标题： 高阶胶粒凝聚在扭曲的$\mathbb T^4$上：最初是半经典 显示英文标题

标题： Higher-order gaugino condensates on a twisted $\mathbb T^4$: In the beginning was semi-classics

摘要： We compute the gaugino condensates, $\left\langle \prod_{i=1}^k \text{tr}(\lambda\lambda)(x_i) \right\rangle $ for $1$ $\leq$ $k$ $\le$ $N-1$ , in $SU(N)$ super Yang-Mills theory on a small four-dimensional torus $\mathbb{T}^4$, subject to 't Hooft twisted boundary conditions. 两项最近的进展对于执行计算和解释结果至关重要：对涉及$1$-形式中心对称性的广义异常的理解，以及在扭曲的$\mathbb T^4$上构建多分数瞬子。这些自对偶的经典配置具有拓扑电荷$k/N$，可以描述为在瞬子液体中紧密堆积的$k$个块的总和。 使用路径积分形式，我们在半经典极限下进行凝聚态计算，并发现，假设 gcd$(k,N)=1$，$\left\langle \prod_{i=1}^k \text{tr}(\lambda\lambda)(x_i) \right\rangle = {\bf n}^{-1} \; N^2\left(16\pi^2 \Lambda^3\right)^k$，其中$\Lambda$是强耦合尺度，${\bf n}$是归一化常数。我们使用路径积分确定归一化常数为${\bf n} = N^2$，这比我们在早期出版物 arXiv:2210.13568 中使用的归一化值大$N$倍。 这一发现解决了在那里的额外因子$N$的差异，使我们的结果与通过直接超对称方法在$\mathbb R^4$上获得的结果一致。 归一化常数${\bf n}$可以在欧几里得路径积分框架内理解为维滕指标$I_W$。 从哈密顿量方法来看，已知$I_W = N$。 虽然值${\bf n} = N^2$正确地再现了凝聚结果，但哈密顿量和路径积分公式之间的这一差异需要调和。 我们尝试提供一个潜在的解决方案，我们在讨论中概述了这一点。 显示英文摘要

摘要： We compute the gaugino condensates, $\left\langle \prod_{i=1}^k \text{tr}(\lambda\lambda)(x_i) \right\rangle $ for $1$ $\leq$ $k$ $\le$ $N-1$, in $SU(N)$ super Yang-Mills theory on a small four-dimensional torus $\mathbb{T}^4$, subject to 't Hooft twisted boundary conditions. Two recent advances are crucial to performing the calculations and interpreting the result: the understanding of generalized anomalies involving $1$-form center symmetry and the construction of multi-fractional instantons on the twisted $\mathbb T^4$. These self-dual classical configurations have topological charge $k/N$ and can be described as a sum over $k$ closely packed lumps in an instanton liquid. Using the path integral formalism, we perform the condensate calculations in the semi-classical limit and find, assuming gcd$(k,N)=1$, $\left\langle \prod_{i=1}^k \text{tr}(\lambda\lambda)(x_i) \right\rangle = {\bf n}^{-1} \; N^2\left(16\pi^2 \Lambda^3\right)^k$, where $\Lambda$ is the strong-coupling scale and ${\bf n}$ is a normalization constant. We determine the normalization constant, using path integral, as ${\bf n} = N^2$, which is $N$ times larger than the normalization used in our earlier publication arXiv:2210.13568. This finding resolves the extra-factor-of-$N$ discrepancy encountered there, aligning our results with those obtained through direct supersymmetric methods on $\mathbb R^4$. The normalization constant ${\bf n}$ can be understood within the Euclidean path-integral framework as the Witten index $I_W$. From the Hamiltonian approach, it is well-established that $I_W = N$. While the value ${\bf n} = N^2$ correctly reproduces the condensate result, this discrepancy between the Hamiltonian and path-integral formulations calls for reconciliation. We attempt to provide a potential solution we outline in our discussion.