Exact classical solutions of nonlinear sigma models on supermanifolds

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 ) ≅ U ( m | n ) / [ U ( p | q ) ⊗ U ( m − p | n − q ) ] for 0 ⩽ p ⩽ m , 0 ⩽ q ⩽ n and 1 ⩽ p + q . The projective superspace 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.