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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,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.then it would perhaps be also possible to combine the two and obtain full GR."

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# New first order Lagrangian for General Relativity

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