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

Abstract

We compute the gaugino condensates, $⟨{\prod }_{i=1}^k\text{tr}(\lambda \lambda )(x_i)⟩$ for $1$ $\le$ $k$ $\le$ $N-1$, 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$(k,N)=1$, $⟨{\prod }_{i=1}^k\text{tr}(\lambda \lambda )(x_i)⟩={\mathbf{n}}^{-1}{N}^2{\left(16{\pi }^2{\Lambda }^3\right)}^k$, where $\Lambda$ is the strong-coupling scale and $\mathbf{n}$ is a normalization constant. We determine the normalization constant, using path integral, as $\mathbf{n}=N^2$, 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 $\mathbf{n}$ 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 $\mathbf{n}=N^2$ 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.

Comments: 29 pages+appendices, 2 figures; clarification added (in a new appendix) regarding the factor-of-$N$ discrepancy, typos fixed, references added, matches the published version