$A_{\infty }$ 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.

48 pages, some pictures; typos fixed, presentation improved

References in corpus (7)

• A Duality Web in 2+1 Dimensions and Condensed Matter Physics
• Critical $O(N)$ Models in $6-\epsilon$ Dimensions
• A Fiber Approach to Harmonic Analysis of Unfolded Higher-Spin Field Equations
• Flows, Fixed Points and Duality in Chern-Simons-matter theories
• Formal Higher Spin Gravities
• Uniformizing higher-spin equations
• Generalized Symmetries of Massless Free Fields on Minkowski Space

Cited by in corpus (7)

• Flat Self-dual Gravity
• Minimal models of field theories: Chiral Higher Spin Gravity
• Minimal model of Chiral Higher Spin Gravity
• Slightly Broken Higher Spin Symmetry: General Structure of Correlators
• On BF-type higher-spin actions in two dimensions
• On (spinor)-helicity and bosonization in $Ad{S}_4/CF{T}_3$
• On correlation functions as higher-spin invariants