Instantons and the information metric

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 $L^2$ 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 $SU(2)$ instantons on $S^4$ of charge one, {\bf 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, {\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 $\C P_2$.