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High Energy Physics - Theory, hep-th
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.
