  In lattice QCD formalism, we derive an analytical gauge-invariant relation
between the Polyakov loop $\langle L_P \rangle$ and the Dirac eigenvalues
$\lambda_n$ in QCD, i.e., $\langle L_P \rangle \propto \sum_n \lambda_n^{N_t
-1} \langle n|\hat U_4|n \rangle$, by considering ${\rm Tr}
(\hat{U}_4\hat{\not{D}}^{N_t-1})$ on a temporally odd-number lattice, where the
temporal lattice size $N_t$ is odd. This formula is a Dirac spectral
representation of the Polyakov loop in terms of Dirac eigenmodes $|n\rangle$.
We here use an ordinary square lattice with the normal (nontwisted) periodic
boundary condition for link-variables $U_\mu(s)$ in the temporal direction.
From this relation, one can estimate each contribution of the Dirac eigenmode
to the Polyakov loop. Because of the factor $\lambda_n^{N_t -1}$ in the Dirac
spectral sum, this analytical relation generally indicates quite small
contribution of low-lying Dirac modes to the Polyakov loop in both confined and
deconfined phases, while the low-lying Dirac modes are essential for chiral
symmetry breaking. Also in lattice QCD calculations in confined and deconfined
phases, we numerically confirm the analytical relation, non-zero finiteness of
$\langle n|\hat U_4|n \rangle$ for each Dirac mode, and negligibly small
contribution of low-lying Dirac modes to the Polyakov loop, i.e., the Polyakov
loop is almost unchanged even by removing low-lying Dirac-mode contribution
from the QCD vacuum generated by lattice QCD simulations. Thus, we conclude
that low-lying Dirac modes are not essential modes for confinement, which
indicates no direct one-to-one correspondence between confinement and chiral
symmetry breaking in QCD.
