Abstract
In this paper we study massless fermions coupled to spherically symmetric SU(N) monopoles without Yukawa couplings between the Higgs and fermion fields. The corresponding Dirac operator is not Fredholm and the associated eigenfunctions are not L2-normalizable. Here we derive a formula for the dimension of the plane-wave normalizable kernel of such a Dirac operator for fermions of any representation of SU(N) in the presence of any spherically symmetric monopole background. Notably, our results also apply to fermions coupled to monopoles that preserve non-abelian gauge symmetry.
Highlights
In this paper we study massless fermions coupled to spherically symmetric SU(N ) monopoles without Yukawa couplings between the Higgs and fermion fields
In the asymptotic limit of moduli space near a wall of marginal stability where the BPS states separate into two relatively unstable clusters, the wall crossing is controlled by a 4D Dirac operator coupled to an abelian monopole that becomes non-Fredholm on the wall of marginal stability
We derive a formula to enumerate the number of plane-wave normalizable zero-energy solutions of the Dirac equation in the presence of a spherically symmetric monopole
Summary
We derive a formula to enumerate the number of plane-wave normalizable zero-energy solutions of the Dirac equation in the presence of a spherically symmetric monopole. A spherically symmetric monopole is a special class of monopole that is rotationally invariant under K = −ir × ∇ + T where {Ti}3i=1 ∈ su(N ) (which satisfy [Ti, Tj] = i ijkTk) generate a SU(2)T subgroup of the gauge group SU(N ) [14] Such a gauge field configuration can be specified by an additional choice of embedding SU(2)I → SU(N ) with generators.
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