# nLab twisted de Rham cohomology

### Context

#### Differential cohomology

differential cohomology

## Ingredients

• cohomology

• differential geometry

• ## Connections on bundles

• connection on a bundle

• curvature

• Chern-Weil theory

• ## Higher abelian differential cohomology

• differential function complex

• differential orientation

• ordinary differential cohomology

• differential K-theory

• differential elliptic cohomology

• differential cobordism cohomology

• ## Higher nonabelian differential cohomology

• Chern-Weil theory in Smooth∞Grpd

• ∞-Chern-Simons theory

• ## Fiber integration

• higher holonomy

• fiber integration in differential cohomology

• ## Application to gauge theory

• gauge theory

• gauge field

• quantum anomaly

• #### Cohomology

cohomology

# Contents

## 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 $\Omega^{even}(X) \oplus \Omega^{odd}(X)$ equipped with the $H$-twisted de Rham differential

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

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

There is also the cohomology of the chain complex whose maps are just multiplication by $H$.

$H \wedge(-) \;\colon\; \Omega^{even/odd}(X) \to \Omega^{odd/even}(X) \,,$

This is also called H-cohomology (Cavalcanti 03, p. 19).

## Properties

###### Proposition

If the de Rham complex $(\Omega^\bullet(X),d_{dr})$ is formal, then for $H \in \Omega_{cl}^3(X)$ a closed differential 3-form, the $H$-twisted de Rham cohomology of $X$ coincides with its H-cohomology, for any closed 3-form $H$.

## References

Last revised on February 14, 2018 at 11:13:38. See the history of this page for a list of all contributions to it.