Spectrum of LargeNGauge Theory from Supergravity

Abstract

Recently, Maldacena proposed that the large N limit of the N 4 supersymmetric gauge theory in four dimensions with U͑N͒ gauge group is dual to the type IIB superstring theory on AdS 5 3 S 5 . We use this proposal to study the spectrum of the large N gauge theory on 4 3 S 3 in a low energy regime. We find that the spectrum is discrete and evenly spaced, and the number of states at each energy level is smaller than the one predicted by the naive extrapolation of the Bekenstein-Hawking formula to the low energy regime. We also show that the gauge theory describes a region of spacetime behind the horizon as well as the region in front. [S0031-9007(98) (2) because of the Bekenstein-Hawking entropy formula for the near extremal 3-brane solution of type IIB theory [3]. In general (1) can be derived by assuming that the entropy is an extensive quantity and that it is invariant under the dilatation.Recently Maldacena made an interesting proposal in [4] that compactifications of M theory and string theory on a sphere to anti-de Sitter space (AdS) are dual to various conformal field theories. In particular, the large N limit of the N 4 U͑N͒ gauge theory in four dimensions is claimed to be described by the type IIB string theory on AdS 5 3 S 5 . This implies that spectra of the two theories and, hence, the entropies are the same.The purpose of this paper is to use this proposal to learn about the spectrum of the large N gauge theory on 4 3 S 3 . We will establish a correspondence between states in this gauge theory and states in string theory on AdS 5 3 S 5 . In the supergravity approximation to string theory, the energy levels are quantized in the units of the AdS radius R ͑4pgN͒ 1͞4 l s where g is the string coupling constant, and l s is the string length,The supergravity approximation is valid, when E is less than both the string scale and the Planck scale,Since l p ϳ g 1͞4 l s in ten dimensions, we can trust the supergravity computation forIn this regime, we find that the entropy of the type IIB string theory scales as a function of E n͞R asfor n ¿ 1. We will show that each supergravity state with energy E n͞R corresponds to a state with energy E n in the gauge theory on 4 3 S 3 with a unit radius sphere. Thus, Maldacena's proposal leads to the prediction that the entropy of the large N gauge theory on 4 3 S 3 for 1 ø n ø ͑ gN͒ 1͞4 is S gauge ϳ n 9͞10 .In particular, the large N spectrum is independent of N.On the other hand, the Bekenstein-Hawking formula (2) at this energy readsSince this formula was originally derived for the 3-branes wrapped on T 3 , it is a prediction for the density of states for the gauge theory on T 3 . However, for sufficiently large n, finite size effects are irrelevant, and we expect that the density of states is independent of the topology of the 3-manifold. In such a case, we can use the formula (6) for the gauge theory on S 3 also. Comparing this with (5), the power of n is different, and there is no factor of p N in (5). Therefore, in the low energy regime (3), S gauge is smaller than the...

Holography beyond the horizon and cosmic censorship

“…In fact, our consideration of the rotating BTZ black hole is very close in spirit to the analysis of [29]. The periodic identification in φ that defines the black hole has played only a spectator role in our discussion.…”

“…We would then say that what we have learnt is that if one considers the the field theory just on those regions of the boundary where ∂ φ is spacelike, it is dually related to the region of the spacetime which is determined by Cauchy data supplemented by the asymptotic boundary conditions on this portion of the boundary. So the discussion here is in a sense a generalisation of [29], considering the restriction to a different region of the boundary.…”

Membranes and fivebranes with lower supersymmetry and their AdS supergravity duals

“…Extending further this relation, it has been argued that in a suitable limit, the generating functional for the boundary correlators of singleton composite field is reproduced by the anti-de Sitter supergravity action [9,10]. The conjecture was further explored in [11,12,13,14,15,16,17,18]. Moreover, models with lower world-volume supersymmetries started to be explored [19,20,21,22].…”

Einstein spaces in five and seven dimensions and non-supersymmetric gauge theories

“…fixed Na 1 a 2 . To understand the large N limit on the supergravity side (in particular, properties of the spectrum [12,13,10]), it is necessary to determine how a 1 a 2 depends on N (cf. eq.…”