Double structures and multiple symmetry groups for the reduced 4-dimensional string background equations

By using the extended double complex method proposed previously, the l-loop string background equations with axion and dilaton fields in 4 dimensions with two commuting Killing symmetries and deltac = 0 are reduced essentiall y to two double Ernst sigma -models. Then the string Geroch group acting on the solution space is extended to a multi-fold version of a semidirect pro duct of the string Geroch group and the Virasoro group. The usual string Ge roch group is just one component of a subgroup in the multi-fold semidirect product group. It is also found that a dobule Z(2) symmetry exists for eve ry case of the 4-dimensional reduced string equations and for most of the c ases here, this Z(2) symmetry is new and cannot be obtained in the usual (n on-double) scheme. These results show that the reduced string background eq uations considered possess more and richer symmetry structures than previou sly expected. Moreover, a double form of string soliton method is briefly d escribed and, as an application, a multiple family of soliton solutions of the considered reduced string equations is given, which shows that the doub le method is more effective.