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.

arXiv commentsubmitted version, 57 pages, abstract and introduction rewritten

References in corpus (27)

• Gromov-Witten classes, quantum cohomology, and enumerative geometry
• The structure of 2D semi-simple field theories
• Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves
• Deformation theory of representations of prop(erad)s
• On operad structures of moduli spaces and string theory
• M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra
• Pre-Lie algebras and the rooted trees operad
• Homotopy theory for algebras over polynomial monads
• Modular operads and Batalin-Vilkovisky geometry
• Pre-Lie deformation theory
• BCOV theory via Givental group action on cohomological field theories
• A tale of three homotopies
• Operads revisited
• On the operad structure of admissible G-covers
• Symmetric homotopy theory for operads
• Representing the deformation $\infty$-groupoid
• P. Etingof's conjecture about Drinfeld associators
• Givental group action on Topological Field Theories and homotopy Batalin--Vilkovisky algebras
• Koszul duality for operadic categories
• Givental Action and Trivialisation of Circle Action
• The twisting procedure
• Formal groups and quantum cohomology
• Grothendieck-Teichmüller and Batalin-Vilkovisky
• Brane actions, Categorification of Gromov-Witten theory and Quantum K-theory
• Grothendieck-Teichmueller group, operads and graph complexes: a survey
• Double ramification cycles and quantum integrable systems
• The Dwyer-Kan model structure for enriched coloured PROPs

Cited by in corpus (2)

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