Vector and chiral gauge theories on the lattice

We combine a pair of independent Weyl fermions to compose a Dirac fermion on the four-dimensional Euclidean lattice. The obtained Dirac operator is antihermitian and does not reproduce anomaly under the usual chiral transformation. To simulate the correct chiral anomaly, we modify the chiral transformation. We also show that chiral gauge theories can be constructed nonperturbatively with exact gauge invariance. The formulation is based on a doubler-free lattice derivative, which is a simple matrix defined as a discrete Fourier transform of momentum with antiperiodic boundary conditions. Long-range fermion hopping interactions are truncated using the Lanczos factor.