in literature, I have found the following notion:
For a smooth manifold $M$, $f \in C(M)$ and a measure $\omega_0$ on M, there is a map $\vee e^{f/\hbar} \omega_0: PV^i(M):=C^\infty(M, \wedge ^i TM) \to \Omega^{n-i}(M)$. Unfortunately, nowhere it is specified, what exactly this map does (it is called contraction somewhere, but I do not think, it has anything to do with the interior derivative).
Also I do not know, what exactly $\hbar$ is, it should have something to do with physics, but is not the Planck constant. $\mathbb{C}[[\hbar]]$ is of relevance as algebras over this ring are considered, and $\hbar=0$ gives classical mechanics.
Here is a guess (assuming $f$ is smooth): Maybe the map means contraction with the $n-$form $\omega=e^{f/\hbar}\omega_0\in C^\infty(M;\Lambda^nT^*M)$. One reasonable interpretation of this would be to map $X_1\wedge \dots\wedge X_i\in C^\infty(M;\Lambda^iTM)$ to the form $\alpha \in C^\infty(M;\Lambda^{n-i}T^*M)$ defined by $$ \alpha(X_{i+1},\dots,X_{n})=\omega(X_1,\dots,X_n),\quad \text{for }X_{i+1}\wedge \dots\wedge X_n\in C^\infty(M;\Lambda^iTM), $$ i.e. we have plugged $X_1,\dots,X_i$ into those slots of $\omega$ with indices $I=\{1,\dots,i\}$. Of course any other set of indices $I$ of cardinality $i$ also gives a well defined map. But either taking the first $i$ indices or the last $i$ indices seems to be the most natural.