Large N behaviour of the two-dimensional Yang–Mills partition function

Source abstract

Abstract We compute the large N limit of the partition function of the Euclidean Yang–Mills measure on orientable compact surfaces with genus $g\geqslant 1$ and non-orientable compact surfaces with genus $g\geqslant 2$ , with structure group the unitary group ${\mathrm U}(N)$ or special unitary group ${\mathrm{SU}}(N)$ . Our proofs are based on asymptotic representation theory: more specifically, we control the dimension and Casimir number of irreducible representations of ${\mathrm U}(N)$ and ${\mathrm{SU}}(N)$ when N tends to infinity. Our main technical tool, involving ‘almost flat’ Young diagram, makes rigorous the arguments used by Gross and Taylor (1993, Nuclear Phys. B 400 (1–3) 181–208) in the setting of QCD, and in some cases, we recover formulae given by Douglas (1995, Quantum Field Theory and String Theory (Cargèse, 1993) , Vol. 328 of NATO Advanced Science Institutes Series B: Physics , Plenum, New York, pp. 119–135) and Rusakov (1993, Phys. Lett. B 303 (1) 95–98).