The index of lattice Dirac operators and K-theory

We mathematically show an equality between the index of a Dirac operator on a flat continuum torus and the $$\eta $$ invariant of a lattice Dirac operator known as the Wilson Dirac operator with a negative mass when the lattice spacing is sufficiently small. Unlike the standard approach, our formulation using K-theory does not require modified chiral symmetry on the lattice. We prove that a one-parameter family of continuum massive Dirac operators and the corresponding Wilson Dirac operators belong to the same equivalence class of the $$K^1$$ group at a finite lattice spacing. Their indices, which are evaluated by the spectral flow or equivalently by the $$\eta $$ invariant at a finite mass, are proved to be equal.