The generalized chiral Schwinger model on the two-sphere

TL;DR

This paper studies a family of theories interpolating between vector and chiral Schwinger models on the two-sphere, addressing gauge invariance issues and deriving Green functions considering topological effects and zero modes.

Contribution

It introduces a fixed background connection to define the generalized Dirac–Weyl operator globally on the two-sphere, enabling analysis of gauge invariance and Green functions in this setting.

Findings

Green functions are derived considering topological charge and zero modes.

In the decompactification limit, flat space Green functions are recovered.

Fermionic condensate in the vacuum vanishes, unlike in the vector Schwinger model.

Abstract

A family of theories which interpolate between vector and chiral Schwinger models is studied on the two--sphere $S^{2}$. The conflict between the loss of gauge invariance and global geometrical properties is solved by introducing a fixed background connection. In this way the generalized Dirac--Weyl operator can be globally defined on $S^{2}$. The generating functional of the Green functions is obtained by taking carefully into account the contribution of gauge fields with non--trivial topological charge and of the related zero--modes of the Dirac determinant. In the decompactification limit, the Green functions of the flat case are recovered; in particular the fermionic condensate in the vacuum vanishes, at variance with its behaviour in the vector Schwinger model.