area element. Due to the space-like nature of the corner metric, this Casimir belongs to the unitary discrete series, and its spectrum is therefore quantized. This result, which reconciles discreteness of the area spectrum with Lorentz invariance, is proven in the continuum and without resorting to a bulk connection. Third, we show that the corner phase space explains why the simplicity constraints become non-commutative on the corner. This fact requires a reconciliation between the bulk and corner symplectic structures, already in the classical continuum theory. Understanding this leads inevitably to the introduction of edge modes., Comment: 70 pages

This is the first paper in a series devoted to understanding the classical and quantum nature of edge modes and symmetries in gravitational systems. The goal of this analysis is to: i) achieve a clear understanding of how different formulations of gravity provide non-trivial representations of different sectors of the corner symmetry algebra, and ii) set the foundations of a new proposal for states of quantum geometry as representation states of this corner symmetry algebra. In this first paper we explain how different formulations of gravity, in both metric and tetrad variables, share the same bulk symplectic structure but differ at the corner, and in turn lead to inequivalent representations of the corner symmetry algebra. This provides an organizing criterion for formulations of gravity depending on how big the physical symmetry group that is non-trivially represented at the corner is. This principle can be used as a "treasure map" revealing new clues and routes in the quest for quantum gravity. Building up on these results, we perform a detailed analysis of the corner symplectic potential and symmetries of Einstein-Cartan-Holst gravity in [1], use this to provide a new look at the simplicity constraints in [2], and tackle the quantization in [3]., Comment: 53 pages, 2 figures. v3: Clarification about the uniqueness of the corner symplectic potential in Section 2.1, 2.6 and Appendix A and its link with boundary conditions

One of the key conceptual challenges in quantum gravity is to understand how quantum theory should modify the very notion of spacetime. One way to investigate this question is to study the alternatives to Schr\"odinger quantum mechanics. The polymer representation, inspired by loop quantum gravity, can be understood as capturing features of discrete spatial geometry. The modular representation, on the other hand, has a built-in unification of position and momentum polarizations via a length scale. In this paper, we introduce the modular polymer representations of the Weyl algebra, in which neither position nor momentum exists as a well-defined operator. As inequivalent representations, they are candidates for describing new physics. We illustrate this by studying the dynamics of the harmonic oscillator as an example, with the prospect of eventually applying this representation to quantum cosmology., Comment: 33 pages

In this work we propose a simple and systematic framework for including edge modes in gauge theories on manifolds with boundaries. We argue that this is necessary in order to achieve the factorizability of the path integral, the Hilbert space and the phase space, and that it explains how edge modes acquire a boundary dynamics and can contribute to observables such as the entanglement entropy. Our construction starts with a boundary action containing edge modes. In the case of Maxwell theory for example this is equivalent to coupling the gauge field to boundary sources in order to be able to factorize the theory between subregions. We then introduce a new variational principle which produces a systematic boundary contribution to the symplectic structure, and thereby provides a covariant realization of the extended phase space constructions which have appeared previously in the literature. When considering the path integral for the extended bulk + boundary action, integrating out the bulk degrees of freedom with chosen boundary conditions produces a residual boundary dynamics for the edge modes, in agreement with recent observations concerning the contribution of edge modes to the entanglement entropy. We put our proposal to the test with the familiar examples of Chern-Simons and BF theory, and show that it leads to consistent results. This therefore leads us to conjecture that this mechanism is generically true for any gauge theory, which can therefore all be expected to posses a boundary dynamics. We expect to be able to eventually apply this formalism to gravitational theories., Comment: 50 pages, 1 figure, published version

We have recently introduced a new and very simple action for three-dimensional massive gravity. This action is written in a first order formulation where the triad and the connection play a manifestly symmetric role, but where internal Lorentz gauge symmetry is broken. The absence of Lorentz invariance, which in this model is the mechanism underlying the propagation of a massive graviton, does however prevent from writing a purely metric non-linear action for the theory. Nonetheless, in this letter, we explain how to disentangle, at the non-linear level, the metric and non-metric degrees of freedom in the equations of motion. Focusing on the metric part, we show that it satisfies modified Einstein equations with higher derivative terms. As a particular case, these equations reproduce a well-studied model known as minimal massive gravity. In the general case, we obtain new metric field equations for massive gravity in three dimensions starting from the simple first order action. These field equations are consistent through a mechanism known as "third way consistency", which our theory therefore provides a new example of., Comment: 5 pages, published version

