Question:
How do we couple $U(1)$ electric (E) and magnetic (M) sources simultaneously in the classical differential geometry language, in a $U(1)$ gauge theory based on $U(1)$ gauge bundle and its $U(1)$ gauge connection (1-form) in a differential form and in a Lagrangian formalism?
Maxwell equations in electric $\vec{E}$ and magnetic $\vec{B}$ field
There is no problem to simultaneously introduce electric and magnetic sources in terms of a spacetime 4-vector $$ J_{e}^\mu=(\rho_e, \vec {J_e}), $$ $$ J_{m}^\mu=(\rho_m, \vec {J_m}). $$ in Maxwell equation in electric and magnetic field. (We will set $\epsilon_0$ and $\mu_0$ as 1) This looks like: $$\left\{\begin{align} \vec\nabla\cdot\vec{E}&=~\rho_e,\\ \vec\nabla\times\vec B~&=~\vec {J_e}+\frac{\partial\vec E}{\partial t},\\ \vec\nabla\times\vec E~&=~-\vec {J_m}+-\frac{\partial\vec B}{\partial t},\\ \vec\nabla\cdot\vec{B}~&=~\rho_m, \end{align}\right.\label{1}\tag{eq.1} $$
There is also a manifest electric and magnetic duality where $$\vec E \to - \vec B.$$ $$\vec B \to \vec E.$$ $$\rho_e \to - \rho_m.$$ $$\rho_m \to \rho_e.$$ This shall be understood as $S$-duality as part of the $SL(2,\mathbb{Z})$ of Maxwell equation.
Lagrangian formulation with 1-form gauge field $A$ and 1-form dual gauge field $V$
However, we know that the field strength curvature (2-form) is not the complete story for Maxwell $U(1)$ gauge theory. We need to introduce gauge field $A$ as (1-form) gauge field. $$ F=d A, \quad F_{\mu\nu} = \partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu} = \begin{bmatrix} 0 & E_x/c & E_y/c & E_z/c \\ -E_x/c & 0 & -B_z & B_y \\ -E_y/c & B_z & 0 & -B_x \\ -E_z/c & -B_y & B_x & 0 \end{bmatrix}. $$ Without the magnetic monopole/source, we have the action: $$ \int d A \wedge \star dA + A \wedge \star J_e, \label{2e} \tag{eq.2e} $$ and variational principle gives the extremal solution as the equation of motion $$ d F = d^2 A =0. $$ $$ \star d \star F = \star d \star dA = J_e.\label{3e} \tag{eq.3e} $$ (may up to plus or minus sign).
Now if we like to introduce a magnetic source, we need to coupled the magnetic source $J_m$ to the dual gauge field $V$ such that $$ F=dA=\star dV=\star F_m, $$ but only with $J_m$ without $J_e$, we have the action $$ \int d V \wedge \star dV + V \wedge \star J_m.\label{2m} \tag{eq.2m} $$ The variational principle gives the extremal solution as the equation of motion $$ d F_m = d^2 V =0. $$ $$ \star d \star F_m =\star d \star dV = J_m.\label{3m} \tag{eq.3m} $$ (may up to plus or minus sign).
Problems
Now there is a problem to introduce $J_e$ and $J_m$ simultaneously in the gauge field action with $A$ or $V$ gauge field. Naively, I can try: $$ \int \frac{1}{2} d A \wedge \star dA + \frac{1}{2} d V \wedge \star dV + A \wedge \star J_e + V \wedge \star J_m + \lambda (dA - \star dV), \label{2em} \tag{eq.2em} $$ such that we have both electric and magnetic gauge field $A$ and $V$, but we impose the Lagrange multiplier $\lambda$ to constrain that $(dA - \star dV)=0$. The hope is that variational principle gives the extremal solution as the equation of motion with both $$ \star d \star F_m = \star d \star d V = J_m \text{ but conflict with } d \star d V \overset{?}{=} d F \overset{?}{=} d^2 A \overset{?}{=}0. $$ $$ \star d \star F =\star d \star dA = J_e. \text{ but conflict with } d \star d A \overset{?}{=} d F_m \overset{?}{=} d^2 V \overset{?}{=}0.\label{3em} \tag{eq.3em} $$ (may up to plus or minus sign).
Hints to solution(s):
- The above formulation \eqref{2em} and \eqref{3em} do not lead to a satisfactory consistent answer.
Possible loopholes to be fixed:
For the classical version of the story, the $A$ and $V$ gauge fields are dual variables. So the $A$ and $V$ cannot be simultaneously formulated as local variables (even at the classical level?). Is there a way to get around with this by taking care of the singularities of E and M sources? in the classical differential geometry?
It is clear that the quantum version of the story for $A$ and $V$ gauge fields as dual variables are similar to that of the order parameter versus the dual order parameter for 2D Ising model. So the analogous question is that can we could the Wilson and 't Hooft gauge fields ($A$ and $V$) and their line operators to E and M sources simultaneously in one formulation? How? (At least \eqref{2em} seems to fail.)
In short, there is an issue of $$ \text{ local vs. nonlocal} $$ $$ \text{ order vs. disorder } $$
