Abstract
We construct holographic backgrounds that are dual by the AdS/CFT correspondence to Euclidean conformal field theories on products of spheres {S}^{d_1}times {S}^{d_2} , for conformal field theories whose dual may be approximated by classical Einstein gravity (typically these are large N strongly coupled theories). For d2 = 1 these backgrounds correspond to thermal field theories on {S}^{d_1} , and Hawking and Page found that there are several possible bulk solutions, with two different topologies, that compete with each other, leading to a phase transition as the relative size of the spheres is modified. By numerically solving the Einstein equations we find similar results also for d2> 1, with bulk solutions in which either one or the other sphere shrinks to zero smoothly at a minimal value of the radial coordinate, and with a first order phase transition (for d1 + d2< 9) between solutions of two different topologies as the relative radius changes. For a critical ratio of the radii there is a (sub-dominant) singular solution where both spheres shrink, and we analytically analyze the behavior near this radius. For d1 + d2< 9 the number of solutions grows to infinity as the critical ratio is approached.
Highlights
Time identified; in this solution the Sd−1 factor shrinks to zero in the interior of space, such that the topology is Rd × S1
We construct holographic backgrounds that are dual by the AdS/CFT correspondence to Euclidean conformal field theories on products of spheres Sd1 × Sd2, for conformal field theories whose dual may be approximated by classical Einstein gravity
By numerically solving the Einstein equations we find similar results for d2 > 1, with bulk solutions in which either one or the other sphere shrinks to zero smoothly at a minimal value of the radial coordinate, and with a first order phase transition between solutions of two different topologies as the relative radius changes
Summary
In terms of this coordinate we write: Gμν dxμdxν This is the convention used in the Fefferman-Graham expansion in order to renormalize the gravitational action [11]. For a given topology and r0 there is a unique metric given by solving (2.6) with the initial conditions (2.3), giving the pair of functions f (z), h(z) This metric is a saddle point of the AdS partition function, and so can be used to extract CFT information. The d’th order term of the expansion (2.9), g(d), gives (up to a constant proportionality number) the expectation value of the CFT stress tensor gi(jd) ∼ Tij (for even dimensions there is a further anomalous contribution to the trace of Tij ) In our case this means the stress tensor expectation values are given by dd dρd
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