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In his article "Axial Anomalies and Index Theorems on Open Spaces" (https://link.springer.com/article/10.1007/BF01202525) C.Callias shows how the index of the Callias-type operator on $R^{n}$ can be used to study properties of fermions in the field of SU(2) Monopoles.

Let M be a complete Riemannian manifold and $D$ be the Dirac operator on $C^{\infty}_{c}(M,E)$ the section of the vector bundle E. Let $A \in End(E)$ a self-adjoint bundle morphism. We define a Callias-type operator by $D+iA$ such that it verifies the following

  1. The commutator $[D,A]$ is an operator of order zero and uniformly bounded.
  2. There exists a compact $K\subset M$ and a positive constant $c>0$ such that $A^{2}≥cId$ on $M\smallsetminus K$.

I would like to know how the index of the Callias operator defined above can be used to count fermions in magnetic monopoles and what is the role of the potential A. And does this operartor has other application in physics?

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