Supergravity and the largeNlimit of theories with sixteen supercharges

Abstract

We consider field theories with sixteen supersymmetries, which includes U(N) Yang-Mills theories in various dimensions, and argue that their large N limit is related to certain supergravity solutions. We study this by considering a system of D-branes in string theory and then taking a limit where the brane worldvolume theory decouples from gravity. At the same time we study the corresponding Dbrane supergravity solution and argue that we can trust it in certain regions where the curvature (and the effective string coupling, where appropriate) are small. The supergravity solutions typically have several weakly coupled regions and interpolate between different limits of string-M-theory. IntroductionString theory contains D-branes which are solitonic objects [1]. When we consider the full theory in the presence of these solitons we have modes that propagate in the bulk and modes that propagate on the solitons. The modes on the soliton interact with each other and with the bulk modes. It is possible, however, to define a limit of the full theory in which the bulk modes decouple from the modes living on the D-brane. This is typically a low energy limit, in which we tune the coupling constant so as to keep only the interactions among the modes living on the D-brane. In this limit the D-brane theory becomes super-Yang-Mills (for p ≤ 3). Separating the branes by some distance corresponds in the field theory to giving Higgs expectation values to some fields. Since we want to keep these expectation values finite when we take the limit, we should consider the branes at substringy distances [2].Since D-branes carry some mass and charge they excite the bulk gravity modes and we can find supergravity solutions carrying the same mass and charges. Naively the supergravity solution describes only the long range fields of the D-branes, since we do not expect supergravity to be valid at short distances. General covariance, however, tells us that we can trust the supergravity solution as long as curvatures are locally small compared to the string scale (or the Planck scale). A more careful analysis shows that for a system with a large number of branes, large N, the curvatures are small and we can trust the supergravity solutions even at the substringy distances involved in the decoupling limit described above. The situation is similar to the one studied in [3] for conformal field theories (see also [4,5]). In particular for the 4D N = 4 U(N) super-Yang-Mills theory associated with N D3-branes, it has been argued in [3] that it is "dual" to type IIB string theory on AdS 5 × S 5 in the large N limit.The aim of this paper is to explore analogous connections in the more general case of non-conformal field theories. The supergravity solutions corresponding to p + 1 super-Yang-Mills are black p-brane solutions. They are extended along p + 1 spacetime dimensions. We interpret the radial variable as being related to the energy scale of the process involved. One of the reasons for this interpretation is the fact that a Dp-brane sitting at so...

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