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Chiral Fermions on the Lattice

The overlap approach to chiral gauge theories on arbitrary $D$--dimensional lattices is studied. The doubling problem and its relation to chiral anomalies for $D=2$ and 4 is examined. In each case it is shown that the doublers can be eliminated and the well known perturbative results for chiral anomalies can be recovered. We also consider the multi--flavour case and give the general criteria for the construction of anomaly free chiral gauge theories on arbitrary lattices. We calculate the second order terms in a continuum approximation to the overlap formula in $D$ dimensions and show that they coincide with the bilinear part of the effective action of $D$--dimensional Weyl fermions coupled to a background gauge field. Finally, using the same formalism we reproduce the correct Lorentz, diffeomorphism and gauge anomalies in the coupling of a Weyl fermion to 2--dimensional gravitational and Maxwell fields.