Dijkgraaf-Vafa theory as large-N reduction

TL;DR

This paper constructs a large-N reduced model of 4D super Yang-Mills theory with matter, linking it to the Dijkgraaf-Vafa matrix model and explaining their correspondence through a non-commutative superspace framework.

Contribution

It explicitly maps a reduced super Yang-Mills model to the Dijkgraaf-Vafa matrix model, providing a new perspective on their equivalence and the treatment of eigenvalues.

Findings

Established a matrix-function mapping in non-commutative superspace

Demonstrated the equivalence between the reduced model and the Dijkgraaf-Vafa matrix model

Provided a natural explanation for the eigenvalue distribution in the matrix model

Abstract

We construct a large-N twisted reduced model of the four-dimensional super Yang-Mills theory coupled to one adjoint matter. We first consider a non-commutative version of the four-dimensional superspace, and then give the mapping rule between matrices and functions on this space explicitly. The supersymmetry is realized as a part of the internal $U(\infty)$ gauge symmetry in this reduced model. Our reduced model can be compared with the Dijkgraaf-Vafa theory that claims the low-energy glueball superpotential of the original gauge theory is governed by a simple one-matrix model. We show that their claim can be regarded as the large-N reduction in the sense that the one-matrix model they proposed can be identified with our reduced model. The map between matrices and functions enables us to make direct identities between the free energies and correlators of the gauge theory and the matrix model. As a by-product, we can give a natural explanation for the unconventional treatment of the one-matrix model in the Dijkgraaf-Vafa theory where eigenvalues lie around the top of the potential.