Double structures and double symmetries for the general symplectic gravity models

Abstract

By using the so-called double-complex function method, a doubleness symmetry for each member of the class of stationary axisymmetric general symplectic gravity models is found and exploited so that some double-complex (n+1)×(n+1) matrix Ernst-like potential for any non-negative integer n can be constructed and the associated motion equations can be extended into a double-complex matrix Ernst-like form. Then double symmetry symplectic groups Sp(2(n+1), R(J)) of the theories are given and verified that their actions can be realized concisely by double-complex matrix form generalizations of the fractional linear transformation on the Ernst potential. These results demonstrate that the theories under consideration possess more and richer symmetry structures. The special cases n=0 and n=1 correspond, respectively, to the pure Einstein gravity and the Einstein–Maxwell-dilaton–axion theories. Moreover, as an application, for each n=0,1,2,…, an infinite chain of double-solutions of the general symplectic gravity model is obtained, which shows that the double-complex method is more effective. Some of the results in this paper cannot be obtained by the usual (nondouble) scheme.