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 g on the moduli spaces M of self-dual connections over Riemannian four-manifolds. Compared with the more widely known L2 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 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 non-degenerate 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.
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Groisser, D., Murray, M.K. Instantons and the Information Metric. Annals of Global Analysis and Geometry 15, 519–537 (1997). https://doi.org/10.1023/A:1006560802410
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DOI: https://doi.org/10.1023/A:1006560802410