We consider the geometric non-linear inverse problem of recovering a Hermitian connection $A$ from source-to-solution map cubic wave equation $\\Box{A}\\phi+\\kappa |\\phi|^{2}\\phi=f$, where $\\kappa\\neq 0$ and $\\Box{A}$ is operator in Minkowski space $\\mathbb{R}^{1+3}$. The arises naturally when considering Yang–Mills–Higgs equations with Mexican hat type potentials. Our proof exploits mic...