The planar limit of mathcal{N} = 2 superconformal quiver theories

Bartomeu Fiol,Alan Rios Fukelman,Jairo Martfnez-Montoya

doi:10.1007/jhep08(2020)161

Abstract

We compute the planar limit of both the free energy and the expectation value of the 1/2 BPS wilson loop for four dimensional mathcal{N} = 2 superconformal quiver theories, with a product of SU(N)s as gauge group and hi-fundamental matter. Supersymmetric localization reduces the problem to a multi-matrix model, that we rewrite in the zero instanton sector as an effective action involving an infinite number of double-trace terms, determined by the relevant extended Cartan matrix. We find that the results, as in the case of mathcal{N} = 2 SCFTs with a simple gauge group, can be written as sums over tree graphs. For the hat{A_1} case, we find that the contribution of each tree can be interpreted as the partition function of a generalized Ising model defined on the tree; we conjecture that the partition functions of these models defined on trees satisfy the Lee-Yang property, i.e. all their zeros lie on the unit circle.

Highlights

Four dimensional N = 2 quiver CFTs have already been studied using localization [7, 10,11,12, 20]
Supersymmetric localization reduces the problem to a multi-matrix model, that we rewrite in the zeroinstanton sector as an effective action involving an infinite number of double-trace terms, determined by the relevant extended Cartan matrix
We find that the results, as in the case of N = 2 SCFTs with a simple gauge group, can be written as sums over tree graphs

Summary

Planar free energy

The products of m + 1 connected correlators that contribute to the planar free energy are those where the 2m traces are distributed in a way that can be characterized by a tree graph [19]: for each correlator introduce a vertex, and join them by an edge if they have operators from the same double-trace. Terms with m values of the ζ function have m pairs of traces, coming from m double-trace terms, which are of the form CIJ Tr aI2(n−k)Tr a2Jk. To find the contribution to the planar free energy at this order, first draw all the trees with m edges. Where vodd is the number of vertices of the tree with odd degree, and p(λ1, λ2) is a symmetric polynomial in λ1 and λ2 with positive coefficients This follows from inspection of the factor attached to each vertex, (2.17). In appendix A we have written the outcome of these sums, up to order λ6

The Lee-Yang property of the planar free energy expansion

Wilson loop in the large N limit

B Wilson loop up to λ7