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equivariant de Rham cohomology in nLab

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Cohomology

cohomology

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equivariant cohomology
- equivariant homotopy theory
- Bredon cohomology
twisted cohomology
- twisted bundle
  - twisted K-theory, twisted spin structure, twisted spin^c structure
- twisted differential c-structures
  - twisted differential string structure, twisted differential fivebrane structure
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homotopy groups of a Lie groupoid

Related topics

∞-Chern-Weil theory

Examples

∞\infty-Lie groupoids

∞\infty-Lie groups

∞\infty-Lie algebroids

∞\infty-Lie algebras

Models

Idea

The equivariant cohomology-generalization of de Rham cohomology.

cohomology	Borel-equivariant cohomology
real ordinary cohomology	real equivariant ordinary cohomology
de Rham cohomology	equivariant de Rham cohomology

Properties

Throughout we consider the following setup:

Models

Given a smooth GG-manifold XX (Def. ) various dg-algebras are used to model the corresponding GG-equivariant de Rham cohomology XX, known as

the Weil model
the Cartan model
the BRST/Kalkman model

The Weil model

Let (W(𝔤),d W)\big( W(\mathfrak{g}), d_W \big) denote the Weil algebra of 𝔤\mathfrak{g}. If {t a}\{t_a\} is a linear basis for 𝔤\mathfrak{g}

(1)𝔤=span({t a}) \mathfrak{g} \;=\; span\big( \{t_a\} \big)

with induced structure constants for the Lie bracket being {f bc a} a,b,c\big\{f_{b c}^a\big\}_{a,b,c}

(2)[t a,t b]=f ab ct b∧t c [t_a, t_b] \;=\; f_{a b}^c t^b \wedge t^c

(using the Einstein summation convention throughout)

then the Weil algebra is the dgc-algebra explicitly given by generators and relations as follows:

(3)W(𝔤)≔ℝ[{t a⏟deg=1} a,{r a⏟deg=2} a]/(d Wt a =−12f bc at b∧t c+r a d Wr a =f bc at b∧r c) W(\mathfrak{g}) \;\coloneqq\; \mathbb{R}\big[ \{ \underset { deg = 1 } { \underbrace{ t^a } } \}_a, \{ \underset { deg = 2 } { \underbrace{ r^a } } \}_a \big] \Big/ \left( \begin{aligned} d_W \, t^a & = - \tfrac{1}{2}f^a_{b c} t^b \wedge t^c + r^a \\ d_W \, r^a & = f^a_{b c} t^b \wedge r^c \end{aligned} \right)

Now consider the tensor product of dgc-algebras of the de Rham algebra of XX with the Weil algebra of 𝔤\mathfrak{g}

(4)(Ω •(X)⊗W(𝔤),d dR+d W). \Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}), \, d_{dR} + d_W \Big) \,.

On this consider the following joint Cartan calculus operations: for each basis element t at_a (1) a graded derivation of degree -1 (contraction)

(5)ι a:(Ω(X)⊗W(𝔤)) •⟶(Ω(X)⊗W(𝔤)) •−1 \iota_a \;\colon\; \Big( \Omega \big( X \big) \otimes W(\mathfrak{g}) \Big)^{\bullet} \longrightarrow \Big( \Omega \big( X \big) \otimes W(\mathfrak{g}) \Big)^{\bullet - 1 }

and a graded derivation of degree 0 (generalized Lie derivative)

(6)ℒ a:(Ω(X)⊗W(𝔤)) •⟶(Ω(X)⊗W(𝔤)) • \mathcal{L}_a \;\colon\; \Big( \Omega \big( X \big) \otimes W(\mathfrak{g}) \Big)^\bullet \longrightarrow \Big( \Omega \big( X \big) \otimes W(\mathfrak{g}) \Big)^{\bullet}

defined on ω∈Ω •(X)\omega \in \Omega^\bullet(X) any differential form and t a,r at^a, r^a as in (3) as follows

(7)ι a:{ω ↦ι v aω t b ↦δ b a r b ↦0 \iota_a \;\colon\; \left\{ \begin{aligned} \omega & \mapsto \iota_{v^a} \omega \\ t^b &\mapsto \delta^a_b \\ r^b & \mapsto 0 \end{aligned} \right.

