Wilson loops and the geometry of matrix models in AdS_4/CFT_3

TL;DR

This paper explores the relationship between supersymmetric Wilson loops in AdS_4/CFT_3 duality and the geometry of matrix models, revealing how critical points of a Hamiltonian function determine supersymmetric configurations and eigenvalue distributions.

Contribution

It establishes a geometric criterion for supersymmetric Wilson loops via the Hamiltonian function h_M and connects critical points to eigenvalue density discontinuities in matrix models.

Findings

Critical points of h_M correspond to supersymmetric M2-brane configurations.

Values of h_M at critical points determine M2-brane actions and Wilson loop values.

The image of h_M(Y_7) influences the eigenvalue support in the dual matrix model.

Abstract

We study a general class of supersymmetric AdS_4 x Y_7 solutions of M-theory that have large N dual descriptions as N = 2 Chern-Simons-matter theories on S^3. The Hamiltonian function h_M for the M-theory circle, with respect to a certain contact structure on Y_7, plays an important role in the duality. We show that an M2-brane wrapping the M-theory circle, giving a fundamental string in AdS_4, is supersymmetric precisely at the critical points of h_M, and moreover the value of this function at the critical point determines the M2-brane action. Such a configuration determines the holographic dual of a BPS Wilson loop for a Hopf circle in S^3, and leads to an effective method for computing the Wilson loop on both sides of the correspondence in large classes of examples. We find agreement in all cases, including for several infinite families, and moreover we find that the image h_M(Y_7) determines the range of support of the eigenvalues in the dual large N matrix model, with the critical points of h_M mapping to points where the derivative of the eigenvalue density is discontinuous.