论文标题
来自非对称四弯曲的仪表凝结物
The gaugino condensate from asymmetric four-torus with twists
论文作者
论文摘要
我们在$ SU（2）$ SUPER YANG-MILLS理论中计算了Gaugino冷凝物，并在不对称的四刺$ \ Mathbb t^4 $中，并带有Hooft的扭曲边界条件。 $ \ mathbb t^4 $不对称由无量纲的引人入胜的参数$δ$控制，与$ l_3 l_4 -l_1 l_2 $成正比，并用$ l_i $表示$ \ mathbb t^4 $时期。我们通过$ \ Mathbb t^4 $的路径积分进行计算。它的大小比反向强尺寸$λ$要小得多，并且该理论在半古典弱耦合方程内。 Instanton背景是为$δ\ ll 1 $ in arXiv中的：hepth/0007113，具有分数拓扑费用$ q = \ frac {1} {2} $，并支持两种gaugino零模式，产生了一种不变的双线凝结物，我们发现这是$ $Δ$δ$δ$ -Dippentent。此外，该理论具有混合的离散性手性/$ 1 $ - 形式中心异常，从而在任何尺寸的圆环上都带有能量特征状态的双重变性。特别是，有两个真空吸尘器，$ | 0 \ rangle $和$ | 1 \ rangle $，在手性转换下交换。使用此信息，凝结液的$δ$独立性，并进一步假设半古典理论连续连接到强耦合的大型 - $ \ Mathbb t^4 $制度，我们确定了Gaugino冷凝物的数值系数：$ \ langle 0 | \ mbox {tr}λλ| 0 \ rangle = | \ langle 1 | \ mbox {tr}λλ| 1 \ rangle | =32π^2λ^3 $，结果等于已知的$ \ mathbb r^4 $值的两倍。我们讨论了可能导致这种差异的连续性方法中可能的漏洞。
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 $Δ$, 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 $Λ$ and the theory is well inside the semi-classical weak-coupling regime. The instanton background, constructed for $Δ\ll 1$ 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 $Δ$-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, $|0\rangle$ and $|1\rangle$, that are exchanged under chiral transformation. Using this information, the $Δ$-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: $\langle 0| \mbox{tr}λλ|0\rangle=|\langle 1| \mbox{tr}λλ|1\rangle|=32π^2 Λ^3$, 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.
