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

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.