Heterotic supergravity with internal almost-Kähler spaces; instantons for SO(32), or E₈ × E₈, gauge groups; and deformed black holes with soliton, quasiperiodic and/or pattern-forming structures

Heterotic supergravity with (1  +  3)-dimensional domain wall configurations and (warped) internal, six dimensional, almost-Kähler manifolds ${}^6\text{X}$ are studied. Considering ten dimensional spacetimes with nonholonomic distributions and conventional double fibrations, 2  +  2  +  ...  =  2  +  2  +  3  +  3, and associated SU(3) structures on internal space, we generalize for real, internal, almost symplectic gravitational structures the constructions with gravitational and gauge instantons of tanh-kink type [1, 2]. They include the first ${\alpha }^{\prime }$ corrections to the heterotic supergravity action, parameterized in a form to imply nonholonomic deformations of the Yang–Mills sector and corresponding Bianchi identities. We show how it is possible to construct a variety of solutions depending on the type of nonholonomic distributions and deformations of 'prime' instanton configurations characterized by two real supercharges. This corresponds to $N=1/2$ supersymmetric, nonholonomic manifolds from the four dimensional point of view. Our method provides a unified description of embedding nonholonomically deformed tanh-kink-type instantons into half-BPS solutions of heterotic supergravity. This allows us to elaborate new geometric methods of constructing exact solutions of motion equations, with first order ${\alpha }^{\prime }$ corrections to the heterotic supergravity. Such a formalism is applied for general and/or warped almost-Kähler configurations, which allows us to generate nontrivial (1  +  3)-d domain walls and black hole deformations determined by quasiperiodic internal space structures. This formalism is utilized in our associated publication [3] in order to construct and study generic off-diagonal nonholonomic deformations of the Kerr metric, encoding contributions from heterotic supergravity. (paper)