Towards the solution to the giant graviton puzzle

In this note we present several ideas toward the solution to the giant graviton puzzle – the apparent multiplicity of supergravity states dual to field theory chiral primary operators. We use the fact that, for certain ranges of the angular momentum, giant gravitons can be mapped into vacua of a dual theory to argue that the sphere and AdS giant gravitons have very different boundary descriptions, and that an unpolarized KK graviton is unphysical in the regime where giant gravitons exist. We also show that a generic boundary state can correspond to different giant graviton configurations, which have non-overlapping ranges of validity. 1 Review of the puzzle In the AdS5 × S 5 supergravity dual of N = 4 SYM we have three distinct types of configurations which correspond to a boundary state with large R-charge. One is a graviton circling the “equator” of the S, the second is a graviton polarized [1] into D3 branes wrapping an S ⊂ S (the giant graviton or sphere giant graviton) [2], and the third is a graviton polarized into D3 branes wrapping an S ⊂ AdS5 (the dual giant graviton or AdS giant graviton.) [3, 4, 5] The field theory interpretation of these states is an interesting issue. The original puzzle was that single-trace chiral primary operators in the field theory with R-charge L should be dual to single particle states with angular momentum L on the S. The natural candidate for these states is the graviton. However, at finite N , there is a cut-off in the field theory since there are no independent single-trace operators with L > N , yet there is no obvious upper bound on the angular momentum of the graviton. The predicted upper bound was thought to be due to stringy effects and dubbed the “stringy exclusion principle” [2]. Instead, the resolution turned out to be an IR effect where the gravity dual was identified as the giant graviton. The size of the S which the brane wraps grows with the angular momentum until the upper bound of L = N is reached where the radius of the S reaches the radius of the S. Unfortunately there are two problems with the above picture. The first is essentially a technical point – for L of order N the single-trace operators are no longer orthogonal (even at large N). However, this does not affect the argument since the correct operators are sub-determinant operators [6] which are also cut off at L = N . The second point, which we address in this paper, is that there is not only the question of whether the extended giant gravitons should be preferred over the point-like gravitons but that the extended AdS giant gravitons also carry the same quantum numbers, appear to have similar properties to the giant gravitons, but crucially have no upper limit on L since the sphere they wrap can be arbitrarily large within the AdS spacetime. So, clearly the giant graviton is the one which should correspond to the field theory state, but how do we rule out the other two states? The presence of these two extra states has long been a puzzle, and different arguments have been made about their fate. One possible explanation is that they correspond to two short multiplets which combine to form a long multiplet, whose dimension is no longer protected [3]. However, as we will see, this is not what seems to happen. The existence of many bulk states with polarized branes dual to only one boundary state is highly reminiscent of a similar problem in gauge–gravity dualities. When one discusses the supergravity dual of the N = 1∗ theory [7], one also encounters three bulk vacua dual to one gauge theory vacuum. As we will review in the first chapter, a generic field theory vacuum can naively have three supergravity duals. One candidate dual bulk contains D3 branes polarized into NS5 branes, another one contains D3 branes polarized into D5 branes, and the third one has a singularity and does not contain any polarized