Instantons and the information metric

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 g on the moduli spaces M of self-dual connections over Riemannian 4-manifolds. Compared with the more widely known L2 metric, the information metric better reflects the conformal invariance of the self-dual YangMills equations, and seems to have better completeness properties. In the case of SU(2) instantons on S4 of charge one, g is known to be the hyperbolic metric on the five-ball. We show more generally that for charge-one SU(2) instantons over 1-connected, positive-definite manifolds, g is nondegenerate and complete in the collar region of M, and is ‘asymptotically hyperbolic’ there; g vanishes at the cone points of M. We give explicit formulae for the metric on the space of instantons of charge one on CP2.