arXiv 2104.062662021-04hep-th

伴随QCD_2中的瞬子及其费米子零模

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

Andrei Smilga

备注：: minor changes, a reference added. 28 pages, 3 figures
发表信息：: SciPost Phys. 10, 152 (2021)
DOI：: 10.21468/SciPostPhys.10.6.152

摘要：

具有规范群$SU(N)/Z_N$的伴随二维$QCD$允许与非平凡$π_1[SU(N)/Z_N] = Z_N$相关的拓扑非平凡规范场组态。拓扑扇区由整数$k=0,\ldots, N-1$标记。然而，与$QED_2$和$QCD_4$相反，该拓扑不与积分不变量（如磁通量或庞特里亚金指数）相关联。这些瞬子可能存在费米子零模，但左手模和右手模的数量总是相等的，因此在其他情况下确定模的数量的阿蒂亚-辛格定理不适用。模2的论证表明，对于一般的规范场组态，要么有一个这样的零模的双重态，要么没有任何模。然而，已知的狄拉克问题的解决方案，为广泛的规范场配置表明$k(N-k)$零模偶的存在，在拓扑扇区$k$。在本文中，我们以明确的方式证明了这些模式在规范背景的一般足够变形下是不鲁棒的，从而证实了模2猜想。这个理论的物理意义（屏蔽与限制问题）进行了简要讨论。

英文摘要：

The adjoint 2-dimensional $QCD$ with the gauge group $SU(N)/Z_N$ admits topologically nontrivial gauge field configurations associated with nontrivial $π_1[SU(N)/Z_N] = Z_N$. 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.

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