Einstein Equations, Non-Linear Sigma Models and Self-Dual Yang-Mills Theory
We analyze the connection between non-linear sigma models, self-dual Yang-Mills theory and General Relativity (Self-dual and non self-dual, with and without Killing vectors), both at the level of the equations and at the level of the different type of solutions (solitons, instantons and calorons) of these theories. Until now no systematic procedure has been found to solve the highly non-linear four dimensional Einstein equations even in the presence of one Killing vector field (“three-dimensional reduced” Gravity). Only the two Killing vector case (two dimensional reduced Gravity) is known to admit a systematic resolution, (although the general solution is not yet known). The complete integrability of the Einstein equations without any Killing vector is still an open problem. In gravity, examples of solitons are provided by black-holes, which are the analogues of electric types monopoles (which may rotate) and Taub-NUT metrics which may be thought of as magnetic type gravitational monopoles. In the Euclidean (imaginary time t = i τ) regime these solutions exist as gravitational instantons (complete non singular solutions of the Einstein equations with (++++) signature).
KeywordsEinstein Equation Killing Vector Imaginary Time Euclidean Action Killing Vector Field
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