Thermal Yang–Mills theory in the Einstein universe

Abstract

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'.

One-loop quantum gravity in the Einstein universe

“…The T 3 dependence is characteristic of the photon gas and, as has been found in our previous paper [12], of the gluon gas as well.…”

“…This heat trace can be computed by using the heat kernel diagonal for the Laplacian ∆ j on the unit sphere S 3 given by the eq. (6.16) of our paper [12]. To get the heat trace we have to multiply the heat kernel diagonal by the volume of the sphere S 3 equal to vol (S 3 ) = 2π 2 and by the dimension of the representation j equal to (2j + 1).…”

“…Then the heat trace of the Laplace type operator L j = −∆ + Q j on the unit sphere S 3 has the form [12] …”

“…For an irreducible representation j the average of a group element over the S 2 is given by eq. (5.55) of [12]: Using the averages of the monomials calculated above we obtain…”

“…10) 12) z is an infrared regularization parameter, and µ 0 is an arbitrary renormalization parameter. The total effective action including the classical term in the one-loop approximation is…”