Gauge invariance of complex general relativity

TL;DR

This paper demonstrates how to ensure gauge invariance in complex general relativity using Ashtekar variables, clarifying boundary conditions and gauge fixing, and showing full gauge invariance at first order in specific topological cases.

Contribution

It introduces a method to reconcile boundary conditions with gauge fixing in complex general relativity and proves gauge invariance of the Hamiltonian action under certain conditions.

Findings

Hamiltonian action is fully gauge-invariant at first order in gauge parameters.

Boundary terms vanish when the spatial slice has no boundary.

Constraints linear in momenta do not contribute to boundary terms in boundaryless topologies.

Abstract

In this paper it is implemented how to make compatible the boundary conditions and the gauge fixing conditions for complex general relativity written in terms of Ashtekar variables using the Henneaux-Teitelboim-Vergara approach. Moreover, it is found that at first order in the gauge parameters, the Hamiltonian action is (on shell) fully gauge-invariant under the gauge symmetry generated by the first class constraints in the case when spacetime $\cal M$ has the topology ${\cal M}= R \times \Sigma$ and $\Sigma$ has no boundary. Thus, the statement that the constraints linear in the momenta do not contribute to the boundary terms is right, but only in the case when $\Sigma$ has no boundary.