Asymptotically hyperbolic connections

TL;DR

This paper reformulates 4D General Relativity using SO(3)-connections, introduces asymptotically hyperbolic connections, and develops a simplified expansion method revealing new features like boundary Chern-Simons terms and quantization conditions.

Contribution

It introduces asymptotically hyperbolic connections, derives an analog of the Fefferman-Graham expansion in this setting, and explores implications for quantum gravity and deformed theories.

Findings

Simpler connection-based expansion analogous to metric case

Obstruction appears at third order in the expansion

Counter terms form the boundary Chern-Simons functional

Abstract

General Relativity in 4 dimensions can be equivalently described as a dynamical theory of SO(3)-connections rather than metrics. We introduce the notion of asymptotically hyperbolic connections, and work out an analog of the Fefferman-Graham expansion in the language of connections. As in the metric setup, one can solve the arising "evolution" equations order by order in the expansion in powers of the radial coordinate. The solution in the connection setting is arguably simpler, and very straightforward algebraic manipulations allow one to see how the obstruction appears at third order in the expansion. Another interesting feature of the connection formulation is that the "counter terms" required in the computation of the renormalised volume all combine into the Chern-Simons functional of the restriction of the connection to the boundary. As the Chern-Simons invariant is only defined modulo large gauge transformations, the requirement that the path integral over asymptotically hyperbolic connections is well-defined requires the cosmological constant to be quantised. Finally, in the connection setting one can deform the 4D Einstein condition in an interesting way, and we show that asymptotically hyperbolic connection expansion is universal and valid for any of the deformed theories.