I’m fascinated by speculative theories of physics in universes with different dimensionality of ours. Among these, a 2D universe with 1 time dimension is a popular one. However, this got me thinking.
In our 3+1D universe, electromagnetism, including light, needs all three dimensions. A current along $x$ would produce a magnetic field pointing in the $z$ direction in any point of the $xy$ plane. Light propagating along $x$ would have its components in the $y$ and $z$ directions. And so on.
This makes me wonder - is electromagnetism even possible in 2+1D? Is there light in a “flat” universe? Does a 2+1D word have a vector field that is analogous to our electromagnetism? How does that work?
As a starting point, I’m aware that in 3+1D, electromagnetism can be derived from the assumption that space is homogeneous and isotropic, and there exists a gauge symmetry. Perhaps similar axioms lead to a similar field to our light?
Note: while I wanted to keep the question focused on 2+1D to avoid the scope getting too wide, an answer that also touches on general dimensionality, or at least 4+1D, would be very interesting.
