Marc Henneaux

Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity

It is shown that the global charges of a gauge theory may yield a nontrivial central extension of the asymptotic symmetry algebra already at the classical level. This is done by studying three dimensional gravity with a negative cosmological constant. The asymptotic symmetry group in that case is eitherR×SO(2) or the pseudo-conformal group in two dimensions, depending on the boundary conditions adopted at spatial infinity. In the latter situation, a nontrivial central charge appears in the algebra of the canonical generators, which turns out to be just the Virasoro central charge.

Geometry of the (2+1) black hole

The geometry of the spinning black holes of standard Einstein theory in 2+1 dimensions, with a negative cosmological constant, and without couplings to matter, is analyzed in detail. It is shown that the black hole arises from identifications of points of anti\char21{}de Sitter space by a discrete subgroup of SO(2,2). The generic black hole is a smooth manifold in the metric sense. The surface r=0 is not a curvature singularity but, rather, a singularity in the causal structure. Continuing past it would introduce closed timelike lines. However, simple examples show the regularity of the metric at r=0 to be unstable: couplings to matter bring in a curvature singularity there. Kruskal coordinates and Penrose diagrams are exhibited. Special attention is given to the limiting cases of (i) the spinless hole of zero mass, which differs from anti\char21{}de Sitter space and plays the role of the vacuum, and (ii) the spinning hole of maximal angular momentum. A thorough classification of the elements of the Lie algebra of SO(2,2) is given in an appendix.

Local BRST cohomology in gauge theories

Abstract The general solution of the anomaly consistency condition (Wess–Zumino equation) has been found recently for Yang–Mills gauge theory. The general form of the counterterms arising in the renormalization of gauge-invariant operators (Kluberg–Stern and Zuber conjecture) and in gauge theories of the Yang–Mills type with non-power counting renormalizable couplings has also been worked out in any number of space–time dimensions. This Physics Report is devoted to reviewing in a self-contained manner these results and their proofs. This involves computing cohomology groups of the differential introduced by Becchi, Rouet, Stora and Tyutin, with the sources of the BRST variations of the fields (“antifields”) included in the problem. Applications of this computation to other physical questions (classical deformations of the action, conservation laws) are also considered. The general algebraic techniques developed in the Report can be applied to other gauge theories, for which relevant references are given.

Nonlinear W∞ as asymptotic symmetry of three-dimensional higher spin AdS gravity

We investigate the asymptotic symmetry algebra of (2+1)-dimensional higher spin, anti-de Sitter gravity. We use the formulation of the theory as a Chern-Simons gauge theory based on the higher spin algebra hs(1, 1). Expanding the gauge connection around asymptotically anti-de Sitter spacetime, we specify consistent boundary conditions on the higher spin gauge fields. We then study residual gauge transformation, the corresponding surface terms and their Poisson bracket algebra. We find that the asymptotic symmetry algebra is a nonlinearly realized W∞ algebra with classical central charges. We discuss implications of our results to quantum gravity and to various situations in string theory.

Local BRST cohomology in the antifield formalism. I. General theorems

We establish general theorems on the cohomologyH*(s/d) of the BRST differential modulo the spacetime exterior derivative, acting in the algebra of localp-forms depending on the fields and the antifields (=sources for the BRST variations). It is shown thatH−k(s/d) is isomorphic toHk(δ/d) in negative ghost degree−k (k>0), where δ is the Koszul-Tate differential associated with the stationary surface. The cohomology groupH1(δ/d) in form degreen is proved to be isomorphic to the space of constants of the motion, thereby providing a cohomological reformulation of Noethers theorem. More generally, the groupHk(δ/d) in form degreen is isomorphic to the space ofn−k forms that are closed when the equations of motion hold. The groupsHk(δ/d)(k>2) are shown to vanish for standard irreducible gauge theories. The groupH2(δ/d) is then calculated explicitly for electromagnetism, Yang-Mills models and Einstein gravity. The invariance of the groupsHk(s/d) under the introduction of non-minimal variables and of auxiliary fields is also demonstrated. In a companion paper, the general formalism is applied to the calculation ofHk(s/d) in Yang-Mills theory, which is carried out in detail for an arbitrary compact gauge group.

Asymptotically anti-De Sitter Spaces

Asymptotically anti-de Sitter spaces are defined by boundary conditions on the gravitational field which obey the following criteria: (i) they are O(3, 2) invariant; (ii) they make the O(3, 2) surface integral charges finite; (iii) they include the Kerr-anti-de Sitter metric. An explicit expression of the O(3, 2) charges in terms of the canonical variables is given. These charges are shown to close in the Dirac brackets according to the anti-de Sitter algebra. The results are extended to the case ofN=1 supergravity. The coupling to gravity of a third-rank, completely antisymmetric, abelian gauge field is also considered. That coupling makes it possible to vary the cosmological constant and to compare the various anti-de Sitter spaces which are shown to have the same energy.

The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant

Liouville theory is shown to describe the asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant. This is because (i) Chern - Simons theory with a gauge group on a spacetime with a cylindrical boundary is equivalent to the non-chiral SL(2,R) WZW model; and (ii) the anti-de Sitter boundary conditions implement the constraints that reduce the WZW model to the Liouville theory.

Consistent couplings between fields with a gauge freedom and deformations of the master equation

Abstract The antibracket in the antifield-BRST formalism is known to define a map H p × H q → H p + q + 1 associating with two equivalence classes of BRST invariant observables of respective ghost number p and q an equivalence class of BRST invariant observables of ghost number p + q + 1. It is shown that this map is trivial in the space of all functionals, i.e., that its image contains only the zeroth class. However, it is generically non-trivial in the space of local functionals. Implications of this result for the problem of consistent interactions among fields with a gauge freedom are then drawn. It is shown that the obstructions to constructing non-trivial such interactions lie precisely in the image of the antibracket map and are accordingly non-existent if one does not insist on locality. However consistent local interactions are severely constrained. The example of the Chern-Simons theory is considered. It is proved that the only consistent, local, Lorentz covariant interactions for the abelian models are exhausted by the non-abelian Chern-Simons extensions.

Dynamics of chiral (self-dual) p-forms

Abstract An action principle describing the dynamics of a p-form gauge field whose field strength is self-dual is given. The action is local, Lorentz invariant and also invariant under the standard gauge transformation of a p-form. The coupling to gravitation is described. The proposed action permits a consistent passage to quantum mechanics. The path integral is briefly discussed.

The Cosmological Constant and General Covariance

In the standard gravitational action without a cosmological constant, one may consider the determinant of the metric as an external field. One then extremizes the action only with respect to variations of the metric that do not change the local volume. The resulting field equations are Einsteins equations with a cosmological constant, which appears as a constant of integration. The dynamics of that theory is analyzed from the point of view of constrained hamiltonian systems. It is observed that contrary to what one might think, the theory is fully covariant and contains only one overall degree of freedom (the cosmological constant) in addition to the two degrees of freedom per point of ordinary Einsteins theory. In the hamiltonian formalism the missing coordinate invariance re-emerges through a tertiary constraint. A Yang-Mills analog is pointed out. The theory is then made manifestly generally covariant by introducing auxiliary fields that are pure gauge except for one overall zero mode. This global mode is a “cosmic time” canonically conjugate to the cosmological constant.