Three-dimensional compact Heterotic solitons with parallel torsion
Andrei Moroianu, Miguel Pino Carmona, C. S. Shahbazi
TL;DR
The paper classifies compact 3-manifolds carrying non-trivial parallel torsion in the 3D Heterotic soliton system, revealing a rigid dichotomy: either a compact quotient of the Heisenberg group with a left-invariant metric or an Einstein manifold of negative curvature (compact hyperbolic quotient). It systematically analyzes two torsion regimes: generic reducible parallel torsion, which is tied to Heisenberg quotients with constant negative scalar curvature, and totally skew-symmetric parallel torsion, which yields a duality between Heisenberg-type and hyperbolic-type solitons with precise curvature relations. A key corollary is a universal bound for the scalar curvature in skew-symmetric parallel torsion cases, and the results extend to both skew-symmetric and twistorial torsion components. Overall, the work connects Heterotic background equations to classical 3D geometries, advancing the understanding of corrected Ricci solitons in differential geometry and string theory contexts.
Abstract
We obtain a rigidity result for compact three-dimensional Heterotic solitons with parallel non-trivial torsion. We show that they are either hyperbolic three-manifolds or compact quotients of the Heisenberg group equipped with a left-invariant metric. In particular, the latter arise both as solitons with completely skew-symmetric torsion as well as with non-vanishing twistorial component. As a corollary, we obtain the universal bound $-24$ for the scalar curvature of Heterotic solitons with parallel skew-symmetric torsion, which prevents it from being arbitrarily large.