Formulation of chiral gauge theories

Abstract

We present a formulation of chiral gauge theories, which admits more general spectra of Dirac operators and reveals considerably more possibilities for the structure of the chiral projections. Our two forms of correlation functions both also apply in the presence of zero modes and for any value of the index. The decomposition of the total set of pairs of bases into equivalence classes is carefully analyzed. Transformation properties are derived. CHIRAL PROJECTIONSStarting from the basic structure of previous approaches to chiral gauge theories with Imλ j = 0 and Im λ k > 0 and whereAssociating j = 0 to zero modes the index of D is given by I = N + 0 − N − 0 . In contrast to the Dirac operators considered previously those in (1) are no longer restricted to one real eigenvalue in addition to zero and also admit more general complex ones. They have nevertheless appropriate realizations which also allow numerical evaluation [4].For the chiral projections P − andP + the fundamental relation is required. Then because of [P − , DD † ] = [P + , DD † ] = 0 we obtain the decompositionin which the projections P R k andP R k are given bywhereFor the other projections, withN − N = I for N = TrP + and N = Tr P − , we getand have for j = 0 the two possibilitiesWith these relations for the chiral projections we see that, introducing Tr 1l = 2d, we havēfor the two choices in (9), respectively, and thatholds, where ± refers to such two choices.