Constant curvature solutions of Grassmannian sigma models: (1) Holomorphic solutions

TL;DR

This paper derives a formula for Gaussian curvature of holomorphic 2-spheres in Grassmannian manifolds, constructs constant curvature solutions, and explores admissible curvatures with explicit examples for specific cases.

Contribution

It provides a general curvature formula and methods to construct constant curvature holomorphic solutions in Grassmannian sigma models, including explicit cases.

Findings

Derived a general Gaussian curvature formula for holomorphic 2-spheres in G(m, n)

Constructed explicit constant curvature solutions for G(2, 4) and G(2, 5)

Proposed conjectures on admissible constant curvatures in Grassmannian manifolds

Abstract

We present a general formula for the Gaussian curvature of curved holomorphic 2-spheres in Grassmannian manifolds G(m, n). We then show how to construct such solutions with constant curvature. We also make some relevant conjectures for the admissible constant curvatures in G(m, n) and give some explicit expressions, in particular, for G(2, 4) and G(2, 5).