A construction of chiral fermion action

According to the necessary requirements for a chirally symmetric Dirac operator, we present a systematic construction of such operators. We formulate a criterion for the hermitian operator which enters the construction such that the doubled modes are decoupled even at finite lattice spacing. PACS numbers: 11.15.Ha, 11.30.Rd, 11.30.Fs Recently we have discussed the necessary requirements [1] for a chirally symmetric Dirac operator Dc to enter the topologically invariant transformation D = Dc(1I + RDc) −1 [2] such that the resulting Dirac operator D could reproduce the continuum physics, where R is a hermitian operator which is local in the position space and trivial in the Dirac space. These constraints for Dc are : (a) Dc agrees with γμ(∂μ + iAμ) in the classical continuum limit. (b) Dc is nonlocal. (c) Dc is free of species doubling. (d) Dc is well defined in topologically trivial background gauge field. (e) Dc has zero modes as well as simple poles ( or equivalently, V has real eigenvalue pairs±1 ) in topologically non-trivial background gauge fields. Furthermore, the zero modes of Dc satisfy the Atiyah-Singer index theorem for any prescribed smooth gauge background. In this paper, we attempt to construct such chirally symmetric Dirac operators Dc which also satisfies the hermiticity condition γ5Dcγ5 = D † c . (1) The hermiticity condition and the chiral symmetry of Dc implies that Dc is antihermitian, thus there exists one to one correspondence between Dc and a unitary operator V , Dc = (1I + V )(1I− V ), V = (Dc − 1I)(Dc + 1I). (2) where V also satisfies the hermiticity condition γ5V γ5 = V . Then the unitary operator V can be expressed in terms of a hermitian operator h, V = γ5h = (