When the fluctuations of a quantum field are 'constrained'-either by boundary conditions or by non-trivial spacetime topology-there occur striking changes in other fields coupled to the constrained quantum field. A particular aspect of this phenomenon will be studied in detail here. Namely, the generation of a 'topological mass' for an Abelian gauge field A_mu , by coupling A_mu to a scalar field phi , with phi constrained by being defined on partially toroidal spacetime T*Eⁿ. Each toroidal component A_a (a=1, 2, . . ., N) of A_mu =(A_a, A) acquires, through its minimal coupling with phi , a quantum mass m_T which the author computes to 1-loop for an equilateral torus T^N. m_T can be real or imaginary, depending on the dimensions N and n into which spacetime is divided (arbitrary N and n are considered). Real m_T parallels electric mass generation in finite-temperature gauge field theory. Imaginary m_T is a signal that the vacuum A_mu =0 is unstable, and A_mu will fluctuate about some non-zero value of its toroidal components A_a, breaking local gauge invariance.