Monopole-vortex continuity of \( \mathcal{N} \) = 1 super Yang-Mills theory on ℝ2 × S1 × S1 with ’t Hooft twist

We study \( \mathcal{N} \) = 1 SU(N) super Yang-Mills (SYM) theory on ℝ² × (S¹)₃ × (S¹)₄ with the ’t Hooft twist. The theory becomes weakly coupled if the length L₄ of (S¹)₄ is sufficiently small, NL₄Λ ≪ 1. We explore the nonperturbative dynamics at the weak-coupling regime by changing the size of L₃ and uncover how 3d monopole/bion-based effective theory for L₃ ≫ L₄ is related to the 2d vortex-based theory for L₃ ≈ L₄. The highlights of our results are (1) the smooth “weak-weak” continuity of the vacuum structure and gluino condensate during the 3d-2d dimensional reduction, (2) the switching of Wilson loop behavior from the area law in 3d to the perimeter law in 2d via a “double-string” picture, (3) the role of mass deformation in breaking discrete chiral symmetry and restoring the area law in 2d, and (4) the microscopic investigation of bions during the reduction from 3d to 2d and the cancellation of the vacuum energy due to the hidden topological angle. We also discuss the generalization of our results for (1)–(3) from \( \mathcal{N} \) = 1 SYM to QCD with adjoint quarks.