We analyze the general solutions of the Ginsparg-Wilson relation for lattice Dirac operators and formulate a necessary condition for such operators to have a nonzero index in the topologically nontrivial background gauge fields. ͓S0556-2821͑99͒03405-0͔Recently there have been very interesting developments in the theoretical understanding of chiral symmetry on the lattice. The idea stems from the Ginsparg-Wilson ͑GW͒ relation ͓1͔ which was derived in 1981 as the remnant of chiral symmetry on the lattice after blocking a chirally symmetric theory with a chirality breaking local renormalization group transformation. The original GW relation iswhere D is lattice Dirac operator, R is a nonsingular Hermitian operator which is local in position space and trivial in Dirac space, and a is the lattice spacing which reminds us that D becomes chirally symmetric in the continuum limit a→0. According to the Nielsen-Ninomiya theorem ͓2͔ the chiral symmetry of a local Dirac operator defined on the regular lattice must be broken in order to avoid species doubling. The main advantage of the GW relation is that it introduces chiral symmetry breaking of D in the mildest way ͓1͔. Although it does not ensure the absence of the species doubling, it does incorporate two remarkable properties. The first is that the action Aϭ D has an exact symmetrywhere is a global parameter, which was discovered by Lüscher ͓3͔. The second is that any operator D satisfying the GW relation possesses a well defined integer index on a finite lattice ͓4,3͔where the left-hand side stands for the fermionic average of ␥ 5 calculated with the infinitesimal mass ⑀ added to the operator D, and n ϩ (n Ϫ ) are the number of the zero modes of D with positive ͑negative͒ chiralities. This is in contrast to the Wilson-Dirac operator for which the left-hand side ͑LHS͒ generally is not an integer on a finite lattice.It is essentially due to these two properties that such formulations of lattice QCD can possess the attractive features pointed out in Refs. ͓3-6͔. However, only the GW relation itself is not sufficient to guarantee that any D satisfying Eq. ͑1͒ must possess exact zero modes with definite chiralities, and reproduce the Atiyah-Singer index theorem on the lattice. In this paper, we analyze general solutions of GW relation and formulate a necessary condition for them to have nonzero indices in topologically nontrivial background gauge fields. We limit our consideration to the operators D satisfying the Hermiticity propertyFirst, we consider the case of nonsingular D which is relevant to topologically trivial gauge field background, except possibly some ''exceptional'' configurations. Then Eq. ͑1͒ is equivalent to the following equation linear in D Ϫ1 :and its general solution can be written in the formwhere D c is the chirally symmetric lattice Dirac operator, i.e.,Thus in the nonsingular case the problem of constructing explicit solutions of D reduces to finding a proper realization of the chirally symmetric operator D c . Note that by virtue of the condition ͑5...

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We analyse the structure of solutions of the Ginsparg-Wilson relation for lattice Dirac operator in topologically trivial gauge sector. We show that the properties of such solutions relating to the perturbative stability of the pole of the fermion propagator as well as to the structure of the Yukawa models based on these solutions are solely determined by the non-local chirally invariant part of these Dirac operators. Depending on the structure of this part, the pole in the fermion propagator may or may not be stable under radiative corrections. We illustrate this by explicit examples.

A set of lattice fermion actions is found which are consistent with canonical quantization of fermion systems. A new type of non-local chirally invariant action determined by the Weyl quantization is found to be inconsistent with gauge invariance. This completes the demonstration of the inconsistency of the non-local actions. The other actions are of the generalized Wilson form and may have the Kogut-Susskind-like symmetry which forbids mass terms.

We develop analytical technique for examining phase structure of Z 2 , U (1), and SU (2) lattice Higgs-Yukawa systems with radially frozen Higgs fields and chirally invariant lattice fermion actions. The method is based on variational mean field approximation. We analyse phase diagrams of such systems with different forms of lattice fermion actions and demonstrate that it crucially depends both on the symmetry group and on the form of the action. We discuss location in the diagrams of possible non-trivial fixed points relevant to continuum physics, and argue that the candidates can exist only in Z 2 system with SLAC action and U (1) systems with naive and SLAC actions.

We examine the lattice boundary formulation of chiral fermions with either an explicit Majorana mass or a Higgs-Majorana coupling introduced on one of the boundaries. We demonstrate that the low-lying spectrum of the models with an explicit Majorana mass of the order of an inverse lattice spacing is chiral at tree level. Within a mean-field approximation we show that the systems with a strong Higgs-Majorana coupling have a symmetric phase, in which a Majorana mass of the order of an inverse lattice spacing is generated without spontaneous breaking of the gauge symmetry. We argue, however, that the models within such a phase have a chiral spectrum only in terms of the fermions that are singlets under the gauge group. The application of such systems to nonperturbative formulations of supersymmetric and chiral gauge theories is briefly discussed.

We define Weyl fermions on a finite lattice in such a way that in the path integral the action is gauge invariant but the functional measure is not. Two variants of such a formulation are tested in perturbative calculation of the fermion determinant in chiral Schwinger model. We find that one of these variants ensures restoring the gauge invariance of the nonanomalous part of the determinant in the continuum limit. A 'perfect' perturbative regularization of the chiral fermions is briefly discussed.

An ansatz for the fermion vacuum functional on a lattice is proposed. It is proved to reproduce correct continuum limit for convergent diagrams of any finite order in smooth external fields, as well as consistent chiral anomalies, and ensures gauge invariance of the absolute value of the functional at any lattice spacing. The ansatz corresponds to a certain non-local fermion action having global chiral invariance. Problems caused by non-smooth gauge fields are discussed.Comment: 10 pages, latex, 2 postscript figures; major improvements, in particular, the fermion action generating the ansatz is presented; to be published in Phys. Lett.