A Heterotic Flux Background and Calibrated Five-Branes

We consider, in flux compactification of heterotic string theory, spacetime-filling five-branes. Stabilizing the fivebrane involves minimizing the combined energy density of the tension and a Coulomb potential associated with an internal 2-dimensional wrapping. After reviewing the generalized calibration under such circumstances, we consider a particular internal manifold based on a $T^2$ bundle over a conformally rescaled $K3$. Here, we find two distinct types of wrapping. In one class, the fivebrane wraps the fibre $T^2$ which belongs to a cyclic homotopy group. The winding number is not extensive, yet it maps to D3-brane number under a U-duality map to type IIB side. We justify this by comparing properties of the two sides in detail. Fivebranes may also wrap a topological 2-cycle of K3, by saturating a standard calibration requirement with respect to a closed K\"ahler 2-form $J_{K3}$ of $K3$. We close with detailed discussion on F-theory dual of these objects and related issues.