Following Lin and Maldacena, we find exact supergravity solutions dual to a class of vacua of the plane wave matrix model by solving an electrostatics problem. These are asymptotically near-horizon D0-brane solutions with a throat associated with NS5-brane degrees of freedom. We determine the precise limit required to decouple the asymptotic geometry and leave an infinite throat solution found earlier by Lin and Maldacena, dual to Little String Theory on S 5 . By matching parameters with the gauge theory, we find that this corresponds to a double scaling limit of the plane wave matrix model in which N → ∞ and the 't Hooft coupling λ scales as ln 4 (N ), which we speculate allows all terms in the genus expansion to contribute even at infinite N . Thus, the double-scaled matrix quantum mechanics gives a Lagrangian description of Little String Theory on S 5 , or equivalently a ten-dimensional string theory with linear dilaton background.1 Unfortunately, the solution contains Ramond-Ramond fields, so string theory is difficult. 2 A similar situation occurs in [6,7], though in the present case, more of the R-symmetry is preserved. See also [8] and [9] for discussions of the type IIB Little String Theory compactified on S 2 and S 3 respectively.

We show that little string theory on S 5 can be obtained as double-scaling limits of the maximally supersymmetric Yang-Mills theories on R×S 2 and R×S 3 /Z k . By matching the gauge theory parameters with those in the dual supergravity solutions found by Lin and Maldacena, we determine the limits in the gauge theories that correspond to decoupling of NS5-brane degrees of freedom. We find that for the theory on R × S 2 , the 't Hooft coupling must be scaled like ln 3 N , and on R × S 3 /Z k , like ln 2 N . Accordingly, taking these limits in these field theories gives Lagrangian definitions of little string theory on S 5 .

We develop a formalism that allows us to write actions for multiple D-branes with manifest general covariance. While the matrix coordinates of the D-branes have a complicated transformation law under coordinate transformations, we find that these may be promoted to (redundant) matrix fields on the transverse space with a simple covariant transformation law. Using these fields, we define a covariant distribution function (a matrix generalization of the delta function which describes the location of a single brane). The final actions take the form of an integral over the curved space of a scalar single-trace action built from the covariant matrix fields, tensors involving the metric, and the covariant distribution function. For diagonal matrices, the integral localizes to the positions of the individual branes, giving N copies of the single-brane action.

We show that the requirement of Poincaré invariance (more specifically invariance under boosts/rotations that mix brane directions with transverse directions) places severe constraints on the form of actions describing multiple D-branes, determining an infinite series of correction terms to the currently known actions. For the case of D0-branes, we argue that up to field redefinitions, there is a unique Lorentz transformation rule for the coordinate matrices consistent with the Poincaré algebra. We characterize all independent Poincaré invariant structures by describing the leading term of each and providing an implicit construction of a Lorentz invariant completion. Our construction employs new matrix-valued Lorentz covariant objects built from the coordinate matrices, which transform simply under the (extremely complicated) Lorentz transformation rule for the matrix coordinates.1 These could be the usual D0-branes of type IIA string theory with d = 9, or any other pointlike D-branes arising from higher dimensional branes wrapped on cycles in a compactification.2 Here, all bulk fields have been set to zero.3 Indeed, to our knowledge none of the actions for multiple D-branes that have appeared previously in the literature are Poincaré invariant apart from the cases p = −1 and p = 9 which are trivial.4 Early work on understanding the structure of non-abelian D-brane actions based on general principles was initiated by Douglas [2,3]. For general reviews discussing the physics of multiple D-branes, including many additional references, see [4,5].

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