The gaugino condensate from asymmetric four-torus with twists

Abstract

We calculate the gaugino condensate in $SU(2)$ super Yang-Mills theory on an asymmetric four-torus ${\mathbb{T}}^4$ with 't Hooft's twisted boundary conditions. The ${\mathbb{T}}^4$ asymmetry is controlled by a dimensionless detuning parameter $\Delta$, proportional to $L_3{L}_4-L_1{L}_2$, with $L_i$ denoting the ${\mathbb{T}}^4$ periods. We perform our calculations via a path integral on a ${\mathbb{T}}^4$. 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 $\Delta \ll 1$ in arXiv:hep-th/0007113, has fractional topological charge $Q=\frac{1}{2}$ and supports two gaugino zero modes, yielding a non-vanishing bilinear condensate, which we find to be $\Delta$-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 $\Delta$-independence of the condensate, and assuming further that the semi-classical theory is continuously connected to the strongly-coupled large-${\mathbb{T}}^4$ regime, we determine the numerical coefficient of the gaugino condensate: $⟨0|\text{mbox}tr\lambda \lambda |0⟩=|⟨1|\text{mbox}tr\lambda \lambda |1⟩|=32{\pi }^2{\Lambda }^3$, a result equal to twice the known ${\mathbb{R}}^4$ value. We discuss possible loopholes in the continuity approach that may lead to this discrepancy.

Comments: 30 pages+appendices; typos corrected, references added, matches published version