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

TL;DR

This paper extends Hitchin's Fisher metric on the moduli space of instantons to include global symmetries, using the $ ext{CP}^N$ sigma model as an example, with implications for gauge/gravity duality.

Contribution

It introduces a method to incorporate global symmetries into the Fisher metric on instanton moduli spaces, enhancing the geometric understanding of these spaces.

Findings

Extended Hitchin's metric to include global symmetries.

Applied the extended metric to the $ ext{CP}^N$ sigma model.

Demonstrated potential relevance to gauge/gravity duality.

Abstract

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 $\mathbb{R}^{2}$.