SciPost Phys. 15, 195 (2023) ·
published 14 November 2023

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We find the spectra and eigenfunctions of both ordinary and supersymmetric quantummechanical models describing the motion of a charged particle over the $\mathbb{CP}^{n1}$ manifold in the presence of a background monopolelike gauge field. The states form degenerate $SU(n)$ multiplets and their wave functions acquire a very simple form being expressed via homogeneous coordinates. Their relationship to multidimensional orthogonal polynomials of a special kind is discussed. By the wellknown isomorphism between the twisted Dolbeault and Dirac complexes, our construction also gives the eigenfunctions and eigenvalues of the Dirac operator on complex projective spaces in a monopole background.
SciPost Phys. 10, 152 (2021) ·
published 22 June 2021

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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.