On the planar free energy of matrix models

Abstract

In this work we obtain the planar free energy for the Hermitian one-matrix model with various choices of the potential. We accomplish this by applying an approach that bypasses the usual diagonalization of the matrices and the introduction of the eigenvalue density, to directly zero in the evaluation of the planar free energy. In the first part of the paper, we focus on potentials with finitely many terms. For various choices of potentials, we manage to find closed expressions for the planar free energy, and in some cases determine or bound their radius of convergence as a series in the ’t Hooft coupling. In the second part of the paper we consider specific examples of potentials with infinitely many terms, that arise in the study of $$ \mathcal{N} $$ N = 2 super Yang-Mills theories on S4, via supersymmetric localization. In particular, we manage to write the planar free energy of two non-conformal examples: SU(N) with Nf < 2N, and $$ \mathcal{N} $$ N = 2*.

A derivation of the planar limit of $$ \mathcal{N} $$ = 2 chiral correlators

“…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.…”

Defect correlators in a $\mathcal{N}=2$ SCFT at strong coupling

“…This way the computation of the vacuum expectation value of an observable in the N = 2 SYM theory which, in principle, should be performed by computing an infinite dimensional path integral, can be carried out via the evaluation of a finite dimensional integration over the elements of a matrix. When the N = 2 theory is conformal, this method led to several results such as the computation of correlation functions among chiral and anti-chiral scalar operators [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], the vacuum expectation value of Wilson loops [17][18][19][20][21][22] and the free energy of the theory [23,24].…”

Wilson loop correlators at strong coupling in $$ \mathcal{N} $$ = 2 quiver gauge theories

“…However, more recently, mainly by exploiting supersymmetric localization techniques [1] also the strong coupling regime of some N = 2 gauge theories in the large-N limit has been investigated. In the context of theories with eight supercharges many BPS observables have been analysed using localization techniques both at weak and strong coupling, such as correlation functions between chiral/anti-chiral scalar operators [2][3][4][5][6][7][8][9][10][11][12], correlators between a chiral operator and a Wilson loop [13][14][15][16][17], the vacuum expectation value of BPS Wilson loops [18][19][20] and the free energy [21][22][23][24].…”