# Poincaré duality for currents and non-closed forms

In page 8 of Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology by Szabo, the author claims that Poincaré duality holds for non-closed forms as long as the other form (pairing with the former one) is distributional, i.e., it's a de Rham current.

Let me be more precise. Let $X$ be a manifold of dimension $n$. Given a de Rham current of compact support $j_e \in H^m_{dR, c} (X)$ that is exact in $H^m_{dR} (X)$ (I don't think this exactness part is relevant, anyway …), the author claims that there exists a homology class $[W_e] \in H_{m - n} (X, \mathbb{R})$ such that $$\int_X a \wedge j_e = \int_{W_e} a$$ for every $a \in \Omega^{n - m} (X)$.

This fact is indeed need in the paper I mentioned since in page 6 he uses this property (without mentioning it previously) to deduce Dirac charge quantization condition from $\exp (-\int_X A\wedge j_e) = \exp (-\int_{W_e} A)$ for possibly non-closed $A$.

I would like a proof or a reference for this fact about currents (mentioned above)

A current is a functional, so it must eat differential forms and spew out numbers. This integral $\int_X a\wedge j_e$ is nothing but notation, to represent $\int_{W_e} a$ (here we implicitly pull-back the form $a$ to $W_e$ and use either compactly-supported forms or characteristic functions). See Chapter III of de Rham's book. Now when $j_e$ is a form itself, then the wedge product (and integral) are defined as usual.
• Thanks for your answer. Apparently you're referring to Poincaré duality of currents as stated in mathoverflow.net/questions/201580/… . The problem is that this only holds for closed $a$, i.e., for a cohomology class. My question concerns the non-closed case, which appears in the paper mentioned above whenever the base space in non-contractible. – user40276 May 26 '16 at 2:46