Abstract
The adjoint 2-dimensional QCD with the gauge group SU(N)/Z_NSU(N)/ZN admits topologically nontrivial gauge field configurations associated with nontrivial \pi_1[SU(N)/Z_N] = Z_Nπ1[SU(N)/ZN]=ZN. The topological sectors are labelled by an integer k=0,\ldots, N-1k=0,…,N−1. However, in contrast to QED_2QED2 and QCD_4QCD4, 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 [1] 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 conjecture of Ref. [1]. The implications for the physics of this theory (screening vs. confinement issue) are briefly discussed.
Highlights
The Lagrangian of the massless 2-dimensional QCD with fermions lying in the adjoint representation of the SU(N ) gauge group reads [5] L = Tr − Fμν + iψγμ Dμ ψ (1.1)where Aμ = Aaμ ta, ψ = ψa ta is a 2-component Majorana spinor, Dμψ = ∂μψ − i g[Aμ, ψ], and the coupling g has the dimension of mass
In contrast to QE D2 and QC D4, this topology is not associated with an integral invariant like the magnetic flux or Pontryagin index
The known solution of the Dirac problem for a wide class of gauge field configurations [2,3,4] indicates the presence of k(N − k) zero mode doublets in the topological sector k
Summary
The Lagrangian of the massless 2-dimensional QCD with fermions lying in the adjoint representation of the SU(N ) gauge group reads [5]. This number may be odd, and in this case at least one such doublet must stay on zero for an arbitrary deformation, or it can be even, and in this case the zero modes are not protected 4, we consider, following [2,3,4], the theory on a cylinder S1 × R, with S1 being a finite spatial circle with antiperiodic boundary conditions for the fermion fields and R the infinite Euclidean time axis, t ∈ (−∞, ∞). We consider more general deformations of the potential and show that in this case most zero modes disappear, leaving only one doublet of the modes when k(N − k) is odd.
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