The large-N limit of QCD and the collective field of the Hitchin fibration

Abstract

By means of a certain exact non-abelian duality transformation, we show that there is a natural embedding, dense in the sense of the distributions in the large-N limit, of parabolic Higgs bundles of rank N on a fiber two-dimensional torus into the QCD functional integral, fiberwise over the base two-dimensional torus of the trivial elliptic fibration on which the four-dimensional theory is defined. The moduli space of parabolic Higgs bundles of rank N is an integrable Hamiltonian system, that admits a foliation by the moduli of holomorphic line bundles over N-sheeted spectral covers (or, what is the same, over a space of N gauge-invariant polynomials), the Hitchin fibration. According to Hitchin, the Higgs bundles can be recovered from the spectral covers and the line bundles. If the N invariant polynomials together with the abelian connection on the line bundles are chosen as the N+1 collective fields of the Hitchin fibration, all the entropy of the functional integration over the moduli of the Higgs bundles is absorbed, in the large-N limit, into the Jacobian determinant of the change of variables to the collective fields of the Hitchin fibration. Hence, the large-N limit is dominated by the saddle-point of the effective action as in vector-like models.