Topological obstruction in block-spin transformations

Block-spin transformations from a fine lattice to a coarse one are shown to give rise to a one-to-one correspondence between the zero-modes of the Ginsparg-Wilson Dirac operator on the fine lattice and those on the coarse lattice. The index is then preserved under the blocking process. Such a one-to-one correspondence is violated and the block-spin transformation becomes necessarily ill-defined when the absolute value of the index is larger than 2rN, where N is the number of the sites on the coarse lattice and r is the dimension of the gauge group representation of the fermion variables.