Magnetic monopoles in 2D

Question

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?

Answer 1

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?

What this is means is that in two spatial dimensions, there is no notion of magnetic charge living inside the world. This becomes clear if we think about this theory as a dimensional reduction of the dual of 3+1 dimensional electromagnetism: If we embed our flatland and expose it to magnetic monopoles moving freely in 3+1 dimensions, then only the monopoles not inside the plane are detectable from inside flatland, because only those monopoles produce magnetic fields that evoke a Lorentz force with a component parallel to the surface of our flatland - a monopole sitting inside flatland produces a force pointing purely perpendicular to it, and so something confined to flatland cannot notice it.

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.