Confinement and Graded Partition Functions for $\mathcal{N}=4$ SYM

TL;DR

This paper investigates how different grading choices in the partition function of $ ext{N}=4$ SYM influence confinement, revealing a continuous range of gradings that maintain confinement at large coupling.

Contribution

It introduces a family of graded partition functions for $ ext{N}=4$ SYM and analyzes their effects on confinement across different coupling regimes.

Findings

Continuous grading parameters preserve confinement at large coupling.

Only discrete gradings preserve confinement at small coupling.

Grading choices can control confinement behavior in supersymmetric gauge theories.

Abstract

Gauge theories with confining phases at low temperatures tend to deconfine at high temperatures. In some cases, for example in supersymmetric theories, confinement can persist for all temperatures provided the partition function includes a grading by $(-1)^F$. When it is possible to define partition functions which smoothly interpolate between no grading and $(-1)^F$ grading, it is natural to ask if there are other choices of grading that have the same effect as $(-1)^F$ on confinement. We explore how this works for $\mathcal{N}=4$ SYM on $S^1\times S^3$ in the large $N$ limit at both small and large coupling. We find evidence for a continuous range of grading parameters that preserve confinement for all temperatures at large coupling, while at small coupling only a discrete set of gradings preserves confinement.