Strong deflection limit-analysis using Picard–Fuchs equation in Einstein–Maxwell–Dilaton spacetime

We consider the deflection of light by a spherically symmetric and electrically charged black hole solution in the Einstein–Maxwell–Dilaton theory with specific values of the dilaton coupling constant where the deflection angle can be represented as elliptic integrals. We show that the deflection angle as a function of two dimensionless variables (s, z), which are related to the background charge and the impact parameter, respectively, satisfies a system of second-order linear partial differential equations called the Picard-Fuchs (PF) equations. For each case of the dilaton coupling, the PF equations lead to first-order ordinary differential equations with respect to the variable s for the constants ā and b̄ in the log-formula for the strong deflection limit. Using the Hamiltonian system associated with Painlevé VI equations, which holds as a result of the integrability of the PF equations, we solve the equations for ā and b̄. By requiring consistency with the Schwarzschild case in the zero-charge limit, ā and b̄ for nonzero charge are uniquely determined.