Schwinger terms, gerbes, and operator residues

These notes explain recent developments concerning chiral anomalies and hamiltonian quantization, their relation to the theory of gerbes, and extensions of generalized loop algebras using the residue calculus of pseudodifferential operators. The renormalization of the Dirac field, leading to Schwinger terms in equal time commutation relations, is treated in a mathematically rigorous manner. The same renormalization can be used to prove the existence of S-operator in background field problems.