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

TL;DR

This work develops a semi-classical framework to study mass-deformed SU($N$) super Yang–Mills on a small twisted four-torus, introducing a detuning parameter $\Delta$ to control nonabelian self-dual fractional instantons. By summing over twisted sectors with fractional topological charges $Q=k/N$ and regulating determinants with Pauli–Villars fields, the authors compute scalar and pseudoscalar condensates to leading order in $|m|^2L^2/\Delta$, recovering the exact gaugino condensate $|\langle\lambda\lambda\rangle|=16\pi^2\Lambda^3$ in the massless limit and showing how a spurious $U(1)$ symmetry removes ${\cal CP}$ violation at $m=0$. The analysis combines a Hamiltonian interpretation, a detailed $\ abla$-expansion in $\Delta$, and a robust path-integral treatment of fractional instantons, enabling explicit expressions for gauge-invariant two-point functions and other correlators. The results provide concrete, lattice-testable predictions for CP-violating observables and extend naturally to non-supersymmetric gauge theories, offering insight into the strong-coupling regime via controlled small-volume semiclassics.

Abstract

We study $SU(N)$ super Yang-Mills theory with a small gaugino mass $m$ and vacuum angle $θ$ on the four-torus $\mathbb{T}^4$ with 't Hooft twisted boundary conditions. Introducing a detuning parameter $Δ$, which measures the deviation from an exactly self-dual $\mathbb{T}^4$, and working in the limits $mLN \ll ΛLN \ll 1$ and $ \frac{(N-1) m^2 L^2}{4 π} \ll Δ\ll 1$, where $L$ is the torus size and $Λ$ the strong-coupling scale, we compute the scalar and pseudo-scalar condensates to leading order in $m^2L^2/Δ$. 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 $|\langle λλ\rangle| = 16π^2 Λ^3$, 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.

Figures (1)

• Figure 1: The action density of a charge-$1/2$ instanton in $SU(2)$ YM theory on an asymmetric ${\mathbb T}^4$ with $L_1 \gg L$ ($L_1=48, L=12$) on the lattice with a unit twist in the small ${\mathbb T}^3$ and a unit twist in the mixed space-time direction. The plots are taken from Wandler:2024hsq. Notice the different axes labeling on the plot, namely $x_0^{plot} \rightarrow x_1$, $x_3^{plot} \rightarrow x_2$, $x_2^{plot} \rightarrow x_3$, $x_1^{plot} \rightarrow x_4$ (thus, $W_3$ on the plot should be understood as $W_2$). The points the plot illustrates are: i.) that the action is localized in a region of size $L$ and spacetime volume $L^4$ and ii.) that the solution interpolates between the two $|k\rangle$ vacua (\ref{['kvacua']}, \ref{['wloop34']}) of the $SU(2)$ theory, the $|k=0\rangle$, $W_2 = 1$, and the $|k=1\rangle$, $W_2=-1$, as described in the text. The dilute gas picture advocated in RTN:1993ilwGonzalez-Arroyo:1995ynx uses these instantons as the basic blocks. Thus, in the $L_1 \rightarrow \infty$ limit, one sums over arbitrary even numbers of these instantons, if the boundary conditions in the path integral are untwisted, i.e. if one is computing a transition amplitude $|0\rangle \rightarrow |0\rangle$ (or $|1 \rangle \rightarrow |1 \rangle$). On the other hand, if the transition amplitude is twisted, as it would be in the numerator of $\Xi_1$ of eqn. (\ref{['xidef2']}), and interpolates from $|0\rangle \rightarrow |1\rangle$ (or v.v.), one sums over an arbitrary odd number of such instantons. This picture, adapted here to $SU(N)$, leads to eqn. (\ref{['energy5']}) for the splitting of the energies of the electric flux sectors.