Higher-order gaugino condensates on a twisted T 4 $$ {\mathbbm{T}}^4

• Published in: The journal of high energy physics Vol. 2025; no. 2; pp. 1 - 57
• Main Authors: Mohamed M. Anber, Erich Poppitz
• Format: Journal Article
• Language: English
• Published: SpringerOpen 01.02.2025
• Subjects: Anomalies in Field and String Theories; Nonperturbative Effects; Supersymmetric Gauge Theory
• ISSN: 1029-8479
• DOI: 10.1007/JHEP02(2025)114

Abstract We compute the gaugino condensates, Π i = 1 k tr λλ x i $$ \left\langle {\Pi}_{i=1}^k\textrm{tr}\left(\uplambda \uplambda \right)\left({x}_i\right)\right\rangle $$ for 1 ≤ k ≤ N − 1, in SU(N) super Yang-Mills theory on a small four-dimensional torus T 4 $$ {\mathbbm{T}}^4 $$ , 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 T 4 $$ {\mathbbm{T}}^4 $$ . 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, Π i = 1 k tr λλ x i $$ \left\langle {\Pi}_{i=1}^k\textrm{tr}\left(\uplambda \uplambda \right)\left({x}_i\right)\right\rangle $$ = N − 1 $$ {\mathcal{N}}^{-1} $$ N 2 (16π 2Λ3) k , where Λ is the strong-coupling scale and N $$ \mathcal{N} $$ is a normalization constant. We determine the normalization constant, using path integral, as N $$ \mathcal{N} $$ = N 2, 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 ℝ 4. The normalization constant N $$ \mathcal{N} $$ 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 N $$ \mathcal{N} $$ = 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.