$A_\infty$ Algebras from Slightly Broken Higher Spin Symmetries

TL;DR

This paper introduces a new class of $A_$-algebras arising from deformations of higher spin symmetries in conformal field theories, connecting symmetry breaking, deformation quantization, and the three-dimensional bosonization conjecture.

Contribution

It defines $A_$-algebras from slightly broken higher spin symmetries and explores their relation to deformation quantization and the bosonization conjecture in three dimensions.

Findings

$A_$-algebras extend associative algebras of higher spin symmetries.

Deformations depend on a parameter related to Chern-Simons level in 3D.

Bosonic and fermionic matter lead to the same $A_$-algebra in 3D.

Abstract

We define a class of $A_\infty$-algebras that are obtained by deformations of higher spin symmetries. While higher spin symmetries of a free CFT form an associative algebra, the slightly broken higher spin symmetries give rise to a minimal $A_\infty$-algebra extending the associative one. These $A_\infty$-algebras are related to non-commutative deformation quantization much as the unbroken higher spin symmetries result from the conventional deformation quantization. In the case of three dimensions there is an additional parameter that the $A_\infty$-structure depends on, which is to be related to the Chern-Simons level. The deformations corresponding to the bosonic and fermionic matter lead to the same $A_\infty$-algebra, thus manifesting the three-dimensional bosonization conjecture. In all other cases we consider, the $A_\infty$-deformation is determined by a generalized free field in one dimension lower.