Instantons and the information metric
arXiv:dg-ga/9611008
Abstract
The information metric arises in statistics as a natural inner product on a space of probability distributions. In general this inner product is positive semi-definite but is potentially degenerate. By associating to an instanton its energy density, we can examine the information metric {\bf g} on the moduli spaces $\M$ of self-dual connections over Riemannian 4-manifolds. Compared with the more widely known metric, the information metric better reflects the conformal invariance of the self-dual Yang-Mills equations, and seems to have better completeness properties. In the case of instantons on of charge one, {\bf g} is known to be the hyperbolic metric on the five-ball. We show more generally that for charge-one instantons over -connected, positive-definite manifolds, {\bf g} is nondegenerate and complete in the collar region of $\M$, and is `asymptotically hyperbolic' there; {\bf g} vanishes at the cone points of $\M$. We give explicit formulae for the metric on the space of instantons of charge one on $\C P_2$.
18 pages, Latex