Deformation theory of Cohomological Field Theories

TL;DR

This paper develops a deformation theory for cohomological field theories, introducing homotopical and quantum extensions, and explores their universal deformation groups using enriched graph complexes.

Contribution

It introduces new deformation frameworks for CohFTs, including homotopical and quantum versions, and studies their universal deformation groups with enriched graph complexes.

Findings

Defined homotopical and quantum extensions of CohFTs

Constructed a new enriched graph complex for moduli spaces

Identified the universal deformation group containing GRT and Givental groups

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--Teichm\"uller group and the Givental group.