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twisted de Rham cohomology

Idea

For $X$ a smooth manifold and $H\in {\Omega }^3(X)$ a closed differential 3-form, the $H$ twisted de Rham complex is the ${\mathbb{Z}}_2$-graded vector space ${\Omega }^{\mathrm{even}}(X)\oplus {\Omega }^{\mathrm{odd}}(X)$ equipped with the $H$-twisted de Rham differential

$$d+H\wedge (-):{\Omega }^{\mathrm{even}/\mathrm{odd}}(X)\to {\Omega }^{\mathrm{odd}/\mathrm{even}}(X),$$d + H \wedge(-) : \Omega^{even/odd}(X) \to \Omega^{odd/even}(X) \,,

Notice that this is nilpotent, due to the odd degree of $H$, such that $H\wedge H=0$, and the closure of $H$, $dH=0$.

Properties

Twisted de Rham cohomology is the recipient of the twisted Chern character in twisted differential K-theory.