N=2 Conformal SYM theories at large N

Abstract

We consider a class of N = 2 conformal SU(N ) SYM theories in four dimensions with matter in the fundamental, two-index symmetric and anti-symmetric representations, and study the corresponding matrix model provided by localization on a sphere S 4 , which also encodes information on flat-space observables involving chiral operators and circular BPS Wilson loops. We review and improve known techniques for studying the matrix model in the large-N limit, deriving explicit expressions in perturbation theory for these observables. We exploit both recursive methods in the so-called full Lie algebra approach and the more standard Cartan sub-algebra approach based on the eigenvalue distribution. The sub-class of conformal theories for which the number of fundamental hypermultiplets does not scale with N differs in the planar limit from the N = 4 SYM theory only in observables involving chiral operators of odd dimension. In this case we are able to derive compact expressions which allow to push the small 't Hooft coupling expansion to very high orders. We argue that the perturbative series have a finite radius of convergence and extrapolate them numerically to intermediate couplings. This is preliminary to an analytic investigation of the strong coupling behavior, which would be very interesting given that for such theories holographic duals have been proposed.

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

“…SCFTs [11][12][13]30]. A very interesting problem would be to prove our conjectures (1.4) and (1.5).…”

“…For concreteness, we will limit ourselves to present all the terms with a single value of the ζ function. Collecting all such terms we deduce of [11] (see also [12]). These ingredients hint at a diagrammatic derivation of (4.10).…”

“…A potentially more illuminating proof might bypass the relation to S 4 correlators, and work directly on R 4 . Indeed, the factor `´λ diagrams considered in closely related work [11,12] (see also [31]), so the form of (1.4) and (1.5) suggest that they can be proven by a combinatorial argument, counting the ways in which those particular Feynman diagrams enter the evaluation of O k Ōk and…”

“…A first approach [2][3][4][5] uses a 4d analog of the tt ˚equations [1]. More recently, it has been shown [6,7] that the evaluation of closely related n-point functions on S 4 can be reduced through supersymmetric localization to matrix model computations; in turn, a Gram-Schmidt orthogonalization procedure applied to these S 4 correlators yields the correlators on R 4 [8][9][10][11][12][13]. Alternatively, the large R-charge limit [14] of these correlation functions has been studied in [15][16][17][18][19][20].…”

“…The path described above has been followed already in a number of papers [3][4][5][8][9][10][11][12][13]. The novel ingredient that we introduce in this work is an alternative way of evaluating the free energy and correlators of the relevant matrix models, which allows us to obtain all-order analytic expressions in the planar limit.…”