Advances in Theoretical and Mathematical Physics

Volume 24 (2020)

Number 3

Homotopy algebras in higher spin theory

Pages: 757 – 819



Si Li (Department of Mathematical Sciences and Yau Mathematical Sciences Center (YMSC), Tsinghua University Beijing, China)

Keyou Zeng (Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada)


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 $L_\infty$-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.

Published 19 August 2020