Biot-Savart law on a torus?

Question

Background: In classical electrodynamics, given the shape of a wire carrying electric current, it is possible to obtain the magnetic field $\mathbf{B}$ via the Biot-savart law. If the wire is a curve $\gamma$ parametrized as $\mathbf{y}(s)$, where $s$ is the arc-length, then

$$ \mathbf{B}(\mathbf{x}) = \beta \int_\gamma \dfrac{ d\mathbf{y}\times(\mathbf{x} - \mathbf{y}) }{|\mathbf{x} - \mathbf{y}|^3} = \beta \int_\gamma ds \dfrac{ \mathbf{y}'(s) \times(\mathbf{x} - \mathbf{y}(s)) }{|\mathbf{x} - \mathbf{y}(s)|^3} \, , $$

where $\beta $ is just a physical constant proportional to the current in the wire.

Another application of the Biot-Savart law is to find the velocity field $\mathbf{v}$ around a bent vortex line in a fluid, in the approximation of incompressible and irrotational fluid flow (i.e. $\nabla \cdot \mathbf{v} =0$ and $\nabla \times \mathbf{v} =0$ almost everywhere) and very-thin diameter of the vortex core. In fact, by demanding that the vorticity of the fluid is concentrated on the vortex core (i.e. it is distributed as a Dirac delta peaked on the vortex core),

$$ \mathbf{w}(\mathbf{x}) = \nabla \times \mathbf{v}(\mathbf{x})= c \int_\gamma ds \, \mathbf{y}'(s)\, \delta( \mathbf{x} - \mathbf{y}(s)) \, , $$

we have that the Helmholtz decomposition and the fact $\delta(\mathbf{y}-\mathbf{x} ) = -\nabla^2 \, (4 \pi |\mathbf{y}-\mathbf{x}|)^{-1}$ tell us that

$$ \mathbf{v}(\mathbf{x})= \frac{c}{4 \pi} \int_\gamma ds \dfrac{ \mathbf{y}'(s) \times(\mathbf{x} - \mathbf{y}(s)) }{|\mathbf{x} - \mathbf{y}(s)|^3}$$

Again, the constant $c$ is just a physical constant that sets the value of the circulation of the field $\mathbf{v}$ around the vortex.

Question: The above construction works in $\mathbb{R}^3$. Imagine now that the wire (or, equivalently, the irrotational vortex) is a curve in the three-dimensional torus $\mathbb{T}^3 = S^1 \times S^1 \times S^1$. How to obtain the equivalent of the Biot-Savart law?

Note on topology: We are changing the base manifold from $\mathbb{R}^3$ to $\mathbb{T}^3 $ but the local differential relations should be unchanged (i.e. the definition of the vorticity 2-form as the external derivative of the velocity 1-form, or the local form of Maxwell equations $dF = J$). The problem is that the Biot-Savart law is non-local, so it is a global problem that "feels" the topology of the manifold. Maybe in the end the question is related to how the Helmholtz decomposition works on a torus.

Important (possible answer): See the following closely related questions in Physics SE: Rotating away a constant gauge field, Periodic boundary conditions for vortex in a square lattice. There is a topological problem in considering Biot-Savart in a torus!

Answer 1

The Biot-Savart law is essentially a case of a Green's function. If we want to solve a scalar, linear, inhomogeneous PDE of the form $$ Df=g $$ Where $D$ is a linear differential operator, $f$ is the unknown function, and $g$ is a "source" function, we can split this into two steps: first, we may find a family of Green's functions $G$ satisfing $$ DG(x,y)=\delta(x-y) $$ where $\delta$ is a Dirac function, and $D$ is understood to treat $G$ as a function of the first argument only. Then, by using the linearity of $D$, we can "split up" the source $g$ into an integral of delta functions and write the solution in terms of the Green's function. $$ f(x)=\int_Mg(y)G(x,y)dy $$ The upshot is in many cases $G(x,y)$ has a very simple form due to the symmetries of the underlying space. One complication is that this problem is inherently global, and there are often tricky questions of existence and uniqueness. If it's a vector valued PDE, so we can choose a basis $e_i$ and think of $G$ as a "matrix" valued function satisfying $D(G^i_j(x,y)e_i)=\delta(x-y)e_j$.

For you're specific problem, the Biot-Savart law can in principal be generalied by obtaining a Green's function of the magnetostatic PDE $\nabla\times B=J$, $\nabla\cdot B=0$ on the torus, with an appropriate set of boundary conditions. I don't happen to know a closed form for such a Green's function, but using Fourier decomposition it should be possible to find at least a series solution.

Answer 2

I will record some considerations of which I am not 100% sure, hence the community wiki.

I agree that the Maxwell equations remain the same on the torus. (What changes are the boundary conditions, but this is not important here, I think). The Biot-Savart law is the solution to the equations of Maxwell with a filiform source term $\gamma$.

Now, identifying the torus with $[0, 1]^3$ with appropriate identification of the boundary points, we see by compactness that $\gamma$ is the sum of a finite number of filaments $\gamma_j$ that are contained in $(0, 1)^3$. For each one of those, the Biot-Savart law is exactly the same. Thus, the Biot-Savart law is the same for $\gamma$, too.