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--Teichm\"uller group and the Givental group.
Keywords
Cite
@article{arxiv.2006.01649,
title = {Deformation theory of Cohomological Field Theories},
author = {Vladimir Dotsenko and Sergey Shadrin and Arkady Vaintrob and Bruno Vallette},
journal= {arXiv preprint arXiv:2006.01649},
year = {2024}
}
Comments
submitted version, 57 pages, abstract and introduction rewritten