Mass-deformed super Yang-Mills theory on T 4: sum over twisted sectors, θ -angle, and CP violation

We study SU(N) super Yang-Mills theory with a small gaugino mass m and vacuum angle θ on the four-torus T4 with ’t Hooft twisted boundary conditions. Introducing a detuning parameter ∆, which measures the deviation from an exactly self-dual T4, and working in the limits mLN ≪ ΛLN ≪ 1 and N−1m2L24π≪∆≪1, where L is the torus size and Λ the strong-coupling scale, we compute the scalar and pseudo-scalar condensates to leading order in m2L2/∆. The twists generate fractional-charge instantons, and we show that summing over all such contributions is crucial for reproducing the correct physical observables in the decompactified strong-coupling regime. From a Hamiltonian perspective, the sum over twisted sectors, already at small torus size, projects in the m = 0 limit onto a definite superselection sector of the ℝ4 theory. In the massless limit, we recover the exact value of the gaugino condensate |〈λλ〉| = 16π2Λ3, and demonstrate how a spurious U(1) symmetry eliminates all CP-violating effects. Our results are directly testable in lattice simulations, and our method extends naturally to non-supersymmetric gauge theories.