2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/51290We study the geometry of determinant line bundles associated to Dirac operators on compact odd dimensional manifolds. Physically, these arise as (local) vacuum line bundles in quantum gauge theory. We give a simplified derivation of the commutator anomaly formula using a construction based on noncyclic trace extensions and associated multiplicative renormalized determinants.clarifications and improvements in the text, added references; results unchangedHigh Energy Physics - TheoryDifferential Geometrytext