Homotopy Manin theories: generalising third-way, Yang–Mills and integrable sigma models

Manin theories are a class of non-topological deformations of Chern–Simons theories that naturally realise the third-way mechanism and furthermore admit localisation despite not being supersymmetric in the usual sense. In this paper, we extend this construction to higher dimensions, thereby producing a large class of examples of third-way-type theories. Furthermore, the construction naturally yields Yang–Baxter integrable deformations of the principal chiral model as well as gravitational models various dimensions.