Heterotic Solutions with G2 and Spin(7) Structures

We study supersymmetric solutions in 7- and 8-dimensional Abelian heterotic supergravity theories. In dimension 7, the solutions are described by $G_2$ with torsion equations. When a $G_2$ manifold has principal orbits $S^{3} \times S^{3}$, the equations are reduced to ordinary differential equations with four radial functions. For these equations we obtain explicit ALC metrics with $S^3$-bolt and $T^{1,1}$-bolt singularities. In dimension 8, we study supersymmetric solutions to $Spin(7)$ with torsion equations associated with 3-Sasakian manifolds by using a similar method to the case $G_2$.