Deformation theory of Cohomological Field Theories

We develop the deformation theory of cohomological field theories (CohFTs), which is done as a special case of a general deformation theory of morphisms of modular operads. This leads us to introduce two new natural extensions of the notion of a CohFT: homotopical (necessary to structure chain-level Gromov–Witten invariants) and quantum (with examples found in the works of Buryak–Rossi on integrable systems). We introduce a new version of Kontsevich’s graph complex, enriched with tautological classes on the moduli spaces of stable curves. We use it to study a new universal deformation group which acts naturally on the moduli spaces of quantum homotopy CohFTs, by methods due to Merkulov–Willwacher. This group is shown to contain both the prounipotent Grothendieck–Teichm"uller group and the Givental group.