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 ?(4) with 't Hooft's twisted boundary conditions. The ?(4) asymmetry is controlled by a dimensionless detuning parameter , proportional to L3L4 - L1L2, with L-i denoting the ?(4) periods. We perform our calculations via a path integral on a ?(4). Its size is taken much smaller than the inverse strong scale Lambda 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=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-Tau(4) regime, we determine the numerical coefficient of the gaugino condensate: < 0|tr lambda lambda|0 >=|< 1|tr lambda lambda|1 >|=32 pi(2)Lambda(3), a result equal to twice the known R-4 value. We discuss possible loopholes in the continuity approach that may lead to this discrepancy.