We present multi-frequency radio observations of the 2010 nova event in the symbiotic binary V407 Cygni, obtained with the Karl G. Jansky Very Large Array and spanning 1-45 GHz and 17-770 days following discovery. This nova---the first ever detected in gamma rays---shows a radio light curve dominated by the wind of the Mira giant companion, rather than the nova ejecta themselves. The radio luminosity grew as the wind became increasingly ionized by the nova outburst, and faded as the wind was violently heated from within by the nova shock. This study marks the first time that this physical mechanism has been shown to dominate the radio light curve of an astrophysical transient. We do not observe a thermal signature from the nova ejecta or synchrotron emission from the shock, due to the fact that these components were hidden behind the absorbing screen of the Mira wind. We estimate a mass loss rate for the Mira wind of Mdot_w ~ 10^-6 M_sun/yr. We also present the only radio detection of V407 Cyg before the 2010 nova, gleaned from unpublished 1993 archival VLA data, which shows that the radio luminosity of the Mira wind varies by a factor of >~20 even in quiescence. Although V407 Cyg likely hosts a massive accreting white dwarf, making it a candidate progenitor system for a Type Ia supernova, the dense and radially continuous circumbinary material surrounding V407 Cyg is inconsistent with observational constraints on the environments of most Type Ia supernovae.
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