On rigid string instantons in four dimensions

TL;DR

This paper explores (anti)self-dual rigid string configurations in four-dimensional space using generalized Weierstrass formulas, revealing they correspond to superminimal immersions and providing explicit solutions and classifications.

Contribution

It introduces explicit formulae for (anti)instantons in four dimensions and links them to superminimal immersions, advancing understanding of rigid string configurations.

Findings

(Anti)instantons correspond to superminimal immersions.

Explicit formulae for generic (anti)instantons are derived.

Certain classes of surfaces are characterized and analyzed.

Abstract

Generalized Weierstrass formulae for surfaces in four-dimensional space $\Bbb{R}^{4}$ are used to study (anti)self-dual rigid string configurations. It is shown that such configurations are given by superminimal immersions into $\Bbb{R}^{4}$. Explicit formulae for generic (anti)instantons are presented. Particular classes of surfaces are also analyzed.