A classical theme in number theory involves thinking of arithmetic objects, like rings of
integers in number elds, as rings of functions on some geometric objects. These objects
behave surprisingly like three-dimensional spaces, which raises the question, what happens
if we try to do physics on these spaces? Along these lines, in recent joint work with Sakellaridis
and Venkatesh we propose that the duality between electricity and magnetism has as an
arithmetic manifestation the relation between L-functions of Galois representations and
integrals of automorphic forms known as the relative Langlands program. I will survey these
ideas without assuming familiarity with either the physics or the number theory.