Homotopy Algebras in Higher Spin Theory

TL;DR

This paper investigates the algebraic structures underlying higher spin theories and Vasiliev equations, employing homotopy algebras to derive new formulations and computational tools for these complex systems.

Contribution

It systematically applies homotopy algebra methods to higher spin theory, deriving a graph formula for vertices and connecting to topological quantum mechanics models.

Findings

Derived a closed combinatorial graph formula for higher spin vertices

Connected higher spin vertices to correlation functions in a topological quantum mechanics model

Provided a homological perturbation theory analysis of Vasiliev equations

Abstract

Motivated by string field theory, we explore various algebraic aspects of higher spin theory and Vasiliev equation in terms of homotopy algebras. We present a systematic study of unfolded formulation developed for the higher spin equation in terms of the Maurer-Cartan equation associated to differential forms valued in L-infinity algebras. The elimination of auxiliary variables of Vasiliev equation is analyzed through homological perturbation theory. This leads to a closed combinatorial graph formula for all the vertices of higher spin equations in the unfolded formulation. We also discover a topological quantum mechanics model whose correlation functions give deformed higher spin vertices at first order.