A new double-scaling limit of super-Yang–Mills theory and pp-wave strings

Abstract

The metric of a spacetime with a parallel plane (pp)-wave can be obtained in a certain limit of the space AdS 5 ×S 5 . According to the AdS/CFT correspondence, the holographic dual of superstring theory on that background should be the analogous limit of N = 4 supersymmetric Yang-Mills theory. In this paper we shall show that, contrary to widespread expectation, non-planar diagrams survive this limiting procedure in the gauge theory. Using matrix model techniques as well as combinatorial reasoning it is demonstrated that a subset of diagrams of arbitrary genus survives and that a non-trivial double scaling limit may be defined. We exactly compute two-and three-point functions of chiral primaries in this limit. We also carefully study certain operators conjectured to correspond to string excitations on the pp-wave background. We find non-planar linear mixing of these proposed operators, requiring their redefinition. Finally, we show that the redefined operators receive non-planar corrections to the planar one-loop anomalous dimension.of the "old matrix models" of non-critical bosonic string theory discovered in [6]. One consequence is that, in order to keep exact orthonormality of the operators (1.2), their correct normalization involves a non-trivial scaling functionAfter establishing that the recipe (1.1) does not fully suppress non-planar diagrams we are immediately led to the following puzzling question: What, then, characterizes the classes of diagrams that are favored, respectively suppressed, in this novel limit? By carefully investigating the pertinent combinatorics we find the following picture: Interpreting the operator TrZ J as a discrete closed string consisting of J "string bits" (see [3] and references therein) non-planar diagrams contributing to this operator correspond to the string splitting into multiple strings in intermediate channels. Taking J large may be interpreted as a continuum limit: The number of string bits diverges and the string becomes long and macroscopic. On the other hand, taking N large acts towards suppressing the string splitting. We then find that the scaling J 2 ∼ N leads to a delicate balance between the two effects such that microscopic strings, made out of only a small number of string bits (small w.r.t. J), are suppressed but macroscopic strings, made out of a large number of bits (i.e. of O(J)), survive.We also look at the three point correlation function of the chiral primaries (1.3), which are protected as well, and find the exact scaling function. As it explicitly encodes information for arbitrary genus, it would be fascinating if its structure could be understood from the string side [7].The present picture is very clear in the case of operators which are protected by supersymmetry, such as (1.3). In the case of unprotected operators, their quantum corrections require further analysis. Interactions involve index loops which produce factors of N. In the 't Hooft limit of Yang-Mills theory, these factors are controlled by making the coupling constant small, g 2 YM ∼ 1...

Quantizing string theory in AdS5×S5: beyond the pp-wave

“…These irrep expansions could equally well have been done using the bosonic R-symmetry group SO (6): this is what is done, with the same results, in [8]. The irreps with larger minimal values of ∆ 0 = K − R have multiplicities that grow as higher powers of K. This is very significant for the eventual string theory interpretation of the anomalous dimensions, but we will not expand on this point here.…”

The complete one-loop dilatation operator of super-Yang–Mills theory

“…We also note that, although L and B are not protected in this subsector, the combination L + B = ∆ 0 is. The weights are 19) where the numbers of individual letters are given by…”

pp-wave light-cone superstring field theory

“…The BMN operators are field theory duals of perturbative string states in the pp-wave background. The SYM theory is believed to be expandable in a double series in the effective genus counting parameter g 2 2 and effective coupling λ ′ [21,22] …”

“…Moreover, the finite quantity ∆−J is a function of λ ′ [20,23,24] and g 2 2 [21,22]. The expansion parameters g 2 and λ ′ are related to string theory parameters by [20] …”

“…Indeed, the exact notion of the Penrose limit in a CFT needs to be better understood [25]. By focusing on a certain subclass of the BMN operators, several proposals for such an extension of the BMN correspondence have been made and were studied in [21,22,26,27,28,29,30,31,32,33]. In this paper we use our interaction vertex to compute three-point functions for various string states.…”