SO(8) and G2×G2 N=1 supergravities in D=6

We construct D=6, N=1 supergravity analogs of the 10D Green-Schwarz and heterotic superstring theories. We find the SO(8) theory previously know from K3 compactification from10D. We also find four inequivalent G2×G2 theories, one with symmetric matter content. It is not obvious how the G2×G2 theories can be gotten from 10D.