Contents

cohomology

# Contents

## Idea

The equivariant version of de Rham cohomology.

## Properties

### Models

Various different dg-algebras are used to model equivariant de Rham cohomology, known as

#### The Weil model

reviews include (Atiyah-Bott 84, Kalkman 93, section 2.1)

#### The Cartan model

reviews include (Mathai-Quillen 86, Kalkman 93, section 2.2)

Let $X$ be a smooth manifold, $G$ a Lie group, and $\rho : X \times G \to X$ a smooth action of $G$ on $X$.

Write

$(\Omega^\bullet(G, \mathfrak{h}^\ast[1])^G \hookrightarrow \Omega(G, \mathfrak{h}^\ast[1])$

for the $G$-invariant differential forms on $G$ with coefficients in the linear dual of the Lie algebra $\mathfrak{h}$ of $H$, shifted up in degree. So for $\{F^a\} \subset \{t^a\}$ a dual basis of $\mathfrak{h}$ inside a dual basis for $\mathfrak{g}$, a general element of this space in degree $2 p + q$ is of the form

$\omega = F^{a_1} \wedge \cdots F^{a_p} \wedge \omega_{a_1,\cdots ,a_q} \,,$

where $\omega_{\cdots}$ are differential q-forms, such that for each $t_a \in \mathfrak{g}$ the Lie derivative of these forms satisfies

$\mathcal{L}_{t_a} \omega_{a_1, a_2 \cdots , a_p} = C_{a a_1}{}^b \omega_{b , a_2, \cdots , a_p} + C_{a a_2}^{}^b \omega_{a_1 , b, \cdots , a_p} + \cdots \,,$

where $\{C_{a b}{}^b\}$ are the structure constants of $\mathfrak{g}$, hence such that $[t_a, t_b] = C_{a b}{}^c t_c$.

Equip this graded vector space $\Omega^\bullet(G, \mathfrak{h}^\ast[1])^G$ with a differential $d_{CE(\mathfrak{g}//\mathfrak{h})}$ by

$d_{CE(\mathfrak{g}//\mathfrak{h})} \colon \omega \mapsto d_{dR}\omega - F^a \iota_{t_a} \omega$

(e.g. Kalkman 93 (1.15)).

The resulting dg-algebra $(\Omega^\bullet(G,\mathfrak{h}^\ast[1])^G, d_{CE}(\mathfrak{g}//\mathfrak{h}))$ is called the Cartan model of $H$-equivariant de Rham cohomology of $G$.

Observe that in the special case that $X = G$ equipped with its canonical right action of $H \coloneqq G$ on itself, this construction reduces to that of the Weil algebra $W(\mathfrak{g})$, whose cohomology is trivial (is concentrated in degree 0, where it is $\mathbb{R}$).

(…)

## References

The Weil model is discussed for instance in

A good account of the Cartan model is in

A review of the Weil model and the Cartan model and the introduction of the “BRST model” (Kalkman model) is in

• Jaap Kalkman, BRST model applied to symplectic geometry, Ph.D. Thesis, Utrecht, 1993 (arXiv:hep-th/9308132 (original ArXiv pdf broken)), published versions at (projectEuclid)

Discussion in the broader context of equivariant differential cohomology is in

• Andreas Kübel, Andreas Thom, Equivariant Differential Cohomology, Transactions of the American Mathematical Society (2018) (arXiv:1510.06392)

Discussion in the context of the gauged WZW model includes

Last revised on October 24, 2018 at 12:30:31. See the history of this page for a list of all contributions to it.