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Gravitational behavior of an effective topological field theory in a gravitational instanton background

Effective topological field theories describe the properties of Dirac fermions in the low-energy regime. In this work, we introduce a new emergent gravity model by considering Dirac fermions invariant under local de Sitter transformations in four-dimensional open manifolds. In the context of Cartan geometry, fermions couple to spacetime through a $Spin(5)$ Cartan connection that can be decomposed in spin connection and tetrads. In presence of a gravitational instanton background, we show that the corresponding effective topological field theory becomes a dynamical gravitational theory with a positive cosmological constant and Barbero-Immirzi parameter. At classical level and in absence of matter, this theory is compatible with general relativity.