Topological Obstruction in Block-spin Transformations

TL;DR

This paper investigates how topological obstructions affect block-spin transformations in lattice gauge theories, showing that the index is preserved under certain conditions but can become ill-defined when topological complexity exceeds a specific threshold.

Contribution

It establishes a topological obstruction criterion for the well-definedness of block-spin transformations and demonstrates the preservation of the Dirac operator index under these transformations.

Findings

Index is preserved under block-spin transformations when conditions are met.

Transformation becomes ill-defined when the index exceeds 2rN.

Topological obstructions limit the applicability of block-spin transformations.

Abstract

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 operators. 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.