# Electromagnetism on 3 torus

We all know Maxwell equations in 3+1 spacetime, where the "space" is $$\mathbb{R}^3$$ and time is $$\mathbb{R}$$. Moreover, it is easy to construct (using differential forms) the corresponding theory in a, say, 1+1 or 2+1 spacetime. For example, in 2+1 we have that $$F=dA$$ is a $$3 \times 3$$ antisymmetric tensor, that tells us that we have one electric field with 2 components and one magnetic field with one component. The Maxwell equations, written in terms of forms, are formally the same as the ones we write in 3+1.

What happens if, instead of modifying the number of spatial dimensions, we just wrap one (or more than one) of those dimensions? In particular, how is it possible to construct the Maxwell equations in a e.g. 1+1 spacetime that is $$S^1 \times \mathbb{R}$$, or in a 3+1 that is $$\mathbb{T}^3\times\mathbb{R}$$ ?

Are differential forms still useful? Is there any reference or book where this is discussed?

• The differential equations are the same. When solving them you apply different boundary conditions than in flat spacetime, such as imposing periodicity in the toroidal directions. – G. Smith Jul 9 '19 at 14:32