From 2d droplets to 2d Yang-Mills

We establish a connection between time evolution of free Fermi droplets and partition function of generalised q-deformed Yang-Mills theories on Riemann surfaces. Classical phases of (0 + 1) dimensional unitary matrix models can be characterised by free Fermi droplets in two dimensions. We quantise these droplets and find that the modes satisfy an abelian Kac-Moody algebra. The Hilbert spaces H+ and H- associated with the upper and lower free Fermi surfaces of a droplet admit a Young diagram basis in which the phase space Hamiltonian is diagonal with eigenvalue, in the large N limit, equal to the quadratic Casimir of u(N). We establish an exact mapping between states in H-+/- and geometries of droplets. In particular, coherent states in H-+/- correspond to classical deformation of upper and lower Fermi surfaces. We prove that correlation between two coherent states in H-+/- is equal to the chiral and anti-chiral partition function of 2d Yang-Mills theory on a cylinder. Using the fact that the full Hilbert space H+ circle times H- admits a composite basis, we show that correlation between two classical droplet geometries is equal to the full U(N) Yang-Mills partition function on cylinder. We further establish a connection between higher point correlators in H-+/- and higher point correlators in 2d Yang-Mills on Riemann surface. There are special states in H-+/- whose transition amplitudes are equal to the partition function of 2d q-deformed Yang-Mills and in general character expansion of Villain action. We emphasise that the q-deformation in the Yang-Mills side is related to special deformation of droplet geometries without deforming the gauge group associated with the matrix model. (C) 2021 The Author(s). Published by Elsevier B.V.

Automated summary

In this study, a connection between the time evolution of free Fermi droplets and the partition function of generalised q-deformed Yang-Mills theories on Riemann surfaces is established. The classical phases of (0 + 1) dimensional unitary matrix models can be characterized by free Fermi droplets in two dimensions. The quantization of these droplets reveals that the modes satisfy an abelian Kac-Moody algebra. The correlation between states in H-+/- is proven to be equal to the chiral and anti-chiral partition function of 2d Yang-Mills theory on a cylinder.