Eigenvalue dynamics for multimatrix models

By performing explicit computations of correlation functions, we find evidence that there is a sector of the two matrix model defined by the $SU(2)$ sector of $\mathcal{N}=4$ super Yang-Mills theory that can be reduced to eigenvalue dynamics. There is an interesting generalization of the usual Van der Monde determinant that plays a role. The observables we study are the Bogomol’nyi-Prasad-Sommerfield operators of the $SU(2)$ sector and include traces of products of both matrices, which are genuine multimatrix observables. These operators are associated with supergravity solutions of string theory.

Figure 1

An example of a graph labeling an operator with a definite scaling dimension. Each node corresponds to an eigenvalue. Edges connect the different nodes so that the eigenvalues are interacting.

Figure 2

An example of a graph labeling a BPS operator. Each node corresponds to an eigenvalue. There are no edges connecting the different nodes so that these eigenvalues are not interacting.