A Green's function for the source-free Maxwell equations on \(AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3\)

We compute a Green's function giving rise to the solution of the Cauchy problem for the source-free Maxwell's equations on a causal domain \(\mathcal{D}\) contained in a geodesically normal domain of the Lorentzian manifold \(AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3\), where \(AdS^5\) denotes the simply connected \(5\)-dimensional anti-de-Sitter space-time. Our approach is to formulate the original Cauchy problem as an equivalent Cauchy problem for the Hodge Laplacian on \(\mathcal{D}\) and to seek a solution in the form of a Fourier expansion in terms of the eigenforms of the Hodge Laplacian on \(\mathbb{S}^3\). This gives rise to a sequence of inhomogeneous Cauchy problems governing the form-valued Fourier coefficients corresponding to the Fourier modes and involving operators related to the Hodge Laplacian on \(AdS^5 \times \mathbb{S}^2\), which we solve explicitly by using Riesz distributions and the method of spherical means for differential forms. Finally we put together into the Fourier expansion on \(\mathbb{S}^3\) the modes obtained by this procedure, producing a \(2\)-form on \(\mathcal{D}\subset AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3\) which we show to be a solution of the original Cauchy problem for Maxwell's equations.