Exact Results in Chern-Simons Supergravity

Abstract

We propose an extension of a recent non-perturbative method suited for solving the N-body problem in (2+1)-gravity to the case of Chern-Simons supergravity. Coupling with supersymmetric point particles is obtained implicitly by extending the DJH matching conditions of gravity. The consistent solution of the interacting case is obtained by building a general non-trivial mapping, extending the superanalytic mapping, between a flat polydromic $X^M$ supercoordinate system and a physical one $x^N$, representing the DJH matching conditions around the superparticles. We show how to construct such a mapping in terms of analytic functions, and we give their exact expressions for the two body case. The extension to the N body case is also discussed. In the Minkoskian coordinates the superparticles move freely, and in particular the fermionic coordinates $Θ(ξ^N_{(i)})$ are constants, whose values can be fixed by using the monodromy properties. While the bosonic part of the supergeodesic equations are obtained, as in gravity, by measuring the bosonic distance in Minkowskian space-time, we find that the fermionic geodesic equations can be defined only by requiring that a non-perturbative divergence of the $X^M = X^M(x^N)$ mapping cancels out on the world-lines of the superparticles.