We propose and study a new action for three-dimensional massive gravity. This action takes a very simple form when written in terms of connection and triad variables, but the connection can also be integrated out to obtain a triad formulation. The quadratic action for the perturbations around a Minkowski background reproduces the action of self-dual massive gravity, in agreement with the expectation that the theory propagates a massive graviton. We confirm this result at the non-linear level with a Hamiltonian analysis, and show that this new theory does indeed possess a single massive degree of freedom. The action depends on four coupling constants, and we identify the various massive and topological (or massless) limits in the space of parameters. This richness, along with the simplicity of the action, opens a very interesting new window onto massive gravity., Comment: 20+9 pages

The proper definition of subsystems in gauge theory and gravity requires an extension of the local phase space by including edge mode fields. Their role is on the one hand to restore gauge invariance with respect to gauge transformations supported on the boundary, and on the other hand to parametrize the largest set of boundary symmetries which can arise if both the gauge parameters and the dynamical fields are unconstrained at the boundary. In this work we construct the extended phase space for three-dimensional gravity in first order connection and triad variables. There, the edge mode fields consist of a choice of coordinate frame on the boundary and a choice of Lorentz frame on the bundle, which together constitute the Lorentz-diffeomorphism edge modes. After constructing the extended symplectic structure and proving its gauge invariance, we study the boundary symmetries and the integrability of their generators. We find that the infinite-dimensional algebra of boundary symmetries with first order variables is the same as that with metric variables, and explain how this can be traced back to the expressions for the diffeomorphism Noether charge in both formulations. This concludes the study of extended phase spaces and edge modes in three-dimensional gravity, which was done previously by the author in the BF and Chern-Simons formulations., Comment: 16+10 pages

Boundaries in gauge field theories are known to be the locus of a wealth of interesting phenomena, as illustrated for example by the holographic principle or by the AdS/CFT and bulk-boundary correspondences. In particular, it has been acknowledged for quite some time that boundaries can break gauge invariance, and thereby turn gauge degrees of freedom into physical ones. There is however no known systematic way of identifying these degrees of freedom and possible associated boundary observables. Following recent work by Donnelly and Freidel, we show that this can be achieved by extending the covariant Hamiltonian formalism so as to make it gauge-invariant under arbitrary large gauge transformations. This can be done at the expense of extending the phase space by introducing new boundary fields, which in turn determine new boundary symmetries and observables. We present the general framework behind this construction, and find the conditions under which it can be applied to an arbitrary Lagrangian. By studying the examples of Abelian Chern-Simons theory and first order three-dimensional gravity, we then show that the new boundary observables satisfy the known corresponding Kac-Moody affine algebras. This shows that this new extended phase space formulation does indeed properly describe the dynamical boundary degrees of freedom, and gives credit to the results which have been previously derived in the case of diffeomorphism symmetry. We expect that this systematic understanding of the boundary symmetries will play a major role for the quantization of gravity in finite regions., Comment: Published version

In loop quantum cosmology, one has to make a choice of SU(2) irreducible representation in which to compute holonomies and regularize the curvature of the connection. The systematic choice made in the literature is to work in the fundamental representation, and very little is known about the physics associated with higher spin labels. This constitutes an ambiguity whose understanding, we believe, is fundamental for connecting loop quantum cosmology to full theories of quantum gravity like loop quantum gravity, its spin foam formulation, or cosmological group field theory. We take a step in this direction by providing here a new closed formula for the Hamiltonian of flat FLRW models regularized in a representation of arbitrary spin. This expression is furthermore polynomial in the basic variables which correspond to well-defined operators in the quantum theory, takes into account the so-called inverse-volume corrections, and treats in a unified way two different regularization schemes for the curvature. After studying the effective classical dynamics corresponding to single and multiple spin Hamiltonians, we study the behavior of the critical density when the number of representations is increased, and the stability of the difference equations in the quantum theory., Comment: 32 pages, colored figures

We show how extended topological quantum field theories (TQFTs) can be used to obtain a kinematical setup for quantum gravity, i.e. a kinematical Hilbert space together with a representation of the observable algebra including operators of quantum geometry. In particular, we consider the holonomy-flux algebra of (2+1)-dimensional Euclidean loop quantum gravity, and construct a new representation of this algebra that incorporates a positive cosmological constant. The vacuum state underlying our representation is defined by the Turaev-Viro TQFT. We therefore construct here a generalization, or more precisely a quantum deformation at root of unity, of the previously-introduced SU(2) BF representation. The extended Turaev-Viro TQFT provides a description of the excitations on top of the vacuum, which are essential to allow for a representation of the holonomies and fluxes. These excitations agree with the ones induced by massive and spinning particles, and therefore the framework presented here allows automatically for a description of the coupling of such matter to (2+1)-dimensional gravity with a cosmological constant. The new representation presents a number of advantages over the representations which exist so far. It possesses a very useful finiteness property which guarantees the discreteness of spectra for a wide class of quantum (intrinsic and extrinsic) geometrical operators. The notion of basic excitations leads to a fusion basis which offers exciting possibilities for constructing states with interesting global properties. The work presented here showcases how the framework of extended TQFTs can help design new representations and understand the associated notion of basic excitations. This is essential for the construction of the dynamics of quantum gravity, and will enable the study of possible phases of spin foam models and group field theories from a new perspective., Comment: 81 pages

