Deformation theory of bialgebras, higher Hochschild cohomology and formality

TL;DR

This paper establishes a deep connection between the homotopy theories of bialgebras and $E_2$-algebras, proving the $E_3$-formality of the deformation complex of symmetric bialgebras and solving longstanding conjectures in deformation theory.

Contribution

It constructs a fully faithful functor linking homotopy bialgebras to $E_2$-algebras and proves the $E_3$-formality of the deformation complex, resolving key conjectures in the field.

Findings

Proved the $E_3$-structure controls bialgebra deformation theory.

Established $E_3$-formality of the deformation complex of symmetric bialgebras.

Provided a new proof of Etingof-Kazdhan deformation quantization, independent of associators.

Abstract

A first goal of this paper is to precisely relate the homotopy theories of bialgebras and $E_2$-algebras. For this, we construct a conservative and fully faithful $\infty$-functor from pointed conilpotent homotopy bialgebras to augmented $E_2$-algebras which consists in an appropriate "cobar" construction. Then we prove that the (derived) formal moduli problem of homotopy bialgebras structures on a bialgebra is equivalent to the (derived) formal moduli problem of $E_2$-algebra structures on this "cobar" construction. We show consequently that the $E_3$-algebra structure on the higher Hochschild complex of this cobar construction, given by the solution to the higher Deligne conjecture, controls the deformation theory of this bialgebra. This implies the existence of an $E_3$-structure on the deformation complex of a dg bialgebra, solving a long-standing conjecture of Gerstenhaber-Schack. On this basis we solve a long-standing conjecture of Kontsevich, by proving the $E_3$-formality of the deformation complex of the symmetric bialgebra. This provides as a corollary a new proof of Etingof-Kazdhan deformation quantization of Lie bialgebras which extends to homotopy dg Lie bialgebras and is independent from the choice of an associator. Along the way, we establish new general results of independent interest about the deformation theory of algebraic structures, which shed a new light on various deformation complexes and cohomology theories studied in the literature.