Classical Fermi Fluid and Geometric Action for $c=1$

TL;DR

This paper models the $c=1$ matrix model as a quantum fluid, explores its classical limit with $$ corrections, and derives a geometric action for the phase space, connecting fluid profiles to string representations.

Contribution

It introduces a geometric action for the $c=1$ matrix model's classical phase space using fluid profiles as coadjoint orbit elements, incorporating string representations and gauge fields.

Findings

Derived a geometric action for the classical phase space of the $c=1$ matrix model.

Connected fluid profiles to string embeddings in phase space.

Showed the collective field action emerges under quadratic profile restrictions.

Abstract

We formulate the $c=1$ matrix model as a quantum fluid and discuss its classical limit in detail, emphasizing the $\hbar$ corrections. We view the fermi fluid profiles as elements of \winf-coadjoint orbit and write down a geometric action for the classical phase space. In the specific representation of fluid profiles as `strings' the action is written in a four-dimensional form in terms of gauge fields built out of the embedding of the `string' in the phase plane. We show that the collective field action can be derived from the above action provided one restricts to quadratic fluid profiles and ignores the dynamics of their `turning points'.