Take a spacetime $M$, with some $k$-manifold embedding
$$X : \Sigma \to M$$
The image of $X$ represents some extended object (a $k$-brane as the string theory people say). If we only care about the dynamic of $X$, we can simply write its action using, say, the Nambu-Goto action :
$$S[X ; U] = \int_{X^{-1}(U)} d\mu[X_*g]$$
which is the volume of the induced metric (ie the pushforward of $g$ by $X$), or in the usual coordinates,
$$S[X ; U] = \int_U \sqrt{\det(g_{\mu\nu}(X(\sigma)) \partial_a X^\mu(\sigma) \partial_b X^\nu(\sigma) )} d^k\sigma$$
But if we want to consider the metric tensor as being itself dynamic (for instance if we're interested in the Nambu-Goto action to treat idealized topological defects, point particles, thin-shell spacetimes, etc), or if we want to couple it to another more traditional field (ie a charged point particle for instance), we generally want everything to be written in terms of an integral over the target space. But how to do this?
The various ways I tried to achieve this are :
The naive way
Simply add the two actions together :
\begin{equation} S[X,g] = \int_M R_g d\mu[g] + \int_\Sigma d\mu[X_* g] \end{equation}
Pros : Somewhat accurate, up to issues that I'll get into later regarding the class of $g$
Cons : Not useful to actually get the Euler-Lagrange equation
The physicist way
When done in physics papers, it's usually represented as
\begin{equation} S[X, g ; U] = \int_U R_g + \left[\int_{X^{-1}(U)} \delta(x - X(\sigma)) \sqrt{\det(g_{\mu\nu}(x) \partial_a X^\mu(\sigma) \partial_b X^\nu(\sigma) )} d^k\sigma\right] d^n x \end{equation}
This lets people use the variation of the metric tensor directly, and gives out the appropriate stress-energy tensor $T \approx m\delta(x(\tau))$ for point particles.
Pros : Good enough for calculations
Cons : Not terribly rigorous regarding the use of distributions
De Rham currents
The common way to deal with distributions on manifolds is the use of currents, which are linear functionals on $k$-forms with compact support. If we take the measure form $\omega = \sqrt{-g} \bigwedge dx_i$ and the integration current $[U]$. The Nambu-Goto action should be, I believe
\begin{equation} S[X,g; U] = \langle X_* [U], \omega[g] \rangle \end{equation}
with $X_* [U]$ the pushforward of the integration over $U$ by $X$. This is indeed equal to $\langle [U], X^* \omega[g] \rangle$ via the properties of currents, which I think is the original Nambu-Goto action. As everything is linear, I think the action should be something like
\begin{equation} S[X,g; U] = \langle [U], R_g \omega[g] + X^* \omega[g] \rangle \end{equation}
Pros : Somewhat rigorous, but see cons
Cons : The Euler-Lagrange equation of this quantity doesn't seem trivial to compute (I'm not quite sure how the fundamental lemma of variational calculus would crop up here). Also by the Geroch-Traschen theorem, $R$ can't be a smooth $0$-form (or even a distribution, depending on the brane), and so shouldn't be on this side
Generalized functions
To work with algebras of distributions, one can use a class of generalized functions (Colombeau generalized functions or asymptotic generalized functions), so that the metric, Riemann tensor and stress-energy tensor can all be expressed as generalized functions that are allowed to be singular.
Pros : All of that
Cons : It's not evident how the action should be written down for a distribution-valued action, or how the variation would work. Also, for physical reasonableness, the standard part of any measurable quantity should exist, and I don't know if that is guaranteed in any reasonable case.
Is there any method to deal with this?