Deformation theory of Cohomological Field Theories

Abstract

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--Teichmüller group and the Givental group.

submitted version, 57 pages, abstract and introduction rewritten

References in corpus (4)

• Modular operads and Batalin-Vilkovisky geometry
• P. Etingof's conjecture about Drinfeld associators
• Koszul duality for operadic categories
• Formal groups and quantum cohomology

Cited by in corpus (2)

• Toward a minimal model for $H_{\ast }(\overline{\mathcal{M}})$
• Volume preservation of Butcher series methods from the operad viewpoint