Adiabatic connection and Schwinger term for two-dimensional Majorana-Weyl fermions in static background geometries

Abstract

We investigate the relation between Schwinger term and adiabatic (=Berry) curvature in the case of the conformal field theory of Majorana-Weyl fermions in 1 + 1 dimensions. The Schwinger term—known as diffeomorphism anomaly—appears in the commutation relations of the light cone components of the energy-momentum tensor. It will be interpreted as the pullback of the curvature of the Pfaffian (=vacuum) line bundle over the space of complex structures. What is new in our analysis is the appearance of nontrivial adiabatic holonomy in one-particle picture which comes from a left invariant connection on the twofold covering of the diffeomorphism group of the circle. This gives rise to an additional term in the anomalous commutator of the parallel transport (“Gauss law”) operators which act on the space of smooth sections in a Fock bundle. As a consequence, many-particle states pick up additional phase besides the phase of the vacuum.