Discrete Faddeev action for the tetrad fields strongly varying along different coordinates

Abstract

Faddeev gravity using a $d$-dimensional tetrad (normally $d = 10$) is classically equivalent to general relativity (GR). The discrete Faddeev gravity on the piecewise flat spacetime normally assumes slowly varying metric and tetrad from vertex to vertex. Meanwhile, Faddeev action is finite (although not unambiguously defined) for discontinuous tetrad fields thus allowing, in particular, to consider a surface as consisting of virtually independent elementary triangles, and its area spectrum as the sum of elementary area spectra. In the discrete connection form, area tensors are canonically conjugate to SO(10) connection matrices, and earlier we have found the elementary area spectrum, which is nonsingular just at large connection or the strongly varying fields constituting a kind of "antiferromagnetic" structure. We appropriately define discrete {\it connection} Faddeev action to unambiguously determine the discrete Faddeev action for the strongly varying fields, but weakly varying metric, equivalent in the continuum limit to the GR action with this metric. Previously, we considered large variations in only one direction, now we use an ansatz in some respects less common, but overall, probably the most common. A unified simplicial connection representation is written out (depending on an auxiliary connection) both for the discrete Faddeev action and for the Regge action.