  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.
