Exact classical solutions of nonlinear sigma models on supermanifolds

TL;DR

This paper constructs explicit exact classical solutions for two-dimensional nonlinear sigma models with super Grassmannian target spaces, generalizing previous non-super models using a Gram-Schmidt orthonormalization approach.

Contribution

It provides a novel explicit construction of harmonic maps into super Grassmannian manifolds, extending classical solutions to the supermanifold context.

Findings

Explicit solutions for super Grassmannian sigma models are derived.

The method generalizes previous non-super solutions using Gram-Schmidt orthonormalization.

The solutions include holomorphic bosonic and fermionic supervector functions.

Abstract

We study two-dimensional nonlinear sigma models with target spaces being the complex super Grassmannian manifolds, that is, coset supermanifolds $G(m,p|n,q)\cong U(m|n)/[U(p|q)\otimes U(m-p|n-q)]$ for $0\leq p \leq m$, $0\leq q \leq n$ and $1\leq p+q$. The projective superspace ${\bf CP}^{m-1|n}$ is a special case of $p=1$, $q=0$. For the two-dimensional Euclidean base space, a wide class of exact classical solutions (or harmonic maps) are constructed explicitly and elementarily in terms of Gramm-Schmidt orthonormalisation procedure starting from holomorphic bosonic and fermionic supervector input functions. The construction is a generalisation of the non-super case published more than twenty years ago by one of the present authors.