Adjoint of the Hodge de Rham star operator under the integral pairing

Given a Riemannian manifold $$(M, g)$$ of dimension $$n$$, the Hodge star operator $$\star: \Omega^k(M) \to \Omega^{n-k}(M)$$ is defined. What is the (formal) adjoint of $$\star$$ under the integration pairing? That is, I'm looking for an operator $$\sharp$$ that assigns to a $$(n-k)$$-dimensional submanifold $$S$$ a $$k$$-dimensional (singular?) submanifold $$\sharp S$$ in such a way that $$\int_S \star \, \alpha = \int_{\sharp S} \alpha$$ holds for all $$\alpha \in \Omega^k(M)$$.

• https://en.wikipedia.org/wiki/Current_(mathematics) – abx Dec 26 '18 at 13:49
• The answer is built-in your question. – Liviu Nicolaescu Dec 26 '18 at 14:32
• Sure, I can always view the last equality as a defining equation for a current. My question is rather if this current has a geometric interpretation or can be constructed by other means. My inspiration came from the fact that the exterior differential has the boundary operator as the adjoint and I was hoping for something comparable for the Hodge operator, perhaps using some kind of flow constructed with the help of the Riemannian metric. – Tobias Diez Dec 26 '18 at 23:13