Working Group VII:

Conformal Field Theory and Black Holes

Members: K. Mecke, C. Meusburger, K.-H. Neeb

In a celebrated work [43], [44] an intriguing correspondence has been discovered between the Lie algebra of vector fields on the event horizon of a black hole in 3 dimensional General Relativity and 2 dimensional Conformal Field Theory. Conformal Field Theory is extremely rich in 2 dimensions and the constraints arising from its conformal gauge symmetry generate an infinite dimensional Lie algebra isomorphic to the Lie algebra of vector fields on the horizon.

Conformal Field Theory of course also plays an important role in Statistical Field Theory because many systems are effectively 2 dimensional and scale invariant at the critical point. To the best of our knowledge, the intriguing observation reported about above has never been made mathematically sound. Moreover, the fascinating question arises whether the powerful methods in 3 dimensions developed in this body of work can be directly extended to the 4 dimensional case and especially to the body of work mentioned under IV. This is because in both cases degrees of freedom are born due to the presence of an inner boundary of spacetime giving rise to non trivial entropy, in both cases Chern-Simons Theory plays a prominent role and therefore in both cases there is a direct link to 2 dimensional Conformal Field Theory through a Wess-Zumino-Witten type of boundary term in 3 dimensions and a corresponding term in 4 dimensions at an instant of time. It would be of high interest to understand the subtle details of this correspondence because black holes not only play an important role in any theory of Quantum Geometry but also provide a sector of the theory with a large amount of symmetry which implies that one is in a simplified situation with better analytic control.

Impressum