We construct in this article a new realization of quantum geometry, which is obtained by quantizing the recently-introduced flux formulation of loop quantum gravity. In this framework, the vacuum is peaked on flat connections, and states are built upon it by creating local curvature excitations. The inner product induces a discrete topology on the gauge group, which turns out to be an essential ingredient for the construction of a continuum limit Hilbert space. This leads to a representation of the full holonomy-flux algebra of loop quantum gravity which is unitarily-inequivalent to the one based on the Ashtekar-Isham-Lewandowski vacuum. It therefore provides a new notion of quantum geometry. We discuss how the spectra of geometric operators, including holonomy and area operators, are affected by this new quantization. In particular, we find that the area operator is bounded, and that there are two different ways in which the Barbero-Immirzi parameter can be taken into account. The methods introduced in this work open up new possibilities for investigating further realizations of quantum geometry based on different vacua., Comment: 72 pages, 6 figures

We study the thermodynamical properties of black holes when described as gases of indistinguishable punctures with a chemical potential. In this picture, which arises from loop quantum gravity, the black hole microstates are defined by finite families of half-integers spins coloring the punctures, and the near-horizon energy measured by quasi-local stationary observers defines the various thermodynamical ensembles. The punctures carry excitations of quantum geometry in the form of quanta of area, and the total horizon area $a_\text{H}$ is given by the sum of these microscopic contributions. We assume here that the system satisfies the Bose-Einstein statistics, and that each microstate is degenerate with a holographic degeneracy given by $\exp\big(\lambda a_\text{H}/\ell_\text{Pl}^2\big)$ and $\lambda>0$. We analyze in detail the thermodynamical properties resulting from these inputs, and in particular compute the grand canonical entropy. We explain why the requirements that the temperature be fixed to the Unruh temperature and that the chemical potential vanishes do not specify completely the semi-classical regime of large horizon area, and classify in turn what the various regimes can be. When the degeneracy saturates the holographic bound ($\lambda=1/4$), there exists a semi-classical regime in which the subleading corrections to the entropy are logarithmic. Furthermore, this regime corresponds to a Bose-Einstein condensation, in the sense that it is dominated by punctures carrying the minimal (or ground state) spin value $1/2$., Comment: 22 pages

We recently introduced a new representation for loop quantum gravity, which is based on the BF vacuum and is in this sense much nearer to the spirit of spin foam dynamics. In the present paper we lay out the classical framework underlying this new formulation. The central objects in our construction are the so-called integrated fluxes, which are defined as the integral of the electric field variable over surfaces of codimension one, and related in turn to Wilson surface operators. These integrated flux observables will play an important role in the coarse graining of states in loop quantum gravity, and can be used to encode in this context the notion of curvature-induced torsion. We furthermore define a continuum phase space as the modified projective limit of a family of discrete phase spaces based on triangulations. This continuum phase space yields a continuum (holonomy-flux) algebra of observables. We show that the corresponding Poisson algebra is closed by computing the Poisson brackets between the integrated fluxes, which have the novel property of being allowed to intersect each other., Comment: 60 pages, 13 figures, published version

We compute the near-horizon radiation of quantum black holes in the context of self-dual loop quantum gravity. For this, we first use the unitary spinor basis of $\text{SL}(2,\mathbb{C})$ to decompose states of Lorentzian spin foam models into their self-dual and anti self-dual parts, and show that the reduced density matrix obtained by tracing over one chiral component describes a thermal state at Unruh temperature. Then, we show that the analytically-continued dimension of the $\text{SU}(2)$ Chern-Simons Hilbert space, which reproduces the Bekenstein-Hawking entropy in the large spin limit in agreement with the large spin effective action, takes the form of a partition function for states thermalized at Unruh temperature, with discrete energy levels given by the near-horizon energy of Frodden-Gosh-Perez, and with a degenerate ground state which is holographic and responsible for the entropy., Comment: 6+2 pages

