String gravitational equations with Hermitian structure

TL;DR

This paper explores string models with complex-structured antisymmetric tensors, deriving static solutions to Einstein's equations with realistic matter content, and investigates their T-dual backgrounds.

Contribution

It introduces a string model with a complex-structured antisymmetric tensor and finds static solutions that satisfy Einstein's equations with physically realistic matter.

Findings

Derived static solutions satisfying Einstein's equations with barotropic matter.

Showed T-duality and reduction commute in these models.

Constructed a string background with complex structure on a manifold.

Abstract

We consider a string model at one-loop related to a $\sigma$-model whose antisymmetric tensor field is constructed as complex structure on the background manifold, specially on a manifold $R\times N$ where $N$ is a complex manifold. As an example, we consider a homogeneous anisotropic $(1+4)$-dimensional $\sigma$-model where space part of the background is a $4$-dimensional complex manifold. By solving the related one-loop $\beta$-functions we obtain a static solution so that by reduction of this solution to $(1+3)$-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 $T$-dual background of the solution is investigated and it is shown that the duality transformation and reduction processes commute with each other.