  Using SU(3) chiral perturbation theory we calculate the density-dependent
complex mean field $U_\Sigma(k_f)+ i W_\Sigma(k_f)$ of a $\Sigma$-hyperon in
isospin-symmetric nuclear matter. The leading long-range $\Sigma N
$-interaction arises from one-kaon exchange and from two-pion exchange with a
$\Sigma$- or a $\Lambda$-hyperon in the intermediate state. We find from the
$\Sigma N\to \Lambda N$ conversion process at nuclear matter saturation density
$\rho_0 = 0.16 $fm$^{-3}$ an imaginary single-particle potential of
$W_\Sigma(k_{f0}) =-21.5$ MeV, in fair agreement with existing empirical
determinations. The genuine long-range contributions from iterated (second
order) one-pion exchange with an intermediate $\Sigma$- or $\Lambda$-hyperon
sum up to a moderately repulsive real single-particle potential of
$U_\Sigma(k_{f0})= 59$ MeV. Recently measured $(\pi^-,K^+$) inclusive spectra
related to $\Sigma^-$-formation in heavy nuclei give evidence for a
$\Sigma$-nucleus repulsion of similar size. Our results suggest that the net
effect of the short-range $\Sigma N$-interaction on the $\Sigma$-nuclear mean
field could be small.