We construct a new vacuum for loop quantum gravity, which is dual to the Ashtekar-Lewandowski vacuum. Because it is based on BF theory, this new vacuum is physical for $(2+1)$-dimensional gravity, and much closer to the spirit of spin foam quantization in general. To construct this new vacuum and the associated representation of quantum observables, we introduce a modified holonomy-flux algebra which is cylindrically consistent with respect to the notion of refinement by time evolution suggested in [1]. This supports the proposal for a construction of a physical vacuum made in [1,2], also for $(3+1)$-dimensional gravity. We expect that the vacuum introduced here will facilitate the extraction of large scale physics and cosmological predictions from loop quantum gravity., Comment: 11 pages, 5 figures

We show the explicit agreement between the derivation of the Bekenstein-Hawking entropy of a Euclidean BTZ black hole from the point of view of spin foam models and canonical quantization. This is done by considering a graph observable (corresponding to the black hole horizon) in the Turaev-Viro state sum model, and then analytically continuing the resulting partition function to negative values of the cosmological constant., Comment: 22+4 pages, 3 figures

We study and compare the spectra of geometric operators (length and area) in the quantum kinematics of two formulations of three-dimensional Lorentzian loop quantum gravity. In the SU(2) Ashtekar-Barbero framework, the spectra are discrete and depend on the Barbero-Immirzi parameter $\gamma$ exactly like in the four-dimensional case. However, we show that when working with the self-dual variables and imposing the reality conditions the spectra become continuous and $\gamma$-independent., Comment: 13 pages. 2 figures

We study the role of the Barbero-Immirzi parameter $\gamma$ and the choice of connection in the construction of (a symmetry-reduced version of) loop quantum gravity. We start with the four-dimensional Lorentzian Holst action that we reduce to three dimensions in a way that preserves the presence of $\gamma$. In the time gauge, the phase space of the resulting three-dimensional theory mimics exactly that of the four-dimensional one. Its quantization can be performed, and on the kinematical Hilbert space spanned by SU(2) spin network states the spectra of geometric operators are discrete and $\gamma$-dependent. However, because of the three-dimensional nature of the theory, its SU(2) Ashtekar-Barbero Hamiltonian constraint can be traded for the flatness constraint of an sl(2,C) connection, and we show that this latter has to satisfy a linear simplicity-like condition analogous to the one used in the construction of spin foam models. The physically relevant solution to this constraint singles out the non-compact subgroup SU(1,1), which in turn leads to the disappearance of the Barbero-Immirzi parameter and to a continuous length spectrum, in agreement with what is expected from Lorentzian three-dimensional gravity., Comment: 36 pages

In this note, we review the canonical analysis of the Holst action in the time gauge, with a special emphasis on the Hamiltonian equations of motion and the fixation of the Lagrange multipliers. This enables us to identify at the Hamiltonian level the various components of the covariant torsion tensor, which have to be vanishing in order for the classical theory not to depend upon the Barbero-Immirzi parameter. We also introduce a formulation of three-dimensional gravity with an explicit phase space dependency on the Barbero-Immirzi parameter as a potential way to investigate its fate and relevance in the quantum theory., Comment: 22 pages. Published version. Choice of gauge at the begining of section II.B. clarified. Published in Gen. Rel. Grav. (2013)

We compute the statistical entropy of a BTZ black hole in the context of three-dimensional Euclidean loop quantum gravity with a cosmological constant $\Lambda$. As in the four-dimensional case, a quantum state of the black hole is characterized by a spin network state. Now however, the underlying colored graph $\Gamma$ lives in a two-dimensional spacelike surface $\Sigma$, and some of its links cross the black hole horizon, which is viewed as a circular boundary of $\Sigma$. Each link $\ell$ crossing the horizon is colored by a spin $j_\ell$ (at the kinematical level), and the length $L$ of the horizon is given by the sum $L=\sum_\ell L_\ell$ of the fundamental length contributions $L_\ell$ carried by the spins $j_\ell$ of the links $\ell$. We propose an estimation for the number $N^\text{BTZ}_\Gamma(L,\Lambda)$ of the Euclidean BTZ black hole microstates (defined on a fixed graph $\Gamma$) based on an analytic continuation from the case $\Lambda>0$ to the case $\Lambda<0$. In our model, we show that $N^\text{BTZ}_\Gamma(L,\Lambda)$ reproduces the Bekenstein-Hawking entropy in the classical limit. This asymptotic behavior is independent of the choice of the graph $\Gamma$ provided that the condition $L=\sum_\ell L_\ell$ is satisfied, as it should be in three-dimensional quantum gravity., Comment: 14 pages. 1 figure. Paragraph added on page 7 to clarify the horizon condition

