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We construct a bosonic analog of a two-dimensional topological DiracSemi-Metal (DSM). The low-energy description of the most basic 2D DSM modelconsists of two Dirac cones at positions $\pm\mathbf{k}_0$ in momentum space.The local stability of the Dirac cones is guaranteed by a composite symmetry$Z_2^{\mathcal{TI}}$, where $\mathcal{T}$ is time-reversal and $\mathcal{I}$ isinversion. This model also exhibits interesting time-reversal and inversionsymmetry breaking electromagnetic responses. In this work we construct abosonic version by replacing each Dirac cone with a copy of the $O(4)$Nonlinear Sigma Model (NLSM) with topological theta term and theta angle$\theta=\pm \pi$. One copy of this NLSM also describes the gapless surfacetermination of the 3D Bosonic Topological Insulator (BTI). We compute thetime-reversal and inversion symmetry breaking electromagnetic responses for ourmodel and show that they are twice the value one gets in the DSM case matchingwhat one might expect from, for example, a bosonic Chern insulator. We alsoinvestigate the stability of the BSM model and find that the composite$Z_2^{\mathcal{TI}}$ symmetry again plays an important role. Along the way weclarify many aspects of the surface theory of the BTI including theelectromagnetic response, the charges and statistics of vortex excitations, andthe stability to symmetry-allowed perturbations. We briefly comment on therelation between the various descriptions of the $O(4)$ NLSM with $\theta=\pi$used in this paper (a dual vortex description and a description in terms offour massless fermions) and the recently proposed dual description of the BTIsurface in terms of $2+1$ dimensional Quantum Electrodynamics with two flavorsof fermion ($N=2$ QED$_3$). In a set of four Appendixes we review some of thetools used in the paper, and also derive some of the more technical results.

A Bosonic Analog of a Topological Dirac Semi-Metal: Effective Theory, Neighboring Phases, and Wire Construction