The Information Metric on the moduli space of instantons with global symmetries

In this note we revisit Hitchin's prescription \cite{Hitchin} of the Fisher metric as a natural measure on the moduli space of instantons that encodes the space-time symmetries of a classical field theory. Motivated by the idea of the moduli space of supersymmetric instantons as an emergent space in the sense of the gauge/gravity duality, we extend the prescription to encode also global symmetries of the underlying theory. We exemplify our construction with the instanton solution of the $\mathbb{CP}^{N}$ sigma model on $\mathbb{R}^{2}$.