Adiabatic connection and Schwinger term for two-dimensional Majorana-Weyl fermions in static background geometries (Q1805289)

This work is related to previous work by \textit{M. N. Sanielevici} and \textit{G. W. Semenoff} [Quantum geometry of a conformal field theory, Phys. Letters B 198, No. 2, 209-214 (1987)]. Both papers consider the theory of free massless Majorana-Weyl fermion fields living on the infinite sheeted covering space \(M\cong \mathbb{R}\times S^1\) of compactified Minkowski space in \(1+1\) dimensions and having static background geometries. The aim in both papers is to understand the diffeomorphism anomaly (Schwinger term) as curvature of the vacuum line bundle. The present author, however, applies completely different and mathematically rigorous methods to the same problem. Unlike Sanielevici and Semenoff the present author does not require the fermionic field operator to be covariantly constant. This leads to a Berry phase in the extended one particle picture whose origin can be traced to a connection on the symmetry group of the model. By concentrating on one chiral component, the author identifies this group with \(\text{Diff}_+(S^1)^\sim\), i.e., the two fold covering group \(\text{Diff}_+(S^1)\) of orientation preserving diffeomorphisms of the circle, and he shows that the connection \(\vartheta\) responsible for the Berry phase has the form \(\vartheta:= \langle \Theta_{\text{CM}}\rangle\), where \(\Theta_{\text{CM}}\) is the Maurer-Cartan form on \(\text{Diff}_+(S^1)^\sim\) and \(\langle\;\rangle\) signifies averaging over \(S^1\).