Quarks and Triality in a Finite Volume

TL;DR

This paper constructs lattice QCD ensembles with non-zero triality in a finite volume, elucidating the role of center fluxes and Gauss's law in understanding the free energy of individual quarks.

Contribution

It introduces a novel method to fix triality in finite lattice volumes, extending effective theories to full lattice QCD and connecting to 't Hooft's electric fluxes and dual formulations.

Findings

Constructed ensembles with non-zero triality in finite volumes.

Extended the Polyakov-loop theory to full lattice QCD.

Derived the same results via dualization and transfer matrix approaches.

Abstract

In order to understand the puzzle of the free energy of an individual quark in QCD, we explicitly construct ensembles with quark numbers $N_V\neq 0\!\mod 3$, corresponding to non-zero triality in a finite subvolume $V$ on the lattice. We first illustrate the basic idea in an effective Polyakov-loop theory for the heavy-dense limit of QCD, and then extend the construction to full Lattice QCD, where the electric center flux through the surface of $V$ has to be fixed at all times to account for Gauss's law. This requires introducing discrete Fourier transforms over closed center-vortex sheets around the spatial volume $V$ between all subsequent time slices, and generalizes the construction of 't Hooft's electric fluxes in the purge gauge theory. We derive this same result from a dualization of the Wilson fermion action, and from the transfer matrix formulation with a local $\mathbb Z_3$-Gauss law to restrict the dynamics to sectors with the required center charge in $V$.