Higher-order gaugino condensates on a twisted T⁴: In the beginning, there was semi-classics

We compute the gaugino condensates, ∏ᵢ₌₁ᵏ tr(λλ)(xᵢ) for $1$ ≤ k ≤ $N-1$, in $SU(N)$ super Yang-Mills theory on a small four-dimensional torus T⁴, 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 T⁴. 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$(k,N)=1$, ∏ᵢ₌₁ᵏ tr(λλ)(xᵢ) = n⁻¹ \; N²(16π² Λ³)ᵏ, where Λ is the strong-coupling scale and n is a normalization constant. We determine the normalization constant, using path integral, as n = N², 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 R⁴. The normalization constant n can be interpreted within the Euclidean path-integral formulation as the Witten index IW. It is well-established that a Hamiltonian calculation of IW yields IW=N, suggesting that while n=N² correctly reproduces the condensate result, it presents a puzzle in reconciling the Witten index computation via the path integral formalism, an issue warranting further investigation.