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Supersymmetric monopole dynamics and duality
Magnetic monopoles arise as classical soliton solutions
in non-abelian gauge theories. The interactive dynamics
of such monopoles is intricate, but has been investigated
to some extent using the so-called adiabatic approximation.
The first goal of this project is to extend this investigation
to the supersymmetric setting, both classically and quantum
mechanically.
Electric-magnetic duality and S-duality conjectures
in gauge theories relate the dynamics of
magnetic monopoles to the strongly coupled dynamics of W-bosons.
There is now considerable evidence for these conjectures, much
of it based on bound state calculations for supersymmetric
monopoles. The goal of the second stage of this project
is to obtain insights into strongly coupled gauge theory
by combining results about supersymmetric monpole dynamics
with duality conjectures.
Observables in 2+1 dimensional quantum gravity
In recent years much progress has been made in the rigorous formulation of 2+1 dimensional quantum gravity. One approach starts with the Chern-Simons formulation of classical gravity in 2+1 dimensions and applies conventional quantisation techniques to the classical (finite-dimensional!) phase space. Another approach, using spin-networks, does not begin with a classical theory but formulates a quantum theory, using discrete variables from the start. In papers published in 2004 and 2003, Freidel and Louapre showed how these approaches are related for Euclidean space-times. The goal of this project is to extend their analysis to the Lorentzian setting and to compare the two approaches by addressing specific physical questions such as gravitational scattering or the ocurrence of a singularity in the evolution of a closed universe.