Prepared for submission to JHEP From 2d Droplets to 2d Yang-Mills Arghya Chattopadhyaya, Suvankar Duttab, Debangshu Mukherjeeb;c, Neetub aInstitute of Mathematical Sciences, Homi Bhaba National Institute (HBNI) IV Cross Road, Taramani, Chennai 600113, Tamil Nadu, India bIndian Institute of Science Education and Research Bhopal Bhopal Bypass, Bhopal 462066, India cIndian Institute of Science Education and Research Thiruvananthapuram Vithura 695551, Kerala, India E-mail:
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[email protected] 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.