Boundary currents and hamiltonian quantization of fermions in background fields

The current algebra generated by fermions coupled to external gauge potentials and metrics on a manifold with boundary is discussed. It is shown that the previous methods, based on index theory arguments and used in the case without boundaries, carry over to the present problem. The resulting current algebra is the same as obtained from a quantization of bosonic Chern-Simons theories on space-times with nonempty boundaries. This means that also in the fermionic setting the construction is `holographicʹ, the Schwinger terms reside on the boundary.