We propose a $mathbb{U}(1) times mathbb{Z}_2$ effective gauge theory for vortices in a $p_x+ip_y$ superfluid in two dimensions. The combined gauge transformation binds $mathbb{U}(1)$ and $mathbb{Z}_2$ defects so that the total transformation remains single-valued and manifestly preserves the the particle-hole symmetry of the action. The $mathbb{Z}_2$ gauge field introduces a complete Chern-Simons term in addition to a partial one associated with the $mathbb{U}(1)$ gauge field. The theory reproduces the known physics of vortex dynamics such as a Magnus force proportional to the superfluid density. More importantly, it predicts a universal Abelian phase, $exp(ipi/8)$, upon the exchange of two vortices. This phase is modified by non-universal corrections due to the partial Chern-Simon term, which are nevertheless screened in a charged superfluid at distances that are larger than the penetration depth.
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