I will present joint work with Yiannis Sakellaridis and Akshay Venkatesh in which we apply insights from physics (the understanding of electric-magnetic duality for boundary conditions in gauge theory) to a problem in number theory (the connection between period integrals and L-functions). The intermediary between the two is a duality between hamiltonian actions of Langlands dual groups, a ``relative" form of the geometric Langlands correspondence.