On the topology of conserved currents (part i)

After a talk this week at Harvard, I got kind of stuck in a few thoughts and I thought I'd write a rough note on the topology of conserved currents. Here the topological feature is tied to the existence of a closed differential form which describes the conserved current, its conservation being equivalent to the statement of closedness. It does not mean that the associated differential form is an integral cohomology class (it can be, but not necessarily).  This mathematical concept is deeply tied to conservation laws in a way which can be made explicit. 

Suppose we have a continuity equation of the form 

$$\nabla_{\mu}j^{\mu} =\frac{\partial \rho}{\partial t} +\nabla_{\frac{\partial}{\partial x^i}}j^i=0$$

on a $(n+1)$-manifold $\mathbb{R}\times M$ and set $M_t= \{t\}\times M$. We use greek indices to denote coordinates $x^{\mu}$, $\mu=0,\dots, n$, on $\mathbb{R}\times M$ with $x^0=t$, also understood to be the time variable, and $x^{i}$ denote local coordinates on $M$.  Take a Riemannian metric of the form  $dt^2+g_t$, where $g_t$ is a metric in $M$.  Above, $\rho$ is a scalar on $\mathbb{R}\times M$ and $j^i \frac{\partial}{\partial x^i}$ is a $t$ dependent vector field on $M$. $\nabla$ is the Levi-Civita connection. We can define a currrent $n$-form

$$J= \frac{1}{n!}j^{\mu}\varepsilon_{\mu \nu_{1}\dots \nu_{n}}\sqrt{g}dx^{\nu_1}\wedge \dots \wedge dx^{\nu_{n}}$$

Observe then that

$$dJ= \nabla_{\mu} j^{\mu} \sqrt{g} dx^0\wedge \dots \wedge dx^n=0$$

In other words, conservation of $j^\mu$ is equivalent to $J$ being closed. $J$ being closed means that $\int_{M_t} J$ is constant. This is a consequence of Stokes' theorem. To see this take the manifold with boundary $M\times [0,t]$. Then

$$\int_{M\times [0,t]}dJ= 0=\int_{M\times\{t\}}J -\int_{M\times\{0\}}J$$

and hence the charge $Q(M_{s})=\int_{M_s}J$ with $s=0,t$ satisfies

$$Q(M_0)=\int_{M_0} \rho \sqrt{g}dx^1\wedge \dots \wedge dx^n=\int_{M_t} \rho \sqrt{g}dx^1\wedge \dots \wedge dx^n=Q(M_t),$$

i.e., it is time independent.