Grothendieck—Teichmüller group action on modular operads

Merkulov—Willwacher (ArXiv:1012.2467) established an action of the Grothendieck—Teichmüller group on the moduli spaces of solutions to the master equation of the BV-algebra associated to any affine supermanifold. We extend their method in order to prove an action of the Grothendieck—Teichmüller group to the moduli space of solutions to the master equation of the BV-algebra associated to any (dg) modular operad. To do so, we develop the deformation theory of morphisms of modular operads. As a byproduct, this gives us the deformation theory of Cohomological Field Theories. This way, we recover and interpret some recent constructions of Cohomological Field Theories due to Pandharipande—Zvonkine (ArXiv:1802.08981). [This is a joint work with Vladimir Dotsenko, Sergei Shadrin, and Arkady Vaintrob.]