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A comment on instantons and their fermion zero modes in adjoint QCD_2
by Andrei Smilga
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Submission summary
Authors (as registered SciPost users):  Andrei Smilga 
Submission information  

Preprint Link:  https://arxiv.org/abs/2104.06266v3 (pdf) 
Date accepted:  20210618 
Date submitted:  20210616 10:13 
Submitted by:  Smilga, Andrei 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The adjoint 2dimensional $QCD$ 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 $k=0,\ldots, N1$. 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 lefthanded and righthanded modes, so that the AtiyahSinger 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 $k(Nk)$ 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.
List of changes
Dear Editors,
Following the request of the second referee, I've introduced the corrections to the article:
1.
I expanded the last paragraph of the paper describing in more detail what was done in Refs. [16,17].
2.
I have also inserted Eq. (6.10) illustrating how the potential between the heavy fractionally charged sources levels off at large distances in the Schwinger model, where the screening phenomenon is well understood.
Published as SciPost Phys. 10, 152 (2021)