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Let $M$ be a smooth manifold and $H$ a closed odd-degree form. Then $(\Omega^{\bullet}(M), d_H)$ defines a complex where $d_H := d + H\wedge$. The cohomology of this complex is called twisted de Rham cohomology.

I am looking for some references which explicitly calculate the twisted de Rham cohomology for some simple examples such as $\mathbb{R}^n$.

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Perhaps trivial remark: if the degree of $H$ is 1 this should be the same as equipping some line bundle over $M$ (the trivial one?) with the flat connection $d_H$ and calculating its cohomology. In particular, if $M = \mathbb R^n$ every line bundle is trivial, so we just recover the de Rham cohomology of $\mathbb R^n$ (the same is clearly true of any $M$ with trivial Picard group). – Gunnar Þór Magnússon Jan 14 '13 at 9:07
Does this answer help? – David Speyer Jan 14 '13 at 14:58

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