Some results for SU(N) gauge-fields on the hypertorus

We show how to prove and to understand the formula for the 'Pontryagin' index P for SU(N) gauge fields on the Hypertorus T4, seen as a four-dimensional euclidean box with twisted boundary conditions. These twists are defined as gauge invariant integers modulo N and labelled by nsub(μν) (=-nsub(νμ)). In terms of these we can write (νelement ofZ) P = ∫Tr(Gsub(μν))anti (Gsub(μν))d4x/16π2 = ν + (N-1/N) x nsub(μν)nsub(μν)/4. Furthermore we settle the last link in the proof of the existence of zero action solutions with all possible twists satisfying nsub(μν)nsub(μν)/4 = kappa(n) = 0(mod N) for arbitrary N. (orig.)