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

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.

arXiv comment48 pages, some pictures; typos fixed, presentation improved

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• Slightly Broken Higher Spin Symmetry: General Structure of Correlators
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