Compactified N=1 supersymmetric Yang-Mills theory on the lattice: Continuity and the disappearance of the deconfinement transition

TL;DR

This study probes confinement in compactified $\mathcal{N}=1$ SYM on $\mathbb{R}^3\times\mathbb{S}^1$ with periodic fermion boundary conditions using lattice Monte Carlo simulations of SU(2) with adjoint gluinos. By varying the bare mass parameter $\kappa$ (soft SUSY breaking) and the lattice coupling $\beta$, the authors map the phase structure via the Polyakov loop and its histograms, uncovering a deconfined region whose size shrinks as $m$ decreases and ultimately vanishes in the supersymmetric limit, indicating continuity of confinement across compactification. The results show three phases at large gluino mass and a collapsing deconfined region at small $R$, with evidence that the two confined regimes are connected, aligning with continuity predictions for PSYM and highlighting stronger-than-expected effects from periodic boundary conditions. The work emphasizes lattice artefacts and sign problems as caveats while outlining plans for continuum and volume scaling studies to refine the phase diagram and explore the chiral sector further.

Abstract

Fermion boundary conditions play a relevant role in revealing the confinement mechanism of N=1 supersymmetric Yang-Mills theory with one compactified space-time dimension. A deconfinement phase transition occurs for a sufficiently small compactification radius, equivalent to a high temperature in the thermal theory where antiperiodic fermion boundary conditions are applied. Periodic fermion boundary conditions, on the other hand, are related to the Witten index and confinement is expected to persist independently of the length of the compactified dimension. We study this aspect with lattice Monte Carlo simulations for different values of the fermion mass parameter that breaks supersymmetry softly. We find a deconfined region that shrinks when the fermion mass is lowered. Deconfinement takes place between two confined regions at large and small compactification radii, that would correspond to low and high temperatures in the thermal theory. At the smallest fermion masses we find no indication of a deconfinement transition. These results are a first signal for the predicted continuity in the compactification of supersymmetric Yang-Mills theory.

Figures (9)

• Figure 1: The phase diagram of SYM according to the theoretical predictions Unsal:2010qh. In the theory with thermal, i. e. antiperiodic, fermion boundary conditions the critical deconfinement radius $R$ is the inverse of the critical temperature $T$. The thermal theory has a larger critical deconfinement radius than the one with periodic fermion boundary. The dark shaded part indicates the deconfined region for both theories.
• Figure 2: The same phase diagram as in Fig. \ref{['fig:theoryphase1']} now for a larger number of Majorana fermions ($N_f > 1$), but still outside the conformal window.
• Figure 3: The phase diagram found in lattice simulations of SYM: $\mathbbm{Z_{2,+}}$/$\xout{\mathbbm{Z_{2,+}}}$ ($\mathbbm{Z_{2,-}}$/$\xout{\mathbbm{Z_{2,-}}}$) stands for confinement/deconfinement in the theory with periodic (antiperiodic) fermion boundary conditions. The green lines show the scans of the parameter range performed in the simulations on an $N_\tau=4$ lattice. The red dots are the position of precise checks of the phases with the histogram of the Polyakov line at different volumes. A large value of $\beta$ corresponds to a small compactification radius $R$.
• Figure 4: The measured modulus of the volume averaged Polyakov loop $|P_L|$ (Eq. (\ref{['eq:PL']})) in scans of the inverse bare coupling $\beta$ at a fixed value of the bare mass parameter $\kappa$ on an $N_\tau\times N_s^3=4\times 8^3$ lattice. With thermal (antiperiodic) fermion boundary conditions the signal for the deconfinement transition, that moves towards lower $\beta$ at larger $\kappa$, is clearly visible in this picture. In the theory with periodic boundary fermion conditions, on the other hand, the picture is completely different. A larger $\langle|P_L|\rangle$ are obtained only at intermediate values of $\beta$.
• Figure 5: The histograms of $|P_L|$ for $\kappa=0.16$ from simulations on $4\times N_s^3$ lattices. The different volumes $N_s^3$ are compared to show the finite size effects. The theory changes from the deconfined phase at $\beta=1.8$ (a) to a confined phase at $\beta=2.0$ (b) and $\beta=2.2$ (c).