6:00 pm Wednesday, October 1, 2014
Back Porch Seminar: A pretty accessible construction of some ALE gravitational instantons! by Michael Lock (UT) in Schedler and Lawn's Back Porch
First, here’s a brief description of what all the stuff in the title means (we’ll make everything totally clear at the beginning of the talk though). By gravitational instanton we mean a complete non-compact hyperkaehler Riemannian four-manifold satisfying “decaying curvature conditions at infinity”. The ALE part stands for asymptotically locally Euclidean, which basically means that, outside of a compact set, the space is diffeomorphic to a quotient of minus a ball by a finite subgroup of , and the metric is locally asymptotic to the Euclidean metric. Gravitational instantons were introduced by Stephen Hawking in the late ’70s in his work on a Euclidean quantum gravity theory (what’s that? a theory of gravity in a space with Euclidean metric signature). These spaces satisfy the vacuum Einstein equations and can be thought of as analogous to instantons in Yang-Mills theory. Gravitational instantons have become relevant in a variety of other mathematical and physical theories, some of which I will briefly mention during the talk. However, the “nature of their existence” is a very interesting question in its own right, and that will be the focus of the talk. In the late ‘80s Kronheimer classified ALE gravitational instantons, and used the hyperkaehler quotient construction to produce all such possible spaces (there was a lot work done earlier which led up to this). This construction is kind of complicated though….. What I’m going to do is ask you to believe a few pretty reasonable things, “build” some spaces, and finally use a sort of special interaction between geometry and topology unique to dimension-four to show that these are in fact ALE gravitational instantons! Submitted by
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