String gravitational equations with Hermitian structure

Abstract

We consider a string model at one-loop related to a [Formula: see text]-model whose antisymmetric tensor field is constructed as complex structure on the background manifold, especially on a manifold [Formula: see text] where [Formula: see text] is a complex manifold. As an example, we consider a homogeneous anisotropic [Formula: see text]-dimensional [Formula: see text]-model where space part of the background is a four-dimensional complex manifold. By solving the related one-loop [Formula: see text]-functions, we obtain a static solution so that by reduction of this solution to [Formula: see text]-dimension, we obtain a static solution of Einstein equation where the matter sector is effectively interpreted as an inhomogeneous, anisotropic and barotropic matter satisfying all the energy conditions. Finally, the [Formula: see text]-dual background of the solution is investigated and it is shown that the duality transformation and reduction processes commute with each other.