  We investigate whether the lightest scalar mesons sigma and kappa have a
large tetraquark component, as is strongly supported by many phenomenological
studies. A search for possible light tetraquark states with J^PC=0^++ and I=0,
2, 1/2, 3/2 on the lattice is presented. We perform the two-flavor dynamical
simulation with Chirally Improved quarks and the quenched simulation with
overlap quarks, finding qualitative agreement between both results. The
spectrum is determined using the generalized eigenvalue method with a number of
tetraquark interpolators at the source and the sink, and we omit the
disconnected contractions. The time-dependence of the eigenvalues at finite
temporal extent of the lattice is explored also analytically. In all the
channels, we unavoidably find lowest scattering states pi(k)pi(-k) or
K(k)pi(-k) with back-to-back momentum k=0, 2*pi/L,... However, we find an
additional light state in the I=0 and I=1/2 channels, which may be interpreted
as the observed resonances sigma and kappa with a sizable tetraquark component.
In the exotic repulsive channels I=2 and I=3/2, where no resonance is observed,
we find no light state in addition to the scattering states.
