# Magnetic monopoles in 2D

Question: Is it possible to have a gapless theory of 2+1 dimensional electromagnetism with magnetic monopoles but no electric monopoles? If such a theory exists, does a Lagrangian description exist?

Background: In 3+1 dimensions, the Maxwell Lagrangian is $$\mathcal L_\text{Maxwell} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu},$$ yielding the equations of motion equations are $$\partial_\mu F^{\mu\nu}=0$$, where $$F_{\mu\nu} = \partial_\mu A_\nu-\partial_\nu A_\mu$$. We also have the Biancci identity $$\partial_\mu \star F^{\mu\nu}=0$$. Since there are no electric monopoles, if we want to couple this theory to magnetic monopoles, we can use the dual photon field $$\tilde A_\mu$$ such that $$\star F_{\mu\nu} = \partial_\mu \tilde A_\nu -\partial_\nu \tilde A_\mu$$. Then the Lagrangian with magnetic monopoles is $$\mathcal L_\text{Dual} = -\frac{1}{4} (d\tilde A)^2 + \tilde A_\mu J^\mu_\text{mag},$$ where $$J^\mu_\text{mag}$$ is the magnetic charge current.

Now I would like to do something similar in 2+1 dimensions. In this case the equations of motion of the free theory are $$\partial_\mu F^{\mu\nu} = 0$$ as before, but the Bianchi identity is now $$\partial_\mu \star F^\mu = 0$$. As a result the dual theory consists of a scalar $$\phi$$ such that $$\star F_\mu = \partial_\mu \phi$$. If I wanted to formulate a Lagrangian in the dual picture coupled to magnetically charged matter I would naïvely write $$\mathcal L_\text{Dual}^{2+1} = -\frac{1}{2} (d\phi)^2 + \phi J_\text{mag}.$$ Although the free dual action is invariant under the shift symmetry $$\phi\to\phi+c$$, the interaction term is not. As a result, I would expect RG flow to produce a non-zero mass term in the IR, thereby giving a gap to the theory.

Are some of my assumptions wrong or is it really the case that including magnetic monopoles in 2+1 dimensional electromagnetism gives the photon a gap?

Taking the dual of 2+1-dimensional electromagnetism and adding a source like in the question does not add magnetic monopoles or "magnetically charged matter" in the usual sense. In general d+1 dimensions, the magnetically charged objects will be $$d-3$$-dimensional objects, which in this case is...-1?
The scalar "current" $$J_\text{mag}(x,y,t)$$ hence does not represent magnetic charge moving inside flatland. In case of the embedding into 3+1, it is related to the amount of magnetic charge that would need to sit "above" $$(x,y)$$ to produce the magnetic field at that point. Note that the magnetic field is also a scalar, meaning it has no directional character - the notion of 2d magnetic point charges doesn't exist because there is simply no such thing as a "radial magnetic field" around such a point charge.