Two signs of a Schwinger term in a commutator of fermionic currents

TL;DR

This paper investigates how the sign of the Schwinger term in fermionic current algebra commutators depends on the ordering of Fourier modes, highlighting differences between fermions and bosons.

Contribution

It demonstrates that reversing the Fourier mode order in fermionic current commutators changes the Schwinger term's sign, a phenomenon not applicable to bosons.

Findings

Sign of Schwinger term depends on mode ordering for fermions

Reversing Fourier modes alters the algebra's central extension sign

Fermionic and bosonic cases differ in mode ordering implications

Abstract

We consider central extensions of two dimensional abelian current algebra and Virasoro algebra and see that the sign of the Schwinger term changes if we arrange the Fourier modes of the fermion in the commutator in the order opposite to the normal. We argue that this is permissible in the case of fermions, but not in the case of bosons.