Working Group IV:

3 dimensional and 4 dimensional General Relativity

Members: C. Meusburger, T. Thiemann

Three-dimensional General Relativity [17] is an ideal testing ground for many conceptual and technical questions arising in four spacetime dimensions. Of particular interest is an alternative road to constraint quantisation which is based on classical symplectic reduction prior to quantisation. The resulting reduced phase space is then quantised via methods from the representation theory of quantum groups. This method works very well in 3 dimensional gravity [18]. Ideas of this type also exist in four dimensions [19], but the situation is more complex. It is desirable to investigate these ideas more rigourously and systematically. A further task is the study of the resulting reduced phase space and the construction of the associated operator constraint quantisation in situations where sufficient analytic control can be gained.

Besides its role as a lower-dimensional model for 4 dimensional gravity, there are certain aspects of 4 dimensional gravity which are linked generically to 3 dimensional gravity and Chern-Simons gauge theory. The first is related to the celebrated connection between 3 dimensional gravity, Chern-Simons theory and knot theory [20]. In Loop Quantum Gravity, the exponential of the Chern-Simons action for the gauge group SU(2) arises as a formal solution of all constraints (for Euclidian signature). To give it a precise mathematical meaning, one may define it as a distribution on a certain dense subspace of the Hilbert space satisfying the defining relation for the Chern-Simons action in the sense of distributions. Using the non-Abelian Stokes Theorem and the Duflo Isomorphism between the invariant symmetric algebra over the Lie algebra and the centre of its universal enveloping algebra, it is possible to derive the expectation value of Wilson loop functionals of the connection without using methods from Conformal Field Theory [21]. The Duflo Isomorphism also appears in a prominent way in a recent quantisation of three-dimensional gravity with positive cosmological constant [22] and points to a new and explicit mechanism for how quantum groups enter the description concretely rather than by abstract arguments. Finally, the description of quantum black holes within Loop Quantum Gravity also naturally leads to SU(2) Chern-Simons theory [23] and 3 dimensional Euclidean Gravity [24] respectively.

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