Exact anomalous dimensions of N=4 Yang–Mills operators with large R charge

Abstract

In a N = 1 superspace formulation of N = 4 Yang-Mills theory we obtain the anomalous dimensions of chiral operators with large R charge J → ∞ keeping g 2 N/J 2 finite, to all orders of perturbation theory in the planar limit. Our result proves the conjecture that the anomalous dimensions are indeed finite in the above limit. This amounts to an exact check of the proposed duality between a sector of N = 4 Yang-Mills theory with large R charge J and string theory in a pp-wave background.

Strings in plane wave backgrounds

“…A computation of the anomalous dimensions of BMN single-trace operators in interacting planar N = 4 SYM [14,49,50] reproduces the mass spectrum of free string theory [16,38]. It is obviously an interesting question how string interactions and the non-planar sector of (interacting) gauge theory will fit into this picture.…”

“…Further support to this conjecture comes from [49] which extended the above computation to two loops and presented arguments for higher loops, again matching the expectation coming from the expansion of the square root in (2.46). In [50] superconformal representation theory was used to argue that the full square root is reproduced; alternatively this was seen to be the case in [14] by exponentiating the quartic vertex; let me sketch how this works. SYM on R×S 3 can be expanded in spherical harmonics on the S 3 .…”

“…This is in contrast to the usual AdS/CFT correspondence, where due to our limited ability to perform calculations for finite λ in SYM -or equivalently in the full string theory on AdS 5 × S 5 -the relation is a strong/weak coupling duality. Note however, while perturbative calculations in λ of BMN operator two-and three-point functions can be reorganized in λ ′ [14,46,47,49,50] -and hence an extrapolation to large λ seems viable -this is no longer the case for higher point functions: computing for example the 4-point function of tr Z J perturbatively in λ, a naive extrapolation to large λ leads to divergences [51].…”

pp-wave light-cone superstring field theory

“…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] …”

“…While some further investigation seems necessary to clarify the exact form of this correspondence, at the planar (interacting) field theory two-point function/free string level, there is increasing evidence [20,23,24] that the BMN operators do correspond to the free string theory. The extension of the duality beyond this limit, by considering string interactions and the non-planar sector of (interacting) gauge theory respectively, needs further clarification.…”