Higher-dimensional Chern-Simons theory appears in the description of isolated horizon boundaries in higher-dimensional General Relativity.

It is a well-known fact that the presence of boundaries (“edges”) leads to the concept of boundary states, which e.g. ensure gauge invariance for parallel transporters ending on the boundary. Most famously, the quantum Hall effect can be explained using such states. In the context of black hole (quantum) physics, boundary states are important since they are microscopic states associated to the horizon of the black hole. Counting such boundary states in agreement with the macroscopic properties of a black hole is thus a good candidate for a microscopic explanation of the Bekenstein-Hawking entropy. This paradigm has been successfully employed in 3+1 dimension in the context of loop quantum gravity, a canonical quantisation of General Relativity.

The applicability of the methods employed in loop quantum gravity to higher dimensions, as well as a desired comparison to string theory, suggest to generalize these results also to higher dimensions. This was in fact the main objective of the paper. In practice, what had to be done is to reinvestigate the classical reformulation of General Relativity in terms of connection variables in the presence of an isolated horizon boundary, e.g. a black hole. It can be shown that the canonical transformation to connection variables induces boundary variables, which can be written either in terms of bi-normals on a spatial slice of the horizon, or as a connection. The symplectic structure on the boundary, written in terms of the connection variables, is that of a higher-dimensional Chern-Simons theory. Furthermore, a boundary condition relating degrees of freedom in the bulk and on the boundary can be derived, which closely resembles the equations of motion of a Chern-Simons theory coupled to particles. In 3+1 dimensions, thus having a 2+1 dimensional boundary, a rigorous quantization of this system is well known. In higher dimensions however, non-Abelian Chern-Simons theory has local degrees of freedom, and a proper quantization is out of technical reach at the moment. However, the afore mentioned bi-normals as boundary variables allow for a quantization. It has been shown by one of the authors in the meantime that the boundary state counting in higher dimensions based on these bi-normals can be reduced to the well-understood 3+1-dimensional one, leading to an area-proportional entropy. While interesting progress in obtaining the prefactor 1/4 in the entropy formula has been made in recently, further research should focus on this problem.

Read the full article: Class. Quantum Grav. 31 055002

About the authors
Norbert Bodendorfer is currently a Postdoc at the Institute of Theoretical Physics, University of Warsaw, Poland, supported by a Feodor Lynen Research Fellowship of the Alexander von Humboldt-Foundation.

Thomas Thiemann is the Chair of the Institute for Quantum Gravity, University of Erlangen-Nuremberg, Germany. Andreas Thurn has been a PhD student at the Institute for Quantum Gravity, University of Erlangen-Nuremberg, Germany.

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