To study the properties of the interstellar medium in the prototypical merging system of the Antennae galaxies ( NGC 4038 and NGC 4039 ) , we have obtained { { } ^ { 12 } CO } ( 1–0 ) , ( 2–1 ) and ( 3–2 ) line maps , as well as a map of the 870 \mu m continuum emission . Our results are analysed in conjunction with data from X-ray to radio wavelengths . In order to distinguish between exact coincidence and merely close correspondence of emission features , we compare the morphological structure of the different emission components at the highest available angular resolution . To constrain the physical state of the molecular gas , we apply models of photon dominated regions ( PDRs ) that allow us to fit CO and [ C ii ] data , as well as other indicators of widespread PDRs in the Antennae system , particularly within the super giant molecular cloud ( SGMC ) complexes of the interaction region ( IAR ) between the two galaxies . The modeled clouds have cores with moderately high gas densities up to 4 10 ^ { 4 } \hbox { { cm } } ^ { -3 } and rather low kinetic temperatures ( \leq 25K ) . At present , all these clouds , including those near the galactic nuclei , show no signs of intense starburst activity . Thermal radio or mid-infrared emission are all observed to peak slightly offset from the molecular peaks . The total molecular gas mass of the Antennae system adds up to \sim 10 ^ { 10 } M _ { \odot } . In the vicinity of each galactic nucleus , the molecular gas mass , 1 – 2 10 ^ { 9 } M _ { \odot } , exceeds that of the Galactic centre region by a factor of almost 100 . Furthermore , the gas does not seem to deviate much from the N _ { H _ { 2 } } / I _ { CO } ratio typical of the disk of our Galaxy rather than our Galactic centre . Alternative heating mechanisms with respect to PDR heating are discussed .