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If the holonomy is generated kinematically by the introduction of gauge invariant sources coupled to Polyakov loops, the contribution of charged (off-diagonal) gluons to the free energy at order $g^3$, ${\\cal F}^{\\left( 3\\rm{:c.g.} \\right)}$, is singular: ${\\cal F}^{\\left( 3\\rm{:c.g.} \\right)} \\neq 0 $ without holonomy, but ${\\cal F}^{\\left( 3 \\rm{:c.g.} \\right)}= 0$ when the holonomy is nonzero, even infinitesimally. 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Korthals Altes, Hiromichi Nishimura, Robert D. Pisarski, Vladimir V. Skokov","submitted_at":"2019-11-22T19:00:02Z","abstract_excerpt":"We compute the term $\\sim g^3$ in the free energy for a $SU(N)$ gauge theory with nonzero holonomy at nonzero temperature. If the holonomy is generated kinematically by the introduction of gauge invariant sources coupled to Polyakov loops, the contribution of charged (off-diagonal) gluons to the free energy at order $g^3$, ${\\cal F}^{\\left( 3\\rm{:c.g.} \\right)}$, is singular: ${\\cal F}^{\\left( 3\\rm{:c.g.} \\right)} \\neq 0 $ without holonomy, but ${\\cal F}^{\\left( 3 \\rm{:c.g.} \\right)}= 0$ when the holonomy is nonzero, even infinitesimally. 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