We investigate four-dimensional Heterotic solitons, defined as a particular class of solutions of the equations of motion of Heterotic supergravity on a four-manifold M or, equivalently, as self-similar points of the renormalization group flow of the NS-NS sector of the Heterotic world-sheet. Heterotic solitons depend on a parameter κ and consist of a Riemannian metric g, a metric connection with skew torsion H on TM and a closed 1-form ϕ on M satisfying a differential system that generalizes the celebrated Hull-Strominger system. In the limit κ → 0, Heterotic solitons reduce to a class of generalized Ricci solitons and can be considered as a higher-order curvature modification of the latter. If the torsion H is equal to the Hodge dual of ϕ, Heterotic solitons consist of either flat tori or Ricci-flat Weyl structures on manifolds of type S¹ x S³ as introduced by P. Gauduchon. We prove that the moduli space of such Ricci-flat Weyl structures is isomorphic to the product of R with a certain finite quotient of the Cartan torus of the isometry group of the typical fiber of a natural fibration M → S¹. We also consider the associated space of essential infinitesimal deformations, which we prove to be obstructed. More generally, we characterize several families of Heterotic solitons as suspensions of certain three-manifolds with prescribed constant principal Ricci curvatures, amongst which we find hyperbolic manifolds, manifolds covered by Sl(2,R) and E(1,1) or certain Sasakian three-manifolds. These solutions exhibit a topological dependence in the string slope parameter κ and yield, to the best of our knowledge, the first examples of Heterotic compactification backgrounds not locally isomorphic to supersymmetric compactification backgrounds.

C.S.S. would like to thank J. Streets and Y. Ustinovskiy for their useful comments on the notion of generalized Ricci soliton. Part of this work was undertaken during a visit of C.S.S. to the University Paris-Saclay under the Deutsch-Franzosische Procope Mobilitat program. C.S.S. would like to thank A. Moroianu and this very welcoming institution for providing a nice and stimulating working environment. The work of A.M. was funded by the Spanish FPU Grant No. FPU17/04964, with additional support from the MCIU/AEI/FEDER UE grant PGC2018-095205-B-I00 and the Centro de Excelencia Severo Ochoa Program grant SEV-2016-0597. The work of C.S.S. was supported by the Germany Excellence Strategy Quantum Universe-390833306.