On the planar free energy of matrix models

Abstract

We consider the Hermitian one-matrix model with a potential with arbitrarily many single-and double-trace terms. We apply 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 N " 2 super Yang-Mills theories on S 4 , via supersymmetric localization. In particular, we manage to write the planar free energy of two non-conformal examples: SU(N) with N f ă 2N , and N " 2 ˚.

Three-point functions in a $\mathcal{N}=2$ superconformal gauge theory and their strong-coupling limit

“…Indeed, as originally shown in [2], a generic N = 2 SYM theory in flat space can be mapped to a matrix model defined on a 4-sphere and the functional path-integral can be reduced to a finite dimensional integration over the elements of a matrix. Using this approach, many interesting results have been obtained in particular when the N = 2 theory is superconformal 1 , like for example the Wilson loop vacuum expectation value [4][5][6][7][8][9][10][11], the chiral/anti-chiral correlators [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28], the correlators of chiral operators and Wilson loops [29][30][31][32][33], the free energy [34][35][36] and the Bremsstrahlung function [37][38][39][40][41]. In the weak-coupling regime it is possible to check at the first perturbative orders that the results obtained with the matrix model agree with those obtained with standard Feynman diagrams (see for example [4,19,22,42,…”

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.…”