Abstract
We define a class of A∞-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∞-algebra extending the associative one. These A∞-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∞-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∞-algebra, thus manifesting the three-dimensional bosonization conjecture. In all other cases we consider, the A∞-deformation is determined by a generalized free field in one dimension lower.
Highlights
Higher spin symmetries generated by abstract conserved tensors Js are powerful enough as to fix all the correlation functions [20,21,22,23] and these turn out to be necessarily given by a free CFT
The purpose of the paper is (i) to define what a slightly broken higher spin symmetry means in mathematical terms since it is not strictly speaking a symmetry in the conventional sense; (ii) to provide an explicit construction; (iii) to explore the simplest consequences including applications to the bosonization duality
The problem of the slightly broken higher spin symmetry is reduced to a much simpler problem of constructing a deformed higher spin algebras (HSA). This brings up the question: how big is the space of all deformations? Quantitatively its size is defined by the second Hochschild cohomology group, whose dimension equals the number of phenomenological parameters entering the correlation functions
Summary
The central statement of the present paper is that the A∞-structure of the previous section, is fully determined by the deformation of the underlying associative algebra. The problem is reduced to defining one function fn of (n − 1) arguments per each set of structure maps mn with only three orderings being nontrivial. Instead of the recursive definition given above it is possible to describe the set of trees that contribute to fn in a more direct way This is easier to do in terms of the sequences of natural numbers One should write down all possible subdiagrams such that the first row is of the same length n − 2 − k Any such pair of Young diagrams gives a sequence of li and mi that are admissible. Solutions to this equation form a three-parameter family of the cochains m(t, , s). While the construction above is quite general, in the sequel we focus upon the case of higher spin algebras and explain why and how these algebras can be deformed
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