Abstract

We consider the gauge/gravity correspondence between maximally supersymmetric Yang–Mills theory in (p + 1) dimensions and superstring theory on the near-horizon limit of the Dp-brane solution. The string-frame metric is AdSp + 2 × S8 − p times a Weyl factor, and there is no conformal symmetry except for p = 3. In a previous paper by one of the present authors, the free-field result of gauge theory has been reproduced from string theory for a particular operator that has angular momentum along S8 − p. In this paper, we extend this result to operators that have angular momenta along AdSp + 2. Our approach is based on a Euclidean formulation proposed by Dobashi et al. [Nucl. Phys. B 665, 94 (2003)] and on the “string bit” picture. We first show that the spinning string solution in Lorentzian AdS, found by Gubser et al. [Nucl. Phys. B 636, 99 (2002)], can be recast in a form that connects two points on the boundary of Euclidean AdS. The transition amplitudes of such strings can be interpreted as gauge theory correlators. We study the case of zero gauge coupling by ignoring interactions among string bits (massless particles in 10D spacetime that constitute a string), and show that the free-field results of gauge theory are reproduced.

Abstract

Gravitational thermodynamics and gravitoscalar thermodynamics with S² × ℝ boundary geometry are investigated through the partition function, assuming that all Euclidean saddle point geometries contribute to the path integral and dominant ones are in the B³ × S¹ or S² × Disc topology sector. In the first part, I concentrate on the purely gravitational case with or without a cosmological constant and show there exists a new type of saddle point geometry, which I call the “bag of gold(BG) instanton,” only for the Λ > 0 case. Because of this existence, thermodynamical stability of the system and the entropy bound are absent for Λ > 0, these being universal properties for Λ ≤ 0. In the second part, I investigate the thermodynamical properties of a gravity-scalar system with a φ² potential. I show that when Λ ≤ 0 and the boundary value of scalar field J_φ is below some value, then the entropy bound and thermodynamical stability do exist. When either condition on the parameters does not hold, however, thermodynamical stability is (partially) broken. The properties of the system and the relation between BG instantons and the breakdown are discussed in detail.

In 1999, R. B. Mann proposed a counterterm that is some sort of
generalization of the well-known Holographic counterterm and that can eliminate
the divergence of the gravitational action of asymptotically AdS and flat
spacetimes (Phys. Rev. D $\boldsymbol{60}$ (1999) 104047 [1]). I show it is not
only for eliminating the divergence of such spacetimes but also for setting the
ground state energy to zero for any $d$-dimensional spacetimes with an $S^{d-2}
\times \mathbb{R} $ boundary geometry, and speculate it is also true for
spacetimes with any (suitable) boundary geometry and topology.