Can nonlocal Dirac operators be topologically proper ?

By examining the analyticity of a sequence of topologically-proper lattice Dirac operators, we show that they tend to a nonlocal Dirac operator. This implies that a nonlocal lattice Dirac operator can have exact zero modes satisfying the Atiyah-Singer index theorem, in gauge backgrounds with nonzero topological charge. PACS numbers: 11.15.Ha, 11.30.Rd, 11.30.Fs It is well known that all fundamental interactions are local. Therefore, it is natural to require that any lattice ( nonperturbative ) formulation of these theories is local in the continuum limit. Although it is straightforward to formulate bosonic fields on the lattice with ultralocal operators, it is nontrivial to formulate the Dirac fermion field on the lattice, which can retain all its vital features in the continuum. In general, a decent lattice Dirac operator D is required to satisfy the following conditions : (i) D is local. ( ||D(x, y)|| ≤ c exp(−|x − y|/l) with l ∼ a; or D(x, y) = 0 for |x − y| larger than a finite distance. ) (ii) In the free fermion limit, D is free of species doublings. ( The free fermion propagator D−1(p) has only one simple pole at the origin p = 0 in the Brillouin zone. ) (iii) In the free fermion limit, D has the correct continuum behavior. ( In the limit a → 0, D(p) ∼ iγμpμ around p = 0. ) (iv) D is γ5-hermitian. ( D † = γ5Dγ5. ) (v) D satisfies the Ginsparg-Wilson relation [1]. (Dγ5 + γ5D = 2aDγ5RD, where R is a positive definite Hermitian operator which commutes with γ5.) However, there exists no proof that any lattice Dirac operator satisfying these five conditions must have exact zero modes satisfying the Atiyah-Singer index theorem [2], in background gauge fields with nonzero topological charge. Though one has no doubts that conditions (ii)-(v) are necessary requirements for any topologically proper D, it is not clear how the locality of D could play any role in the index of D. For example, consider the chirally symmetric Dirac operator