Yes, there is.

## QED

The gauge group for electromagnetism is $U(1)$ in general. In Quantum Electrodynamics, electromagnetism is the curvature of the $U(1)$ bundle (similarly to how the strong interaction is the curvature of the $SU(3)$ bundle, gravity is the curvature of 4-dimensional spacetime, etc.).

In the same context, it's worth noting the covariant derivatives is general relativity are such that $\nabla_\mu-\partial_\mu$ is a relativistic analog of the Newtonian gravitational potential. This is also true in QED, and QFT in general, where to suitable natural some constants, $\nabla_\mu-\partial_\mu=ig_sA_\mu$ is the vector potential (multiplied by the coupling constant and $i$ of course) of the interaction, in this case, of electromagnetism.

## Kaluza-Klein theory

In Kaluza-Klein theory, the 5-dimensional metric tensor can be written as

\[g_{ab}= \left[ \begin{array} ag_{\mu\nu} +\phi^2A_\mu A_\nu&& \phi^2 A_\mu \\ \phi^2 A_\nu &&\phi^2 \end{array} \right]\]

The vector potentials are manifest in the metric itself, scaled by the radion.

More interestingly (for the context of this question), the electric charge is the momentum of the particle in the 5th dimension (which is compactified onto a circle), in other words, the electric charge is a measure of how much the particle revolve around on the compactified dimension.

## String theory

In string theory, in addition to both the above interpretations, the electric charge is also equivalent to (in a system of suitable natural units, if you like) the winding number of the string around a compactified dimension. It is to be noted that this is not so in the Type IIB string theory, which has no gauge group. This is what gives electrodynamics it's *U(*1) symmetry (the symmetry is however unified into a larger gauge group \(SO(32)\) or \(E_8\times E_8\) in the respective heterotic theories).

What's interesting is that this description is T-dual to the Kaluza-Klein interpretation (which also holds true in string theory, since Kaluza-Klein theory is just an effective action of classical string theory, without supersymmetry and the nuclear forces).