THE LARGE-N LIMIT OF QCD AND THE COLLECTIVE FIELD OF THE HITCHIN
FIBRATION
M. Bochicchio
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 on 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 line bundles
over branched N-sheeted coverings of the fiber torus, the Hitchin fibration.
While for parabolic Higgs bundles of rank N the number of moduli per
parabolic point is of order N squared, for the Hitchin fibration the number
of moduli per branch point is at most of order N, since the ramification index
of any branch point cannot exceed N on a N-sheeted covering.
A collective field, from which, according to Hitchin, the Higgs bundle may be
recovered, consists of N holomorphic functions defining the covering and a
section of the line bundle.
As a consequence, all the entropy of the functional integration is absorbed,
in the large-N limit, into the Jacobian determinant of the change of variables
to the collective field of the Hitchin fibration.
Hence, the large-N limit is dominated by the saddle-point of the effective
action as in vector-like models.
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