Structure of Higher Spin Gauge Interactions

TL;DR

This paper connects the abstract algebraic structure of higher spin gauge interactions to conventional field theory, clarifies the recursive construction of interaction vertices, and broadens the framework to include all possible interactions.

Contribution

It establishes a link between homotopy Lie algebra structures and Fock complex implementations in higher spin gauge theories, enhancing the understanding of their recursive interaction construction.

Findings

Reproduces recursive equations for higher order vertices.

Discards tracelessness constraints, simplifying equations.

Framework encompasses all possible interaction terms.

Abstract

In a previous paper, higher spin gauge field theory was formulated in an abstract way, essentially only keeping enough machinery to discuss "gauge invariance" of an "action". The approach could be thought of as providing an interface (or syntax) towards an implementation (or semantics) yet to be constructed. The structure then revealed turns out to be that of a strongly homotopy Lie algebra. In the present paper, the framework will be connected to more conventional field theoretic concepts. The Fock complex vertex operator implementation of the interactions in the BRST-BV formulation of the theory will be elaborated. The relation between the vertex order expansion and homological perturbation theory will be clarified. A formal non-obstruction argument is reviewed. The syntactically derived sh-Lie algebra structure is semantically mapped to the Fock complex implementation and it is shown that the recursive equations governing the higher order vertices are reproduced. Global symmetries and subsidiary conditions are discussed and as a result the tracelessness constraints are discarded. Thus all equations needed to compute the vertices to any order are collected. The framework is general enough to encompass all possible interaction terms. Finally, the abstract framework itself will be strengthened by showing that it can be naturally phrased in terms of the theory of categories.