Matrix models without scaling limit

TL;DR

This paper demonstrates that the NLS and KdV hierarchies naturally arise from hermitean one-matrix models without requiring a continuum limit, highlighting their topological nature and connections to topological field theories.

Contribution

It shows that matrix models inherently encode integrable hierarchies without the need for scaling limits, revealing their topological significance.

Findings

NLS and KdV hierarchies emerge exactly from matrix models

No continuum limit is necessary to obtain these hierarchies

Connections to topological field theories and gravity are discussed

Abstract

In the context of hermitean one--matrix models we show that the emergence of the NLS hierarchy and of its reduction, the KdV hierarchy, is an exact result of the lattice characterizing the matrix model. Said otherwise, we are not obliged to take a continuum limit to find these hierarchies. We interpret this result as an indication of the topological nature of them. We discuss the topological field theories associated with both and discuss the connection with topological field theories coupled to topological gravity already studied in the literature.