# Roberge-Weiss periodicity, canonical sector and modified Polyakov-loop

## TL;DR

This paper explores the Fourier decomposition of the QCD grand-canonical partition function at finite temperature, highlighting the role of the modified Polyakov-loop in canonical sectors and its implications for understanding QCD properties.

## Contribution

It introduces the importance of the modified Polyakov-loop in the canonical ensemble and proposes a systematic method to compute the dual quark condensate at finite imaginary chemical potential.

## Key findings

- Modified Polyakov-loop helps evade the Polyakov-loop paradox.
- Systematic approach to compute dual quark condensate.
- Enhanced understanding of QCD at finite temperature.

## Abstract

To obtain deeper understanding of QCD properties at finite temperature, we consider the Fourier decomposition of the grand-canonical partition function based on the canonical ensemble method via the imaginary chemical potential. Expectation values are, then, represented by summation over each canonical sector. We point out that the modified Polyakov-loop can play an important role in the canonical ensemble; for example, the Polyakov-loop paradox which is known in the canonical ensemble method can be evaded by considering the quantity. In addition, based on the periodicity issue of the modified Polyakov-loop at finite imaginary chemical potential, we can construct the systematic way to compute the dual quark condensate which has strong unclearness in its foundation in the presence of dynamical quarks so far.