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 $AdS_3$. 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, which form 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 $AdS_3$. The class of SFT solutions found can, in turn, be interpreted as the "enveloping of enveloping", or the enveloping of $AdS_3$ 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.