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Gravity as an $\mathfrak{su}(1, 1)$ gauge theory in four dimensions

Abstract : We start with the Hamiltonian formulation of the first order action of pure gravity with a full $\mathfrak{sl}(2, \mathbb C)$ internal gauge symmetry. We make a partial gauge-fixing which reduces $\mathfrak{sl}(2, \mathbb C)$ to its sub-algebra $\mathfrak{su}(1, 1)$ . This case corresponds to a splitting of the space-time ${\mathcal M}=\Sigma \times \mathbb R$ where $\Sigma$ inherits an arbitrary Lorentzian metric of signature (−, +, +). Then, we find a parametrization of the phase space in terms of an $\mathfrak{su}(1, 1)$ commutative connection and its associated conjugate electric field. Following the techniques of loop quantum gravity, we start the quantization of the theory and we consider the kinematical Hilbert space on a given fixed graph $\Gamma$ whose edges are colored with unitary representations of $\mathfrak{su}(1, 1)$ . We compute the spectrum of area operators acting on the kinematical Hilbert space: we show that space-like areas have discrete spectra, in agreement with usual $\mathfrak{su}(2)$ loop quantum gravity, whereas time-like areas have continuous spectra. We conclude on the possibility to make use of this formulation of gravity to construct a holographic description of black holes in the framework of loop quantum gravity.
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Hongguang Liu, Karim Noui. Gravity as an $\mathfrak{su}(1, 1)$ gauge theory in four dimensions. Class.Quant.Grav., 2017, 34 (13), pp.135008. ⟨10.1088/1361-6382/aa7348⟩. ⟨hal-01582725⟩

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