Heriot-Watt Mathematics Report Series
HWM05-9, 16 May 2005
Phase space structure of Chern-Simons theory with a non-standard puncture
C Meusburger and B J Schroers
Abstract
We explicitly determine the symplectic structure on the phase space of
Chern-Simons theory with gauge group $G \ltimes g^*$
on a three-manifold of topology $\RR \times S $, where $S $ is a surface of genus $g$ with $n+1$ punctures. At each puncture additional variables
are introduced and coupled minimally to the Chern-Simons gauge field.
The first $n$ punctures are treated in the usual way and
the additional variables lie on coadjoint
orbits of $G \ltimes g^* $. The $(n+1)$st puncture plays a distinguished role and the associated variables lie in the cotangent bundle of $G\ltimes g^*$. This allows us to impose a curvature singularity for the Chern-Simons gauge field at the
distinguished puncture with an arbitrary Lie algebra valued
coefficient. The treatment of the distinguished
puncture is motivated by the desire to construct a simple model for
an open universe in the Chern-Simons formulation of $(2+1)$-dimensional
gravity.
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