Faddeev formulation of gravity in discrete form

TL;DR

This paper develops a discrete formulation of Faddeev gravity, representing the metric with vector fields constant within simplices, and connects it with a first-order connection approach to facilitate calculations on simplicial complexes.

Contribution

It introduces a discrete form of the Faddeev gravity action incorporating connection variables and demonstrates its consistency with the continuous formulation for slowly varying fields.

Findings

Discrete Faddeev action is consistent with continuous form for slow field variations.

The connection representation aids in fixing the action on piecewise constant fields.

The approach provides a framework for analyzing gravity in a simplicial setting.

Abstract

We study Faddeev formulation of gravity, in which the metric is composed of vector fields. We consider these fields constant in the interior of the 4-simplices of a simplicial complex. The action depends not only on the values of the fields in the interior of the 4-simplices but on the details of (regularized) jump of the fields between the 4-simplices. Though, when the fields vary arbitrarily slowly from the 4-simplex to 4-simplex, the latter dependence is negligible (of the next-to-leading order of magnitude). We put the earlier proposed in our work first order (connection) representation of the Faddeev action into the discrete form. We show that upon excluding the connections it is consistent with the above Faddeev action on the piecewise constant fields in the leading order of magnitude. Thus, using the discrete form of the connection representation of the Faddeev action can serve a way to fix the value of this action on the piecewise constant ansatz on simplices.