Entropy bounds and Cardy - Verlinde formula in Yang - Mills theory

Using gauge formulation of gravity the three-dimensional SU(2) YM theory equations of motion are presented in equivalent form as FRW cosmological equations. With the radiation, the particular (periodic, big bangbig crunch) three-dimensional universe is constructed. Cosmological entropy bounds (so-called Cardy-Verlinde formula) have the standard form in such universe. Mapping such universe back to YM formulation we got the thermal solution of YM theory. The corresponding holographic entropy bounds (Cardy-Verlinde formula) in YM theory are constructed. This indicates to universal character of holographic relations. [email protected] 2 [email protected] 1 It becomes popular in theoretical physics to apply different formulations (and even theories) in the description of the same phenomenon. For example, it is expected that Einstein theory presented in Yang-Mills (YM) form is easier to quantize. Moreover, its renormalizability properties seem to be better in YM-like form [1]. From another point, there exists the YM theory presentation in Einstein-like form. In this relation the natural question is: can one achieve some new results in YM theory using recent studies of cosmology based on holographic principle? In this paper we start the investigation in this direction. Using threedimensional YM theory as an example we first rewrite it in the gravitational form. The YM equations of motion are then rewritten as FRW cosmological equations where natural definition of Hubble, Bekenstein and BekensteinHawking entropies maybe done. As a result FRW equation is represented in the form similar to two-dimensional CFT entropy (so-called Cardy-Verlinde formula [2]). Introducing the radiation (matter) the explicit solution of such three-dimensional periodic universe is obtained. The cosmological entropy bounds (holographic Cardy-Verlinde formula) are quite simple for such universe. In particular, the Bekenstein entropy is constant. Such threedimensional universe is then mapped back to YM theory form. (Such procedure maybe considered as an interesting way of generating of YM solutions from FRW cosmological solutions). The correspondent thermal YM solution on which YM action diverges is constructed. The divergence of action is absorbed into the renormalization of the gauge coupling constant. After the identification of YM entropy with Bekenstein entropy the Cardy-Verlinde formula defines the entropy bounds in YM theory. The action of Yang-Mills theory with the gauge group SU(2) is given by SYM = 1 4g YM ∫