Deformation theory of cohomological field theories

Abstract: I will explain how to develop the deformation theory of cohomological field theories as a special case of a general deformation theory of morphisms of modular operads. Two cases will be considered: a classical and a quantum one. Using ideas of Merkulov–Willwacher based on graphs complexes, I will introduce and develop a new universal deformation group which acts functorially via explicit formulas on the moduli spaces of gauge equivalence classes of morphisms of modular operads. In the classical case, the action is trivial; but in the quantum case, this group contains the prounipotent Grothendieck–Teichmüller group and its action is highly non-trivial even in the simplest case. Then, I will enrich these graph complexes with characteristic classes coming from the geometry of the moduli spaces of curves and obtain in this way (rather surprisingly) a natural homotopy extension to Givental group action in the classical case, and in the quantum case, a huge group that includes both Givental and Grothendieck–Teichmüller groups.

It is a joint work with Volodya Dotsenko, Sergey Shadrin, and Arkady Vaintrob (arXiv:2006.01649).

algebraic topologycategory theoryquantum algebra

Audience: learners

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