In loop quantum gravity, the number $N_\Gamma(A,\gamma)$ of microstates of a black hole for a given discrete geometry $\Gamma$ depends on the so-called Barbero-Immirzi parameter $\gamma$. Using a suitable analytic continuation of $\gamma$ to complex values, we show that the number $N_\Gamma(A,\pm i)$ of microstates behaves as $\exp(A/(4\ell_\text{Pl}^2))$ for large area $A$ in the large spin semiclassical limit. Such a correspondence with the semiclassical Bekenstein-Hawking entropy law points towards an unanticipated and remarkable feature of the original complex Ashtekar variables for quantum gravity., Comment: 5 pages. New point of view on the analytic continuation, which is now made rigorous by analytically-continuing the SU(2) spins in addition to the Barbero-Immirzi parameter

We introduce a three-dimensional Plebanski action for the gauge group SO(4). In this model, the $B$ field satisfies quadratic simplicity constraints similar to that of the four-dimensional Plebanski theory, but with the difference that the $B$ field is now a one-form. We exhibit a natural notion of "simple one-form", and identify a gravitational sector, a topological sector and a degenerate sector in the space of solutions to the simplicity constraints. Classically, in the gravitational sector, the action is shown to be equivalent to that of three-dimensional first order Riemannian gravity. This enables us to perform the complete spin foam quantization of the theory once the simplicity constraints are solved at the classical level, and to compare this result with the various models that have been proposed for the implementation of the constraints after quantization. In particular, we impose the simplicity constraints following the prescriptions of the so-called BC and EPRL models. We observe that the BC prescription cannot lead to the proper vertex amplitude. The EPRL prescription allows to recover the expected result when, in this three-dimensional model, it is supplemented with additional secondary second class constraints., Comment: 30 pages. 18 figures. References added. New paragraph in the conclusion

This review is devoted to the analysis of the mutual consistency of the spin foam and canonical loop quantizations in three and four spacetime dimensions. In the three-dimensional context, where the two approaches are in good agreement, we show how the canonical quantization \`a la Witten of Riemannian gravity with a positive cosmological constant is related to the Turaev-Viro spin foam model, and how the Ponzano-Regge amplitudes are related to the physical scalar product of Riemannian loop quantum gravity without cosmological constant. In the four-dimensional case, we recall a Lorentz-covariant formulation of loop quantum gravity using projected spin networks, compare it with the new spin foam models, and identify interesting relations and their pitfalls. Finally, we discuss the properties which a spin foam model is expected to possess in order to be consistent with the canonical quantization, and suggest a new model illustrating these results.

In this paper, we study the discrete classical phase space of loop gravity, which is expressed in terms of the holonomy-flux variables, and show how it is related to the continuous phase space of general relativity. In particular, we prove an isomorphism between the loop gravity discrete phase space and the symplectic reduction of the continuous phase space with respect to a flatness constraint. This gives for the first time a precise relationship between the continuum and holonomy-flux variables. Our construction shows that the fluxes depend on the three-geometry, but also explicitly on the connection, explaining their non commutativity. It also clearly shows that the flux variables do not label a unique geometry, but rather a class of gauge-equivalent geometries. This allows us to resolve the tension between the loop gravity geometrical interpretation in terms of singular geometry, and the spin foam interpretation in terms of piecewise flat geometry, since we establish that both geometries belong to the same equivalence class. This finally gives us a clear understanding of the relationship between the piecewise flat spin foam geometries and Regge geometries, which are only piecewise-linear flat: While Regge geometry corresponds to metrics whose curvature is concentrated around straight edges, the loop gravity geometry correspond to metrics whose curvature is concentrated around not necessarily straight edges., Comment: published version

In this work, we study the classical and quantum properties of the unique commutative Lorentz-covariant connection for loop quantum gravity. This connection has been found after solving the second-class constraints inherited from the canonical analysis of the Holst action without the time-gauge. We show that it has the property of lying in the conjugacy class of a pure $\su(2)$ connection, a result which enables one to construct the kinematical Hilbert space of the Lorentz-covariant theory in terms of the usual $\SU(2)$ spin-network states. Furthermore, we show that there is a unique Lorentz-covariant electric field, up to trivial and natural equivalence relations. The Lorentz-covariant electric field transforms under the adjoint action of the Lorentz group, and the associated Casimir operators are shown to be proportional to the area density. This gives a very interesting algebraic interpretation of the area. Finally, we show that the action of the surface operator on the Lorentz-covariant holonomies reproduces exactly the usual discrete $\SU(2)$ spectrum of time-gauge loop quantum gravity. In other words, the use of the time-gauge does not introduce anomalies in the quantum theory., Comment: 28 pages. Revised version taking into account referee's comments