and

(8)ℒ a:{ω ↦ℒ v aω t b ↦f ac bt c r b ↦f ac br c \mathcal{L}_a \;\colon\; \left\{ \begin{aligned} \omega & \mapsto \mathcal{L}_{v^a} \omega \\ t^b &\mapsto f_{a c}^b t^c \\ r^b & \mapsto f_{a c}^b r^c \end{aligned} \right.

where

v a:X↪((e,t a),0)TG×TX≃T(G×X)⟶dρTX v^a \;\colon\; X \overset{ \big( (e,t_a), 0 \big) }{\hookrightarrow} T G \times T X \simeq T ( G \times X ) \overset{ d \rho } {\longrightarrow} T X

is the vector field on XX which is the derivative of the action ρ\rho of GG along the Lie algebra-element t a∈𝔤≃T eGt_a \in \mathfrak{g} \simeq T_e G,

and where ι v a\iota_{v^a} is ordinary contraction of vector fields into differential forms and ℒ v a=[d dR,ι−v a]\mathcal{L}_{v^a} = [d_{dR}, \iota-{v^a}] is Lie derivative of differential forms.

With this one defines the sub-chain complex of horizontal differential forms as the joint kernel of the contraction operators (5)

(9)(Ω •(X)⊗W(𝔤)) hor↪ker({ι a} a)(Ω •(X)⊗W(𝔤)) \Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \Big)_{hor} \overset{ ker\big( \{\iota_a\}_a \big) }{\hookrightarrow} \Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \Big)

(this subspace need not be preserved by the differential, but the following further subspace is)

and the further sub-dgc-algebra of basic differential forms, which are in addition in the kernel of the Lie derivatives (6)

(10)(Ω •(X)⊗W(𝔤)) basic↪ker({ℒ a} a)(Ω •(X)⊗W(𝔤)) hor↪ker({ι a} a)(Ω •(X)⊗W(𝔤)) \Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \Big)_{basic} \overset{ ker\big( \{\mathcal{L}_a\}_a \big) }{\hookrightarrow} \Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \Big)_{hor} \overset{ ker\big( \{\iota_a\}_a \big) }{\hookrightarrow} \Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \Big)

Since the differential d dR+d Wd_{dR} + d_W (4) graded-commutes with ι a\iota_a to ℒ a\mathcal{L}_a (by definition and by Cartan's magic formula) and hence graded-commutes with the Lie derivative ℒ a\mathcal{L}_a itself, it restricts to this joint kernel, thus defining a sub-dgc-algebra (just no longer semi-free, in generaL)

(11)((Ω •(X)⊗W(𝔤)) basic,d dR+d W) \Big( \Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \Big)_{basic} , d_{dR} + d_W \Big)

This dgc-algebra is called the Weil model for GG-equivariant de Rham cohomology of XX.

(see Atiyah-Bott 84, Mathai-Quillen 86, Sec. 5 Kalkman 93, Sec.2.1, Miettinen 96, Sec. 2).

The Cartan model

The Cartan model follows from the Weil model above by algebraically solving the horizontality constraint (9). This we discuss first below. Then we describe the resulting dgc-algebra further below.

Reviews include (Mathai-Quillen 86, Sec. 5, Kalkman 93, section 2.2)

Via horizontal projection of the Weil model

The Cartan model arises form the Weil model above by the observation that the first of the two constraints defining basic differential forms (10), namely the constrain for horizontal differential forms (9), may be uniformly solved:

Lemma

(projection operator onto horizontal differential forms)

Consider the normal ordered exponential of minus the sum of the contraction derivations (5) followed by wedge product with the corresponding degree-1 generator (3)

(12):exp(−θ aι a):=1−∑aθ aι a+12∑a,bθ aθ bι bι a−⋯:Ω •(X)⊗W(𝔤)⟶Ω •(X)⊗W(𝔤) : \exp \big( - \theta^a \iota_a \big) : \;=\; 1 - \underset{a}{\sum} \theta^a \iota_a + \tfrac{1}{2} \underset{a,b}{\sum} \theta^a \theta^b \iota_b \iota_a - \cdots \;\;\colon\;\; \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \longrightarrow \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g})

We have:

