Monopole-vortex continuity of $\mathcal{N}=1$ super Yang-Mills theory on ${\mathbb{R}}^2\times S^1\times S^1$ with 't Hooft twist

We study $\mathcal{N}=1$ $SU(N)$ super Yang-Mills (SYM) theory on ${\mathbb{R}}^2\times (S^1{)}_3\times (S^1{)}_4$ with the 't Hooft twist. The theory becomes weakly coupled if the length $L_4$ of $(S^1{)}_4$ is sufficiently small, $N{L}_4\mathrm{\Lambda }\ll 1$. We explore the nonperturbative dynamics at the weak-coupling regime by changing the size of $L_3$ and uncover how $3$d monopole/bion-based effective theory for $L_3\gg L_4$ is related to the $2$d vortex-based theory for $L_3\approx L_4$. The highlights of our results are (1) the smooth “weak-weak” continuity of the vacuum structure and gluino condensate during the $3$d-$2$d dimensional reduction, (2) the switching of Wilson loop behavior from the area law in $3$d to the perimeter law in $2$d via a “double-string” picture, (3) the role of mass deformation in breaking discrete chiral symmetry and restoring the area law in $2$d, and (4) the microscopic investigation of bions during the reduction from $3$d to $2$d 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.