Mass-deformed Super Yang-Mills theory on ${\mathbb{T}}^4$: sum over twisted sectors, $\theta$-angle, and CP violation

Abstract

We study $SU(N)$ super Yang-Mills theory with a small gaugino mass $m$ and vacuum angle $\theta$ on the four-torus ${\mathbb{T}}^4$ with 't Hooft twisted boundary conditions. Introducing a detuning parameter $\Delta$, which measures the deviation from an exactly self-dual ${\mathbb{T}}^4$, and working in the limits $mLN\ll \Lambda LN\ll 1$ and $\frac{(N-1)m^2{L}^2}{4\pi }\ll \Delta \ll 1$, where $L$ is the torus size and $\Lambda$ the strong-coupling scale, we compute the scalar and pseudo-scalar condensates to leading order in $m^2{L}^2/\Delta$. The twists generate fractional-charge instantons, and we show that summing over all such contributions is crucial for reproducing the correct physical observables in the decompactified strong-coupling regime. From a Hamiltonian perspective, the sum over twisted sectors, already at small torus size, projects in the $m=0$ limit onto a definite superselection sector of the ${\mathbb{R}}^4$ theory. In the massless limit, we recover the exact value of the gaugino condensate $|⟨\lambda \lambda ⟩|=16{\pi }^2{\Lambda }^3$, and demonstrate how a spurious $U(1)$ symmetry eliminates all $CP$-violating effects. Our results are directly testable in lattice simulations, and our method extends naturally to non-supersymmetric gauge theories.

Comments: 52 pages+appendices; typos corrected, matches the published version