We compute the gaugino condensates, \( \left\langle {\Pi}_{i=1}^k\textrm{tr}\left(\uplambda \uplambda \right)\left({x}_i\right)\right\rangle \) for 1 ≤ k ≤ N − 1, in SU(N) super Yang-Mills theory on a small four-dimensional torus \( {\mathbbm{T}}^4 \) , 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 \( {\mathbbm{T}}^4 \) . 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, \( \left\langle {\Pi}_{i=1}^k\textrm{tr}\left(\uplambda \uplambda \right)\left({x}_i\right)\right\rangle \) = \( {\mathcal{N}}^{-1} \) N2 (16π2Λ3)k, where Λ is the strong-coupling scale and \( \mathcal{N} \) is a normalization constant. We determine the normalization constant, using path integral, as \( \mathcal{N} \) = N2, which is N times larger than the normalization used in our earlier publication [1]. This finding resolves the extra-factor-of-N discrepancy encountered there, aligning our results with those obtained through direct supersymmetric methods on ℝ4. The normalization constant \( \mathcal{N} \) can be understood within the Euclidean path-integral framework as the Witten index IW. From the Hamiltonian approach, it is well-established that IW = N. While the value \( \mathcal{N} \) = N2 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.
