From 2d droplets to 2d Yang-Mills

Abstract

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+⊗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.