In string theory, when dealing with branes, the following happens: We rewrite a worldvolume action $S = \int_{\Sigma_{p+1}} \omega^{(p+1)}$ of a $D_p$-brane as an integral over the whole $\mathbb{R}^{1,9}$ as $$ \int_{\mathbb{R}^{1,9}}\omega^{p+1}\wedge\delta^{(10-p-1)}(\Sigma_{p-1})\tag{1}$$ where the defining property of $\delta^{(10-p-1)}(\Sigma_{p-1})$ is precisely that this object is equal to the original woldvolume action. But what, exactly, is this "delta function p-form"? I have the following fragments of an explanation:
We might be content to just take $(1)$ as suggestive notation, just like we can take $\int \delta(x)f(x) = f(0)$ as suggestive notation for the functional $f\mapsto f(0)$. However, this is unsatisfactory: This form actually turns out to be a charge density, and further more, at some points, one wants to consider things like the exterior derivative of the delta function p-form, and for this, one would need a proper theory of what this object actually is (just like one needs the theory of distributions to define the derivatives of the delta function). Likewise, something like the wedge of this thing with itself needs to be given meaning for $p< 6$ (something that horribly fails for the usual $\delta(x)^2$, I might note!).
It is common for the people that use this object to say that it's the "Poincaré dual" of $\Sigma_{p+1}$. This has several problems: First, $\mathbb{R}^{1,9}$ is not compact, and neither is $\Sigma_{p+1}$, so we cannot apply this duality directly. Even if we could (e.g. by compactifying), the duality is between (co)homology classes, not cycles and cocycles. There is an explicit chain isomorphism (the cap product with the fundamental class) for the singular (co)homological case, but if we want to get a p-form, we would need a dualizing map that sends a (smooth?) $p$-cycle to a smooth $n-p$-differential form. I am unable to find an explicit description of such a map. However, on a compact manifold, we could simply choose the cycle $\Sigma_{p+1}$ and pick any form from its dual cohomology class as $\delta^{(10-p-1)}(\Sigma_{p+1})$. Then the physical theory would be forced to enjoy yet another higher gauge symmetry $\delta^{(10-p-1)}(\Sigma_{p+1})\mapsto \delta^{(10-p-1)}(\Sigma_{p+1}) + \mathrm{d}\Lambda^{(10-p-2)}$, which seems a bit bad because the delta function form appears as the "electric/magnetic charge density" of the brane and a charge density should not be gauge variant. Is that actually the case?
There is a "Poincaré duality" for non-compact manifolds that says that the map that $\Omega^p(X) \to \Omega_c^{n-p}(X)^\ast, \omega\mapsto (\eta\mapsto\int_X \omega\wedge \eta)$ induces on $H^p(X)\to H^{n-p}_c(X)^\ast$ is an isomorphism. (Continuous elements in) $\Omega_c^{n-p}(X)^\ast$ would be deRham currents on X, the proper generalization of distributions to manifolds. So if we had a way to associate a $p$-form to a smooth $p$-cycle, this would give the desired duality. However, although we have that $H_p(X,\mathbb{R})\cong H^p(X,\mathbb{R})$ because the latter is the dual of the former and they are finite-dimensional vector spaces, this isomorphism is not natural, giving us no unique way to identify classes of cycles with classes of cocycles.
Does anyone know how one tells the story of how to construct $\delta^{(10-p-1)}(\Sigma_{p+1})$ coherently?
This unanswered math.SE question seems closely related.
