Boundary currents and hamiltonian quantization of fermions in background fields

TL;DR

This paper explores the algebraic structure of fermions coupled to gauge fields on manifolds with boundaries, revealing a holographic relationship where boundary effects encode bulk properties, extending previous index theory methods.

Contribution

It demonstrates that fermionic current algebra with boundaries can be quantized similarly to bosonic Chern-Simons theories, establishing a holographic correspondence.

Findings

Current algebra with boundaries matches bosonic Chern-Simons quantization

Schwinger terms are localized on the boundary

Methods extend index theory to boundary cases

Abstract

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 'holographic', the Schwinger terms reside on the boundary.