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Online: Deformation theory of cohomological field theories

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Sergey Shadrin
University of Amsterdam
Tue, 30/06/2020 - 14:00 - 15:00

I want to explain how develop the deformation theory of cohomological field theories (CohFTs) as a special case of a general deformation theory of morphisms of modular operads. This way our construction goes naturally through an interplay of classical (homotopy) CohFTs and their quantum extensions, quantum (homotopy) CohFTs. 

Then we use an idea of Merkulov--Willwacher that allows to attach some natural graph complexes to the deformation theory of the underlying shifted Lie structures in order to introduce and develop a new universal deformation group and which acts functorially via explicit formulas on the moduli spaces of gauge equivalence classes of morphisms of modular operads and their quantum extensions (the action is trivial for the former but contains the prounipotent Grothendieck--Teichm\"uller group and is highly non-trivial even in the simplest cases for the latter). 

Exploring the limits of application of the Merkulov-Willwacher idea in the particular cases of classical homotopy CohFTs and quantum homotopy CohFTs, we extend these graph complexes by natural classes and morphisms specific for the moduli spaces of curves, and this way we obtain (rather surprisingly) a natural extension to homotopy structures of the Givental group action in the classical case, and a huge group that includes both Givental-like elements and Grothendieck--Teichm\"uller-like elements in the quantum case. 

It is a joint work with Volodya Dotsenko, Arkady Vaintrob, and Bruno Vallette.

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