The SAGEX review on scattering amplitudes Chapter 8: Half BPS correlators

Abstract

Half BPS correlators in N=4 super Yang-Mills theory are key quantities both in the AdS/CFT correspondence as well as in scattering amplitudes research. They are dual at strong coupling to quantum gravity amplitudes. At weak coupling on the other hand they contain all N=4 SYM amplitudes. They have been found to possess a number of hidden symmetries, for example non-trivial permutation symmetry of perturbative four-point integrands and a higher dimensional conformal symmetry. Their study has enjoyed continuous progress since the discovery of AdS/CFT and they are now some of the best understood quantities of any four-dimensional quantum field theory. In this review we outline the current knowledge of half BPS correlators, emphasising these two co-existing relations to scattering amplitudes at strong and weak coupling.

One-loop string amplitudes in AdS5×S5: Mellin space and sphere splitting

“…Moreover, it does not depend explicitly on the charges p i at all. In a similar way, M (1,3) is just a constant, but for the factor of (Σ − 1) 3 (which is singlet under crossing). The crucial observation is that both M (1,0) and M (1,3) can be understood as 10d objects in the following precise sense.…”

“…The combination of analytic bootstrap techniques and the knowledge about the spectrum of supergravity, which consists of protected half-BPS single-particle states and multi-particle states (but no excited string states), allows one to solve the problem of computing the tree-level contribution to the four-point correlators. The expression for M (1,0) was given in [4,5], building on previous work [33]. 5 In our notation…”

“…The appearance of these bold-font variables has a precise meaning [22], which we explain in the next paragraph. The first tree-level (α ) 3 string correction is given simply by [27] M (1,3) = 2ζ 3 (Σ − 1) 3 .…”

“…To various orders, fully explicit results have been computed in [28][29][30]. Both M (1,0) and M (1,3) come with interesting properties which are simple to see in our formalism: We would expected M (1,0) to be function of all variables {ŝ, t, š, ť, p 1 , p 2 , p 3 , p 4 } but it happens to depend only on the specific combinations s = ŝ + š and t = t + ť, u = −s − t − 4. Moreover, it does not depend explicitly on the charges p i at all.…”

“…An ongoing challenge in theoretical physics is to understand the dynamics of quantum gravity and its relation to string theory. Recent advances within the analytic bootstrap program have shown that the study of the AdS/CFT correspondence, beyond the classical bulk approximation, leads to concrete and quantitative results for both quantum gravity and string theory in AdS, see [1,2] for a review. In this context, the problem of quantum gravity is well posed and the CFT approach is able to tackle AdS gravity with a variety of computational tools.…”