Constant curvature holomorphic solutions of the supersymmetric grassmannian sigma model: the case of $G(2,4)$

TL;DR

This paper investigates constant curvature holomorphic solutions in the supersymmetric Grassmannian sigma model, specifically focusing on the G(2,4) case, and extends known solutions from the bosonic to the supersymmetric setting.

Contribution

It generalizes known solutions for the supersymmetric Grassmannian sigma model and provides necessary and sufficient conditions for constant curvature solutions in G(2,N).

Findings

Constructed supersymmetric invariant solutions.

Identified additional solutions beyond known cases.

Detailed analysis of the G(2,4) case.

Abstract

We explore the constant curvature holomorphic solutions of the supersymmetric grassmannian sigma model $G(M,N)$ using in particular the gauge invariance of the model. Supersymmetric invariant solutions are constructed via generalizing a known result for ${C}P^{N-1}$. We show that some other such solutions also exist. Indeed, considering the simplest case of $G(2,N)$ model, we give necessary and sufficient conditions for getting the constant curvature holomorphic solutions. Since, all the constant curvature holomorphic solutions of the bosonic $G(2,4)$ $\sigma$-model are known, we treat this example in detail.