We develop an analytic formalism to describe dipole radiation near the Dirac cone of a two-dimensional photonic crystal slab. In contrast to earlier work, we account for all polarization effects and derive a closed-form expression for the dyadic Green's function of the geometry. Using this analytic Green's function, we demonstrate that the dipolar interaction mediated by the slab exhibits winding phases, which are key ingredients for engineering topological systems for quantum emitters, as discussed in detail in J. Perczel et al., (2018), arXiv:1810.12299. As an example, we study the coherent atomic interactions mediated by the Dirac cone, which were recently shown to be unusually long-range with no exponential attenuation. These results pave the way for further, rigorous analysis of emitters interacting in photonic crystals via photonic Dirac cones.