Solutions in bosonic string field theory and higher spin algebras in AdS

We find a class of analytic solutions in open bosonic string field theory, parametrized by the chiral copy of higher spin algebra in ${\mathrm{AdS}}_3$. The solutions are expressed in terms of the generating function for the products of Bell polynomials in derivatives of bosonic space-time coordinates $X^m(z)$ of the open string, the form of which is determined in this work. The products of these polynomials form a natural operator algebra realizations of $w_{\infty }$ (area-preserving diffeomorphisms), enveloping algebra of SU(2) and higher spin algebra in ${\mathrm{AdS}}_3$. The class of string field theory solutions found can, in turn, be interpreted as the “enveloping of enveloping,” or the enveloping of ${\mathrm{AdS}}_3$ higher spin algebra. We also discuss the extensions of this class of solutions to superstring theory and their relations to higher spin algebras in higher space-time dimensions.