A comment on instantons and their fermion zero modes in adjoint QCD_2

TL;DR

This paper examines the nature of instantons and fermion zero modes in 2D adjoint QCD, showing that zero modes are not robust under gauge background deformations and discussing implications for confinement.

Contribution

It provides an explicit demonstration that fermion zero modes in adjoint QCD_2 are not stable under gauge background deformations, confirming the mod 2 conjecture.

Findings

Zero modes are not robust under gauge background deformations.

The mod 2 argument suggests either a single doublet or no zero modes.

Implications for screening versus confinement in the theory.

Abstract

The adjoint 2-dimensional $QCD$ with the gauge group $SU(N)/Z_N$ admits topologically nontrivial gauge field configurations associated with nontrivial $\pi_1[SU(N)/Z_N] = Z_N$. The topological sectors are labelled by an integer $k=0,\ldots, N-1$. However, in contrast to $QED_2$ and $QCD_4$, this topology is not associated with an integral invariant like the magnetic flux or Pontryagin index. These instantons may admit fermion zero modes, but there is always an equal number of left-handed and right-handed modes, so that the Atiyah-Singer theorem, which determines in other cases the number of the modes, does not apply. The mod. 2 argument suggests that, for a generic gauge field configuration, there is either a single doublet of such zero modes or no modes whatsoever. However, the known solution of the Dirac problem for a wide class of gauge field configurations indicates the presence of $k(N-k)$ zero mode doublets in the topological sector $k$. In this note, we demonstrate in an explicit way that these modes are not robust under a generic enough deformation of the gauge background and confirm thereby the mod. 2 conjecture. The implications for the physics of this theory (screening vs. confinement issue) are briefly discussed.