To each local field (including the real or complex numbers) we associate a quantum dilogarithm and show that it satisfies a pentagon identity and some symmetries. Using an angled version of these quantum dilogarithms, we construct three generalized TQFTs in 2+1 dimensions, one given by a face state-integral and two given by edge state-integrals. Their partition functions rise to distributional invariants of 3-manifolds with torus boundary, conjecturally related to point counting of the A-polynomial curve. The partition function of one of these face generalized TQFTs for the case of the real numbers can be expressed either as a multidimensional Barnes-Mellin integral or as a period on a curve which is conjecturally the A-polynomial curve.
