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We compute a Green's function giving rise to the solution of the Cauchyproblem for the source-free Maxwell's equations on a causal domain$\mathcal{D}$ contained in a geodesically normal domain of the Lorentzianmanifold $AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$, where $AdS^5$ denotesthe simply connected $5$-dimensional anti-de-Sitter space-time. Our approach isto formulate the original Cauchy problem as an equivalent Cauchy problem forthe Hodge Laplacian on $\mathcal{D}$ and to seek a solution in the form of aFourier expansion in terms of the eigenforms of the Hodge Laplacian on$\mathbb{S}^3$. This gives rise to a sequence of inhomogeneous Cauchy problemsgoverning the form-valued Fourier coefficients corresponding to the Fouriermodes and involving operators related to the Hodge Laplacian on $AdS^5 \times\mathbb{S}^2$, which we solve explicitly by using Riesz distributions and themethod of spherical means for differential forms. Finally we put together intothe 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 forMaxwell's equations.

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