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arXiv · 2006.01649

Deformation theory of Cohomological Field Theories

Vladimir Dotsenko·Sergey Shadrin·Arkady Vaintrob·Bruno Vallette
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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.

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Vladimir Dotsenko, Sergey Shadrin, Arkady Vaintrob, Bruno Vallette. 2020-06-02. Deformation theory of Cohomological Field Theories. https://doi.org/10.1515/crelle-2023-0098

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