This is the projection operator onto the sub-space of horizontal differential forms (9).
The restriction of this projector to Ω •(X)\Omega^\bullet\big(X \big) is a graded algebra-isomorphism onto the horizontal forms in CE(𝔤)⊗Ω •(X)CE(\mathfrak{g}) \otimes \Omega^\bullet\big(X \big)

Ω •(X)⟶≃:exp(−θ aι a):(CE(𝔤)⊗Ω •(X)) hor \Omega^\bullet\big(X \big) \underoverset{\simeq}{ :\exp\big( - \theta^a \iota_a \big): }{\longrightarrow} \big( CE(\mathfrak{g}) \otimes \Omega^\bullet(X) \big)_{hor}
Hence the further tensor product with ℝ[{r a} a]\mathbb{R}\big[ \{r^a\}_a \big] is an algebra isomorphism onto the full subspace of horizontal differential forms (9)

ℝ[{r a} a]⊗Ω •(X)⟶≃:exp(−θ aι a):(CE(𝔤)⊗Ω •(X)) hor \mathbb{R}\big[ \{r^a\}_a \big] \otimes \Omega^\bullet\big(X \big) \underoverset{\simeq}{ :\exp\big( - \theta^a \iota_a \big): }{ \longrightarrow } \big( CE(\mathfrak{g}) \otimes \Omega^\bullet(X) \big)_{hor}
The operator commutes with the Lie derivative (6) and hence restricts to an isomorphism onto the sub-dgc-algebra of basic differential forms (10)

(13)(ℝ[{r a} a]⊗Ω •(X)) G⟶≃:exp(−θ aι a):(CE(𝔤)⊗Ω •(X)) bas \big( \mathbb{R}\big[ \{r^a\}_a \big] \otimes \Omega^\bullet\big(X \big) \big)^G \underoverset{\simeq}{ \;\;\; :\exp\big( - \theta^a \iota_a \big): \;\;\; }{ \longrightarrow } \big( CE(\mathfrak{g}) \otimes \Omega^\bullet(X) \big)_{bas}
The inverse of (13)

(14)(ℝ[{r a} a]⊗Ω •(X)) G⟵≃ϵ(CE(𝔤)⊗Ω •(X)) bas \big( \mathbb{R}\big[ \{r^a\}_a \big] \otimes \Omega^\bullet\big(X \big) \big)^G \underoverset{\simeq}{ \;\;\; \epsilon \;\;\; }{ \longleftarrow } \big( CE(\mathfrak{g}) \otimes \Omega^\bullet(X) \big)_{bas}

is the algebra homomorphism given setting all generaotors θ a\theta^a in (3) to zero

(15)ϵ:{ω ↦ω θ a ↦0 r a ↦r a \epsilon \;\colon\; \left\{ \array{ \omega & \mapsto \omega \\ \theta^a & \mapsto 0 \\ r^a & \mapsto r^a } \right.
The induced differential on the left, which hence makes :exp(−θ aι a):: \exp\big( - \theta^a \iota_a\big) : a dgc-algebra-isomorphism and hence in particular a quasi-isomorphism is

(16)ϵ∘(d dR+d W)∘:exp(−θ aι a):=d dR+r aι v a. \epsilon \circ \big( d_{dR} + d_W\big) \circ : \exp\big( -\theta^a \iota_a \big) : \;=\; d_{dR} + r^a \iota_{v^a} \,.

This is the Mathai-Quillen isomorphism (Mathai-Quillen 86, around (5.9)).

Proof

Observe that the operator (12) is equal to the product

:exp(−θ aι a):=(id−θ 1ι 1)(id−θ 2ι 2)⋯(id−θ dim(𝔤)ι dim(𝔤)). : \exp \big( - \theta^a \iota_a \big) : \;=\; \big( id - \theta^1 \iota_1 \big) \big( id - \theta^2 \iota_2 \big) \cdots \big( id - \theta^{dim(\mathfrak{g})} \iota_{dim(\mathfrak{g})} \big) \,.

Here all factors commute with each other, and each factor is itself a projection operator, with image the kernel of the corresponding single contraction operator, e.g.

im(1−θ 1ι 1)≃ker(ι 1) im \big( 1 - \theta^1 \iota_1 \big) \;\simeq\; ker\big( \iota_1\big)

etc.

