Virtually planar groups (that is, those groups with a finite-index subgroup admitting a planar Cayley graph) exhibit many fairly unique coarse geometric properties. Often, we find that any one of these properties completely characterises this class of groups. 

Abelian gauge theories in 2+1 dimensions are very interesting QFTs: they are strongly coupled and exhibit non-trivial dynamics. However, they are somewhat more tractable than non-Abelian theories in 3+1 dimensions. In this talk, I will first review the known properties of fermions in 2+1 dimensions and some conjectures about QED_3 with a single Dirac fermion. I will then present the recent proposal from [arXiv:2409.17913] regarding the phase diagram of QED_3 with two fermions. The findings reveal surprising (yet compelling) features: while semiclassical analysis would suggest two trivially gapped phases and a single phase transition, the actual dynamics indicate the presence of two distinct phase transitions separated by a "quantum phase." This intermediate phase exists over a finite range of parameters in the strong coupling regime and is not visible semiclassically. Moreover, these phase transitions are second-order and exhibit symmetry enhancement. The proposal is supported by several non-trivial checks and is consistent with results from numerical bootstrap, lattice simulations, and extrapolations from the large-Nf expansion.