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.
Highlights
Gauge theories tend to deconfine at high temperatures
In pure SUðNÞ YM theory, it is known that htrΩi 1⁄4 0 at low temperatures, see, e.g., [3], but it is nonzero at high temperatures [1]
We found that the large N theory is in the confined phase on S3R × S1β when θ 1⁄4 π=3, π, 5π=3 both at small and large coupling
Summary
Gauge theories tend to deconfine at high temperatures. This statement can be made precise in theories that have a center symmetry, such as SUðNÞ Yang-Mills (YM) theory, see, e.g., [1,2]. In the large N limit, the deconfinement transition is forced by the fact that the confining phase features a Hagedorn density of states This means that the density of states, ρðEÞ, grows exponentially with the energy ρðEÞ ∼ eþβHE, where it is assumed that E ∼ N0 is large compared to the characteristic mass scale of the theory, and βH > 0, the Hagedorn temperature, depends on the matter content of the theory, but it can be estimated as βH ∼ minðΛ−1; RÞ where Λ is the large scale and R is the characteristic size of the spatial box. ZðβÞ can avoid the Hagedorn instability and the deconfinement phase transition This naturally occurs in supersymmetric gauge theories, where the bosonic and fermionic states with energies E > 0 necessarily come in degenerate multiplets in the infinitevolume limit. We define this grading below and study the resulting phase structure at small and large ’t Hooft coupling
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