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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.

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It is indeed compactly-supported Poincare duality via currents, for which the standard reference is de Rham's Differentiable Manifolds. For example, the Dirac delta function is the dual of a point! Anyway, to get to a quick understanding, see the beginning of Section 7.3 of Nicolaescu's notes http://www3.nd.edu/~lnicolae/Lectures.pdf

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  • $\begingroup$ If I read that right, I run in two troubles with this construction: The first is the one I already talked about in the question - if the $\delta(\Sigma)$ is any representant of the dual cohomology class, then the charge density of the brane containing that expression is not unique, which is troubling since the flux associated to it determines angles. The second is that Nicolaescu seems to say the property $\int_M\omega\wedge\delta(\Sigma) = \int_\Sigma\omega$ only needs to hold for closed $\omega$, but I don't see that the Lagrangian form of the worldvolume action is closed in general. $\endgroup$ – ACuriousMind May 25 '16 at 19:44
  • $\begingroup$ I unfortunately don't understand the jargon you use. Anyway, $\omega$ must be closed so that it doesn't matter about the choice of $\delta(\Sigma)$ by Stokes' theorem. Otherwise, you define $\delta(\Sigma)$ to be the particular form (for a given form $\omega$) such that the integral equality holds. $\endgroup$ – Chris Gerig May 25 '16 at 21:35
  • $\begingroup$ I elaborated on the MathStackExchange post that you linked, which may further help you. $\endgroup$ – Chris Gerig May 26 '16 at 6:03
  • $\begingroup$ Well, my problem is this: One part of the derivation using this object uses $\delta^{(4)}(\Sigma_6)\wedge\delta^{(4)}(\Sigma_6)$ and then says that "in flat space" this is 0. I can't see how to make sense of that statement using currents, and I can't see how to make sense of other parts of the derivation if $\delta$ is a Poincaré dual form only defined up to cohomology class. So I'm not sure whether the string theorist simply messed up the math here, or whether they have some very specific definition in mind that they didn't bother to explicitly write down. $\endgroup$ – ACuriousMind May 26 '16 at 11:14

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