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Heterotic supergravity with (1+3)--dimensional domain wall configurations and(warped) internal, six dimensional, almost-K\"ahler manifolds $\ ^6\mathbf{X}$are studied. Considering ten dimensional spacetimes with nonholonomicdistributions and conventional double fibrations, 2+2+...=2+2+3+3, andassociated $SU(3)$ structures on internal space, we generalize for real,internal, almost symplectic gravitational structures the constructions withgravitational and gauge instantons of tanh-kink type. They include the first$\alpha ^{\prime}$ corrections to the heterotic supergravity action,parameterized in a form to imply nonholonomic deformations of the Yang-Millssector and corresponding Bianchi identities. We show how it is possible toconstruct a variety of solutions depending on the type of nonholonomicdistributions and deformations of 'prime' instanton configurationscharacterized by two real supercharges. This corresponds to $\mathcal{N}=1/2$supersymmetric, nonholonomic manifolds from the four dimensional point of view.Our method provides a unified description of embedding nonholonomicallydeformed tanh-kink-type instantons into half-BPS solutions of heteroticsupergravity. This allows us to elaborate new geometric methods of constructingexact solutions of motion equations, with first order $\alpha ^{\prime}$corrections to the heterotic supergravity. Such a formalism is applied forgeneral and/or warped almost-K\"ahler configurations, which allows us togenerate nontrivial (1+3)-d domain walls and black hole deformations determinedby quasiperiodic internal space structures. This formalism is utilized in ourassociated publication [EPJC 77 (2017) 17, arXiv: 1608.01980] in order toconstruct and study generic off-diagonal nonholonomic deformations of the Kerrmetric, encoding contributions from heterotic supergravity.

classical and quantum gravity

Heterotic Supergravity with Internal Almost-K\"ahler Spaces, Instantons  for $SO(32)$, or $E_8 \times E_8$, Gauge Groups, and Deformed Black Holes  with Soliton, Quasiperiodic and/or Pattern-forming Structures