  We derive an analytical gauge-invariant formula between the Polyakov loop
$L_P$ and the Dirac eigenvalues $\lambda_n$ in QCD, i.e., $L_P \propto \sum_n
\lambda_n^{N_t -1} \langle n|\hat U_4|n \rangle$, in ordinary periodic square
lattice QCD with odd-number temporal size $N_t$. Here, $|n\rangle$ denotes the
Dirac eigenstate, and $\hat U_4$ temporal link-variable operator. This formula
is a Dirac spectral representation of the Polyakov loop in terms of Dirac
eigenmodes $|n\rangle$. Because of the factor $\lambda_n^{N_t -1}$ in the Dirac
spectral sum, this formula indicates negligibly small contribution of low-lying
Dirac modes to the Polyakov loop in both confinement and deconfinement phases,
while these modes are essential for chiral symmetry breaking. Next, we find a
similar formula between the Wilson loop and Dirac modes on arbitrary square
lattices, without restriction of odd-number size. This formula suggests a small
contribution of low-lying Dirac modes to the string tension $\sigma$, or the
confining force. These findings support no crucial role of low-lying Dirac
modes for confinement, i.e., no direct one-to-one correspondence between
confinement and chiral symmetry breaking in QCD, which seems to be natural
because heavy quarks are also confined even without light quarks or the chiral
symmetry.
