Effective Action and Phase Transitions in Yang-Mills Theory on Spheres

Abstract

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 .

One-loop quantum gravity in the Einstein universe

“…We will need the asymptotics of the function Ω(t) obtained in [11]. We have as t → 0 14) and as t → ∞,…”

“…In [9] we applied these methods to study quantum gravity and Yang-Mills theory on any symmetric space. Further, we applied these methods to study the thermal Yang-Mills theory on product of spheres, such as S 1 × S 1 × S 2 and S 1 × S 3 in [11,12].…”

Thermal Yang-Mills Theory In the Einstein Universe

“…In four dimensions there are not many choices of spaces of non-negative constant curvature, only products of spheres and circles; the non-flat ones are S 1 × S 1 × S 2 and S 1 × S 3 . The case of S 1 × S 1 × S 2 was studied in our previous paper [11] and in this paper we studied the case of S 1 × S 3 . One of our results is the calculation of the heat kernel of the Laplacian acting on arbitrary fields on S 3 in arbitrary representation of SU (2).…”

“…In our recent paper [11] we considered the finite temperature Yang-Mills theory on S 1 × S 1 × S 2 with an Abelian covariantly constant background on S 2 . We showed that despite the positive spatial curvature the gluon operator still has negative modes for any compact semi-simple gauge group, which means that the vacuum with covariantly constant chromomagnetic fields on S 2 does not represent the true vacuum of Yang-Mills theory.…”

“…The present paper is a continuation of the study of Yang-Mills theory on spheres; our primary goal here is to extend this analysis to the Einstein universe S 1 × S 3 with a non-Abelian covariantly constant background on S 3 with a compact simple gauge group G that has the group SU (2) as a subgroup. We refer to the paper [11] for general introduction and notation.…”