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Green's function of the Vector fields on DeSitter Background

In this paper we study the propagator of a vector fields on a euclidean maximally-symmetric background in arbitrary space-time dimensions. We study two cases of interest: Massive and massless vector fields. In each case we computed the propagator of the vector fields on euclidean deSitter background, isolating the transverse and longitudinal part. In both case of massive and massless vector fields, the short distance limit of the full propagator agrees with the flat space-time propagator. In the case of massive propagator, the transverse part has a well defined massless limit, and in this limit it goes to the transverse propagator for the massless fields. The transverse propagator for antipodal point separation is nonzero and negative, but vanishes in the flat space-time limit (Ricci scalar going to zero). The longitudinal part of the massive propagator diverges as $1/m^2$, where $m$ is the mass of the field. The longitudinal part of the massless propagator is gauge dependent and in particular is proportional to the gauge parameter used in the gauge fixing condition. It vanishes in the Landau gauge. Comparison with the past literature is made.