Solving loop equations by Hitchin systems via holography in large-N QCD_4

TL;DR

This paper establishes a holographic approach to solving large-N QCD_4 loop equations by mapping them to an effective action on Hitchin systems, simplifying the computation to residue calculations and reproducing the beta function coefficient.

Contribution

It introduces a novel holographic framework connecting large-N QCD_4 loop equations with Hitchin systems, enabling explicit solutions via residue calculations.

Findings

Loop equations are derived from an effective Hitchin system action.

Residue calculation simplifies the loop equations for self-avoiding loops.

Reproduction of the first beta function coefficient confirms the approach's validity.

Abstract

For (planar) closed self-avoiding loops we construct a "holographic" map from the loop equations of large-N QCD_4 to an effective action defined over infinite rank Hitchin bundles. The effective action is constructed densely embedding Hitchin systems into the functional integral of a partially quenched or twisted Eguchi-Kawai model, by means of the resolution of identity into the gauge orbits of the microcanonical ensemble and by changing variables from the moduli fields of Hitchin systems to the moduli of the corresponding holomorphic de Rham local systems. The key point is that the contour integral that occurs in the loop equations for the de Rham local systems can be reduced to the computation of a residue in a certain regularization. The outcome is that, for self-avoiding loops, the original loop equations are implied by the critical equation of an effective action computed in terms of the localisation determinant and of the Jacobian of the change of variables to the de Rham local systems. We check, at lowest order in powers of the moduli fields, that the localisation determinant reproduces exactly the first coefficient of the beta function.