We study the stability of a non-Abelian chromomagnetic vacuum in Yang-Mills theory in Euclidean Einstein universe S 1 × S 3 . We assume that the gauge group is a simple compact group G containing the group SU (2) as a subgroup and consider static covariantly constant gauge fields on S 3 taking values in the adjoint representation of the group G and forming a representation of the group SU (2). We compute the heat kernel for the Laplacian acting on fields on S 3 in an arbitrary representation of SU (2) and use this result to compute the heat kernels for the gluon and the ghost operators and the one-loop effective action. We show that the only configuration of the covariantly constant Yang-Mills background that is stable is the one that contains only spinor (fundamental) representations of the group SU (2); all other configurations contain negative modes and are unstable. For the stable configuration we compute the asymptotics of the effective action, the energy density, the entropy and the heat capacity in the limits of low/high temperature and small/large volume and show that the energy density has a non-trivial minimum at a finite value of the radius of the sphere S 3 . This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'.

We study the covariantly constant Savvidy-type chromomagnetic vacuum in finite-temperature Yang-Mills theory on the four-dimensional curved spacetime. Motivated by the fact that a positive spatial curvature acts as an effective gluon mass we consider the compact Euclidean spacetime S 1 × S 1 × S 2 , with the radius of the first circle determined by the temperature a 1 = (2π T ) −1 . We show that covariantly constant Yang-Mills fields on S 2 cannot be arbitrary but are rather a collection of monopole-antimonopole pairs. We compute the heat kernels of all relevant operators exactly and show that the gluon operator on such a background has negative modes for any compact semisimple gauge group. We compute the infrared regularized effective action and apply the result for the computation of the entropy and the heat capacity of the quark-gluon gas. We compute the heat capacity for the gauge group SU (2N ) for a field configuration of N monopole-antimonopole pairs. We show that in the high-temperature limit the heat capacity per unit volume is well defined in the infrared limit and exhibits a typical behavior of second-order phase transition ∼ (T − T c ) −3/2 with the critical temperature T c = (2πa) −1 , where a is the radius of the 2-sphere S 2 .

Abstract:We study quantum gravity with the Einstein-Hilbert action including the cosmological constant on the Euclidean Einstein universe S 1 × S 3 . We compute exactly the spectra and the heat kernels of the relevant operators on S 3 and use these results to compute the heat trace of the graviton and ghost operators and the exact one-loop effective action on S 1 × S 3 . We show that the system is unstable in the infrared limit due to the presence of the negative modes of the graviton and the ghost operators. We study the thermal properties of the model with the temperature T = (2πa 1 ) −1 determined by the radius a 1 of the circle S 1 . We show that the heat capacity C v is well defined and behaves like ∼ T 3 in the high temperature limit and has a singularity of the type ∼ (T − T c ) −1 , indicating a second-order phase transition, with the critical temperature T c determined by the cosmological constant Λ and the radius a of the sphere S 3 . We also discuss some peculiar properties of the model such as the negative heat capacity as well as possible physical applications.

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