The planar limit of ${\cal N}=2$ chiral correlators

Abstract

We derive the planar limit of 2-and 3-point functions of single-trace chiral primary operators of N " 2 SQCD on S 4 , to all orders in the 't Hooft coupling. In order to do so, we first obtain a combinatorial expression for the planar free energy of a hermitian matrix model with an infinite number of arbitrary single and double trace terms in the potential; this solution might have applications in many other contexts.We then use these results to evaluate the analogous planar correlation functions on R 4 . Specifically, we compute all the terms with a single value of the ζ function for a few planar 2-and 3-point functions, and conjecture general formulas for these terms for all 2-and 3-point functions on R 4 .

The planar limit of $$ \mathcal{N} $$ = 2 chiral correlators

“…Our approach to compute these extremal correlators -initiated in our previous work [26] -also relies squarely on supersymmetric localization, but it contains a couple of twists that we now detail. The first new ingredient is an alternative approach to compute the relevant integrals [9,27,28] -dubbed the full Lie algebra approach in [29] -that doesn't restrict the integration domain to a Cartan subalgebra, thus avoiding the introduction of Vandermonde determinants.…”

“…For N " 2 theories, this leads to a rewriting of the matrix integrals in terms of an effective action with an infinite number of single and double trace terms [28,30]. The second ingredient is a combinatorial solution of the planar limit of such matrix models with single and double trace terms [26,[30][31][32], given by a sum over tree graphs.…”

On the planar free energy of matrix models

“…On the other hand, potentials of the form (1.2) with an infinite number of terms appear in statistical physics, in Chern-Simons theories with matter, or in duals to M-theory on certain backgrounds as reviewed in [5]. More recently, it has been realized [6][7][8][9] that supersymmetric localization [10] reduces the evaluation of certain observables of four dimensional N " 2 super Yang-Mills (SYM) theories on S 4 to matrix models that can be recast to take the form of (1.2).…”

“…The second ingredient, which builds upon the first, is a recently found combinatorial expression for the planar free energy of matrix models with potentials of the form (1.2) [7,9] as a sum over a particular type of graphs, known as tree graphs…”

“…where V i is the planar connected correlator, given by (1.5), on the i-th vertex on the tree, that contains the following operators: tr a is if the directed edge labelled s leaves that vertex; tr a js if the directed edge labelled s arrives at that vertex; any single trace operators inserted on that vertex. While in this work we will focus on the planar free energy, we have shown in previous work how to apply this approach to the planar limit of other observables, like the Wilson loop [7,8] or extremal correlators [9].…”