New infinite-dimensional hidden symmetries for the stationary axisymmetric Einstein–Maxwell equations with multiple Abelian gauge fields

Abstract

By proposing a so-called extended hyperbolic complex (EHC) function method, an Ernst-like (p+2)×(p+2) matrix EHC potential is introduced for the stationary axisymmetric (SAS) Einstein–Maxwell theory with p Abelian gauge fields (EM-p theory, for short), then the field equations of the SAS EM-p theory are written as a so-called Hauser–Ernst-like self-dual relation for the EHC matrix potential. Two Hauser–Ernst-type EHC linear systems are established, based on which some new parametrized symmetry transformations for the SAS EM-p theory are explicitly constructed. These hidden symmetries are found to constitute an infinite-dimensional Lie algebra, which is the semidirect product of the Kac–Moody algebra su(p+1,1)⊗R(t,t−1) and Virasoro algebra (without centre charges). All of the SAS EM-p theories for p = 0,1,2,... are treated in a unified formulation, p = 0 and p = 1 correspond, respectively, to the vacuum gravity and the Einstein–Maxwell cases.