  Quantum magnets provide the simplest example of strongly interacting quantum
matter, yet they continue to resist a comprehensive understanding above one
spatial dimension (1D). In 1D, a key ingredient to progress is Luttinger liquid
theory which provides a unified description. Here we explore a promising
analogous framework in two dimensions, the Dirac spin liquid (DSL), which can
be constructed on several different lattices. The DSL is a version of Quantum
Electrodynamics ( QED$_3$) with four flavors of Dirac fermions coupled to
photons. Importantly, its excitations also include magnetic monopoles that
drive confinement. By calculating the complete action of symmetries on
monopoles on the square, honeycomb, triangular and kagom\`e lattices, we answer
previously open key questions. We find that the stability of the DSL is
enhanced on the triangular and kagom\`e lattices as compared to the bipartite
(square and honeycomb) lattices. We obtain the universal signatures of the DSL
on the triangular and kagom\`e lattices, including those that result from
monopole excitations, which serve as a guide to numerics and to experiments on
existing materials. Interestingly, the familiar 120 degree magnetic orders on
these lattices can be obtained from monopole proliferation. Even when unstable,
the Dirac spin liquid unifies multiple ordered states which could help organize
the plethora of phases observed in strongly correlated two-dimensional
materials.
