1/2 BPS structure constants and random matrices

Abstract

We study three point functions of half BPS operators in $$ \mathcal{N} $$ N = 4 super Yang-Mills theory focusing on correlators of two of the operators with dimension of order ∆ ~ N2 and a light single trace operator. These describe vacuum expectation values of type IIB supergravity modes in LLM backgrounds that do not necessarily preserve the same symmetries as the background solution. We propose a class of complex matrix models that fully capture the combinatorics of the problem, and describe their solution in the large N limit. In simple regimes when the dual description is in terms of widely separated condensates of giant gravitons we find that the models are solvable in the large N and can be approximated by unitary Jacobi ensembles; we describe how these distributions are reproduced in the dual bubbling geometry picture for large droplets. In the case of two eigenvalue droplets the model is exactly solvable at finite N. As a result we compute all half-BPS structure constants of heavy-heavy-light type, and reproduce the formulas found via holographic renormalization in the large N limit. We also comment on structure constants of three heavy operators.

Multi-matrix correlators and localization

“…Although the droplet description of such states in supergravity is compelling, a precise mapping between the weak coupling BPS states is not fully developed. 1 The coherent states (3.7) have a more natural connection to such geometries [15]. A worthwhile exercise would be to study correlators of single trace chiral primaries in the background of heavy coherent states corresponding to both giant gravitons or bubbling geometries; see [45] for some finite N results.…”

“…More recently, there have been works showing that certain generating functions can be used to perform computations in the free-field theory limit [8][9][10][11][12]. This technique has been succesfully implemented in the computation of three-point correlators involving large operators made out of a single matrix field [13][14][15], as in the half-BPS sector of N = SYM [16,17], where the dual gravitational description is explicitly realized from the gauge theory. An explicit mapping between BPS states made out of more than one matrix and asymptotically AdS 5 × S 5 geometries is still lacking, though a compelling description in terms of bubbling geometries seems to exist [18][19][20].…”