Unifying Structures in Higher Spin Gauge Symmetry
Consider a set of free gauge fields together with their field equations and possibly a unifying action integrated over all of them. Then considering the problem of introducing interactions among such fields, there has historically been two broad avenues of approach. One approach entails "gauging" a non-Abelian global symmetry algebra, in the process making it local. The other approach entails "deforming" an already local but Abelian gauge algebra, in the process making it non-Abelian. In cases where both avenues have been explored, such as for spin 1 and 2 gauge fields, the results agree (barring conceptual and technical issues) with Yang-Mills theory and Einstein gravity. In the case of an infinite tower of higher spin gauge fields, the first approach has been thoroughly developed and explored by M. Vasiliev, whereas the second approach, after having lain dormant for a long time, has received new attention by several authors lately. In the present paper we briefly review some aspects of the history of higher spin gauge fields as a backdrop to a first attempt at comparing the gauging vs. deforming approaches. A common unifying structure of strongly homotopy Lie algebras underlying both approaches will be elicited. The modern deformation approach, using BRST-BV methods, will be decribed as far as it is developed at the present time.