Formal Higher-Spin Theories and Kontsevich-Shoikhet-Tsygan Formality

Abstract

The formal algebraic structures that govern higher-spin theories within the unfolded approach turn out to be related to an extension of the Kontsevich Formality, namely, the Shoikhet-Tsygan Formality. Effectively, this allows one to construct the Hochschild cocycles of higher-spin algebras that make the interaction vertices. As an application of these results we construct a family of Vasiliev-like equations that generate the Hochschild cocycles with $sp(2n)$ symmetry from the corresponding cycles. A particular case of $sp(4)$ may be relevant for the on-shell action of the $4d$ theory. We also give the exact equations that describe propagation of higher-spin fields on a background of their own. The consistency of formal higher-spin theories turns out to have a purely geometric interpretation: there exists a certain symplectic invariant associated to cutting a polytope into simplices, namely, the Alexander-Spanier cocycle.

typos fixed, many comments added, 36 pages + 20 pages of Appendices, 3 figures

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Cited by in corpus (16)

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• Chern-Simons Matter Theories and Higher Spin Gravity
• Formal Higher Spin Gravities
• Type-B Formal Higher Spin Gravity
• More on Chiral Higher Spin Gravity and Convex Geometry
• Minimal models of field theories: Chiral Higher Spin Gravity
• Chiral Higher Spin Gravity and Convex Geometry
• Characteristic Cohomology and Observables in Higher Spin Gravity
• Higher Spin Gravities and Presymplectic AKSZ Models
• Conformal Higher-Spin Gravity: Linearized Spectrum = Symmetry Algebra
• Hochschild cohomology of the Weyl algebra and Vasiliev's equations
• $A_{\infty }$ Algebras from Slightly Broken Higher Spin Symmetries
• Homotopy Algebras in Higher Spin Theory
• On deformations of $A_{\infty }$-algebras
• Deformations, renormgroup, symmetries, AdS/CFT
• Chiral Higher Spin Gravity in (A)dS${}_4$ and secrets of Chern-Simons Matter Theories