The Shape of Instantons : Cross-Section of Supertubes and Dyonic Instantons

TL;DR

This paper investigates the relationship between Yang-Mills instantons, algebraic curves, and supertubes in string theory, establishing a correspondence between instanton data and geometric profiles with implications for charge and angular momentum.

Contribution

It introduces a novel correspondence between ADHM instanton data and algebraic curves, linking physical instanton configurations to geometric curve properties.

Findings

Maximized supertube angular momentum occurs for circular Higgs zero loci.

Identified profiles of instanton strings with supercurves and D-helices.

Established a dictionary connecting instanton charges with algebraic curve degrees.

Abstract

We explore the correspondence between Yang-Mills instantons and algebraic curves. The curve is defined by Higgs zero locus of dyonic instantons in 1+4 dimensional Yang-Mills-Higgs theory, and it is identified in string theory with the cross-section of supertubes connecting parallel D4-branes. To present evidence for the identification, we show that with total charges fixed, the supertube angular momentum computed from the Higgs zero locus is maximized when the locus is circular, which has been proven for the cross-section of the supertubes. This leads to a consistent dictionary between the charges in two pictures. We also consider a T-dual version of the story, identifying the profiles of the wavy instanton strings with those of the supercurves/D-helices. Based on this observation, we then argue a novel correspondence between ADHM data of instantons and algebraic curves defining the locus. The degree of the curve is related to the instanton number, and splitting property of the curve is physically manifested by well-separated instantons.