# Gribov Copy and Complex Phase of Chiral Determinant

## TL;DR

This paper investigates the complex phase of the chiral determinant in 2D U(1) gauge fields, showing how gauge fixing methods influence phase fluctuations and the uniqueness of the phase determination.

## Contribution

It demonstrates that Laplacian gauge reduces phase fluctuations and allows for a unique phase determination, improving gauge fixing procedures in lattice gauge theory.

## Key findings

- Phase fluctuates over Gribov copies in Landau gauge.
- Laplacian gauge reduces phase fluctuation and yields a unique phase.
- Using Laplacian gauge as preconditioning improves Landau gauge fixing.

## Abstract

We calculate the complex phase of chiral determinant by the vacuum overlap formula with configurations of two-dimensional U(1) gauge field fixed in Landau and Laplacian gauge. The complex phase fluctuates over the Gribov copies, which appear in the process of Landau gauge fixing and contain vortex-like singularities. In the Laplacian gauge, the fluctuation can be reduced and the phase can be determined uniquely. If it is used as a preconditioning for Landau gauge fixing, the most smooth configuration is obtained among the copies generated.