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 pre-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].

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.

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 $$ \mathfrak{sl} $$ sl (2, ℂ) component to the corner symmetry algebra of tetrad gravity. Second, we also reveal that the corner metric satisfies a local $$ \mathfrak{sl} $$ sl (2, 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.

In loop quantum gravity, the number NΓ(A, γ) of microstates of a black hole for a given discrete geometry Γ depends on the so-called Barbero-Immirzi parameter γ. Using a suitable analytic continuation of γ to complex values, we show that the number NΓ(A, ±i) of microstates behaves as exp(A/(4ℓ 2 Pl )) 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. PACS numbers: 04.70.Dy, 04.60.-mThe Barbero-Immirzi parameter γ was originally introduced [1] as a way to circumvent the problem of imposing the reality constraints in the complex (self-dual) Ashtekar formulation of gravity [2]. Historically, γ appeared as a parameter labeling a family of canonical transformations turning the ADM phase space into the so-called Ashtekar-Barbero phase space, parametrized by a real su(2) connection and its conjugate momentum. Later on, Holst [3] realized that this Hamiltonian formulation of gravity could be obtained by adding to the standard Hilbert-Palatini Lagrangian a topological term with γ as a coupling constant. This term vanishes due to the Bianchi identities when one resolves the spin connection in terms of the tetrad, and for this reason γ is not relevant at the classical level.In the quantum theory however, the Barbero-Immirzi parameter plays a crucial role since the spectrum of the geometric operators is discrete in units of the loop quantum gravity (LQG) scale ℓ LQG = √ γG = √ γℓ Pl , where ℓ Pl is the Planck length [4]. Moreover, this γ-dependency of the fundamental physical cut-off is inherited by the value of the black hole entropy in the LQG calculation. Compatibility with the expected semiclassical value S = A/(4ℓ 2 Pl ) (where A is the area of the horizon) requires that γ be fixed to a particular real value. In fact, a lot of different techniques have been developed in order to obtain the value of γ [5, 6] (see also [7]).In LQG, the horizon of a black hole has the topology of a 2-sphere, with colored punctures coming from the spin network links that cross the horizon. Each puncture carries a quantum of area, and the sum of these microscopic areas gives the macroscopic area A of the horizon. In the

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.