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Asymptotically hyperbolic connections. (English) Zbl 1349.83008

Summary: General relativity in four-dimensions can be equivalently described as a dynamical theory of \(\mathrm{SO}(3)\sim \mathrm{SU}(2)\)-connections rather than metrics. We introduce the notion of asymptotically hyperbolic connections, and work out an analogue 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 unconstrained by Einstein equations ‘stress-energy tensor’ 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.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
53Z05 Applications of differential geometry to physics
58J28 Eta-invariants, Chern-Simons invariants
81S40 Path integrals in quantum mechanics
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