The gaugino condensate from asymmetric four-torus with twists

We calculate the gaugino condensate in SU(2) super Yang-Mills theory on an asymmetric four-torus 𝕋⁴ with ’t Hooft’s twisted boundary conditions. The 𝕋⁴ asymmetry is controlled by a dimensionless detuning parameter ∆, proportional to L₃L₄ − L₁L₂, with L_i denoting the 𝕋⁴ periods. We perform our calculations via a path integral on a 𝕋⁴. Its size is taken much smaller than the inverse strong scale Λ and the theory is well inside the semi-classical weak-coupling regime. The instanton background, constructed for ∆ ≪ 1 in [1], has fractional topological charge $$ Q=\frac{1}{2} $$Q = 1 2 and supports two gaugino zero modes, yielding a non-vanishing bilinear condensate, which we find to be ∆-independent. Further, the theory has a mixed discrete chiral/1-form center anomaly leading to double degeneracy of the energy eigenstates on any size torus with ’t Hooft twists. In particular, there are two vacua, ∣0〉 and ∣1〉, that are exchanged under chiral transformation. Using this information, the ∆-independence of the condensate, and assuming further that the semi-classical theory is continuously connected to the strongly-coupled large-𝕋⁴ regime, we determine the numerical coefficient of the gaugino condensate: 〈0|trλλ|0〉 = ∣〈1|trλλ|1〉∣ = 32π²Λ³, a result equal to twice the known ℝ⁴ value. We discuss possible loopholes in the continuity approach that may lead to this discrepancy.