# New first order Lagrangian for General Relativity

1. May 15, 2015

### MTd2

http://arxiv.org/abs/1503.08640

New first order Lagrangian for General Relativity

Yannick Herfray, Kirill Krasnov
(Submitted on 30 Mar 2015)
We describe a new BF-type first-order in derivatives Lagrangian for General Relativity. The Lagrangian depends on a connection field as well as a Lie-algebra valued two-form field, with no other fields present. There are two free parameters, which translate into the cosmological constant and the coefficient in front of a topological term. When one of the parameters is set to zero, the theory becomes topological. When the other parameter is zero, the theory reduces to the (anti-) self-dual gravity. Thus, our new Lagrangian interpolates between the topological and anti-self-dual gravities. It also interprets GR as the (anti-) self-dual gravity with an extra quadratic in the auxiliary two-form field term added to the Lagrangian, precisely paralleling the situation in Yang-Mills theory.

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"Our work is also of relevance for the spin foam approach to quantum gravity [15]. The spin foam description of the topological theory, whose Lagrangian is (2) with α = 0, is considered to be understood. Thus, if it was possible to give a spin foam description of the ASD gravity (2) with λ = 0, then it would perhaps be also possible to combine the two and obtain full GR. Given that the theory with α = 0 is believed to give rise to the quantum group SUq(2), our description thus points in the direction of full GR being about ”q-deformed instantons”, whatever that may be."

Last edited: May 15, 2015
2. May 16, 2015

### MTd2

3. May 16, 2015

### MTd2

Other papers by Krasnov which led directly to this results are :

http://arxiv.org/pdf/1103.4498.pdf
New Action Principle for General Relativity
General Relativity can be reformulated as a diffeomorphism invariant SU(2) gauge theory. A new action principle for this ”pure connection” formulation of GR is described

http://arxiv.org/pdf/1101.4788.pdf
Gravity as a diffeomorphism invariant gauge theory

A general diffeomorphism invariant SU(2) gauge theory is a gravity theory with two propagating polarizations of the graviton. We develop this description of gravity, in particular for future applications to the perturbative quantization. Thus, the linearized theory, gauge symmetries, gauge fixing are discussed in detail, and the propagator is obtained. The propagator takes a simple form of that of Yang-Mills theory with an additional projector on diffeomorphism equivalence classes of connections inserted. In our approach the gravitational perturbation theory takes a rather unusual form in that the Planck length is no longer fundamental.

http://arxiv.org/pdf/1205.7045.pdf
Pure Connection Formalism for Gravity: Linearized Theory

Abstract We give a description of gravitons in terms of an SL(2, C) connection field. The gauge-theoretic Lagrangian for gravitons is simpler than the metric one, in particular because the Lagrangian only depends on 8 components of the field per spacetime point as compared to 10 in the Einstein-Hilbert case. Particular care is paid to the treatment of the reality conditions that guarantee that one is dealing with a system with a hermitian Hamiltonian. We give general arguments explaining why the connection cannot be taken to be real, and then describe a reality condition that relates the hermitian conjugate of the connection to its (second) derivative. This is quite analogous to the treatment of fermions where one describes them by a second-order in derivatives Klein-Gordon Lagrangian, with an additional first-order reality condition (Dirac equation) imposed. We find many other parallels with fermions, e.g. the fact that the action of parity on the connection is related to the hermitian conjugation. Our main result is the mode decomposition of the connection field, which is to be used in forthcoming works for computations of graviton scattering amplitudes.