  For the Antennae interacting galaxy pair, we have obtained high quality,
fully sampled $^{12}$CO J=1-0, 2-1 and 3-2 maps of the regions surrounding the
nuclei and the area of overlap between the two galaxies, with an angular
resolution of 15" or 1.5 kpc. The $^{12}$CO J=1-0, 2-1, 3-2 emission all peak
in the overlap region, which contains ~ 4 x 10^9 solar mass of molecular gas
mass if the conventional X factor is used. Line emission at $^{13}$CO J=2-1 and
3-2 is detected at selected points in the two nuclei and the overlap region.
Both the ^{12}CO/^{13}CO J=2-1 and 3-2 integrated intensity ratios are
remarkably high in the overlap region. This is the first published case in
which such high $^{12}$CO/$^{13}$CO J=2-1 and 3-2 ratios are found outside a
galactic nucleus. Detailed LVG modeling indicates that the $^{12}$CO and
^{13}CO emission originate in different spatial components. Assuming a CO-to-H2
abundance ratio of $10^{-4}$, the X factor given by the LVG model is an order
of magnitude lower than the conventional value for molecular clouds in the
Milky Way. Accordingly, we suggest the possibility that the strong CO emission
in the overlap region of the Antennae galaxies is associated with increased
radiative efficiency, possibly caused by a large velocity dispersion within the
individual molecular clouds. A comparison of the CO J=3-2 emission with the
SCUBA 850 um continuum in the Antennae galaxies shows that the CO line emission
contributes globally 46% of the 850 um continuum flux and that the ratio of
$^{12}$CO J=3-2 to SCUBA 850 um flux varies by a factor of two across the
system.
