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Amphithéâtre Léon Motchane (Institut des Hautes Etudes Scientifiques)
Amphithéâtre Léon Motchane
Institut des Hautes Etudes Scientifiques
35, route de Chartres
I will explain how General Relativity in four space-time dimensions (with cosmological constant) can be reformulated as an SU(2) gauge theory of a certain type. The action is a (diffeomorphism invariant) functional of an SU(2) connection; no metric is present in the formulation of the theory. This formulation of GR is in many ways analogous to the one proposed many years ago by Eddington (Eddington's Lagrangian is a function of just the affine connection and is given by the square root of the determinant of the Ricci tensor). The new formulation has some remarkable properties. First, on a 4-manifold, the space of SU(2) connections modulo gauge transformations has just 9 components per space-time point, as compared to 10 components of a metric tensor. Correspondingly, the action of the new formulation can be interpreted as a functional on the space of conformal classes of metrics. Thus, the conformal factor is not free to propagate in this formulation even off-shell. This has the effect that the action functional is (at least locally) convex - there is no conformal mode problem of the usual metric formulation of GR. Another remarkable property of the new formulation is that diffeomorphisms are very easy to deal with. The most natural gauge-fixing of these does not make the corresponding components of the connection propagate. As a result, in (linearized) quantum theory only 8 components of the connection propagate as compared to 10 metric components in the usual formulation. All these 8 components fit into a single irreducible representation of the Lorentz group, which makes the propagator very simple. There are also some remarkable simplifications in the structure of the interaction vertices. The full off-shell 3-vertex (in de Sitter space) contains just 3 terms as compared to a couple of dozen terms in the metric formulation. The 4-vertex is a couple of lines as compared to a couple of pages in the standard description. As an illustration of the formalism I will describe how the graviton scattering amplitudes are computed in this approach.
The new gauge-theoretic reformulation of GR also leads to (an infinite-parameter) family of modified gravitational theories, all propagating just two polarizations of the graviton, as GR. This leads to a rather strong claim that, in spite of the standard GR uniqueness theorems, General Relativity is not the only interacting theory of massless spin two particles. However, GR appears to be the only parity-invariant gravitational theory, as all the "deformations of GR" can be shown to be parity violating. I will also describe how matter is coupled in this approach, and give some speculations as to a possible UV completion of gravity.