Hence the joint image is the joint kernel of the contraction operators.

It is clear by inspection that ϵ\epsilon in (14) is a linear inverse to :exp(−θ aι a):: \exp\big( - \theta^a \iota_a\big) :. Therefore, since ϵ\epsilon is manifestly an algebra homomorphism, so is :exp(−θ aι a):: \exp\big( - \theta^a \iota_a\big) :.

This implies that the induced differential (16) is a graded derivation and hence that it may be identified by its action on generators. Direct inspection indeed yields

for all generators r ar^a

ϵ∘(d dR+d W)∘:exp(−θ aι a):(r a) =0 =(d dR+r aι v a)(r a) \begin{aligned} \epsilon \circ \big( d_{dR} + d_W\big) \circ : \exp\big( -\theta^a \iota_a \big) : \big( r^a \big) & = 0 \\ & = \big( d_{dR} + r^a \iota_{v^a} \big) ( r^a ) \end{aligned}

and for all differential forms ω∈Ω •(X)\omega \in \Omega^\bullet\big( X \big):

ϵ∘(d dR+d W)∘:exp(−θ aι a):(ω) =ϵ∘(d dR+d W)(ω−θ aι v aω+⋯) =ϵ∘(d dRω+∑aθ ad dRι v aω+(r a−12f bc at b∧t c)ι v aω+⋯) =d dRι v a(ω)+r aι v a(ω) \begin{aligned} \epsilon \circ \big( d_{dR} + d_W\big) \circ : \exp\big( -\theta^a \iota_a \big) : \big( \omega \big) & = \epsilon \circ \big( d_{dR} + d_W\big) \big( \omega - \theta^a \iota_{v^a} \omega + \cdots \big) \\ & = \epsilon \circ \big( d_{dR} \omega + \underset{a}{\sum} \theta^a d_{dR} \iota_{v^a} \omega + \big( r^a - \tfrac{1}{2}f^a_{b c} t^b \wedge t^c \big) \iota_{v^a} \omega + \cdots \big) \\ & = d_{dR} \iota_{v^a} ( \omega) + r^a \iota_{v^a} (\omega) \end{aligned}

because ϵ\epsilon annihilates, by (15), all summands containing a θ a\theta^a-factor.

The left hand side graded algebra of the isomorphism (13) equipped with the induced differential (16) is called the Cartan model, and that isomorphism exhibits it as equivalent to the Weil model:

(17)((Ω •(X)⊗ℝ[{r a} a]) G,d dR+r aι v a) ⟶≃:exp(−θ aι a): ((Ω •(X)⊗W(𝔤)) basic,d dR+d W) Cartan model Weil model \array{ \Big( \Big( \Omega^\bullet\big(X \big) \otimes \mathbb{R}\big[ \{r^a\}_a \big] \Big)^G \,,\, d_{dR} + r^a \iota_{v^a} \Big) & \underoverset{\simeq}{ : \exp\big( - \theta^a \iota_a \big) : }{\longrightarrow} & \Big( \Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \Big)_{basic} , d_{dR} + d_W \Big) \\ \text{Cartan model} && \text{Weil model} }

This statement is originally due to Cartan 50, Sec. 6.

In summary, the Cartan model is explicitly the following dgc-algebra:

Direct definition

Write

(Ω •(G,𝔤 *[1]) G↪Ω(G,𝔤 *[1]) (\Omega^\bullet(G, \mathfrak{g}^\ast[1])^G \hookrightarrow \Omega(G, \mathfrak{g}^\ast[1])

for the GG-invariant differential forms on GG with coefficients in the linear dual of the Lie algebra 𝔤\mathfrak{g}, shifted up in degree. So for {F a}\{F^a\} a dual basis, a general element of this space in degree 2p+q2 p + q is of the form

ω=F a 1∧⋯F a p∧ω a 1,⋯,a q, \omega \;=\; F^{a_1} \wedge \cdots F^{a_p} \wedge \omega_{a_1,\cdots ,a_q} \,,

where ω ⋯\omega_{\cdots} are differential q-forms, such that for each t a∈𝔤t_a \in \mathfrak{g} the Lie derivative of these forms satisfies

ℒ v aω a 1,a 2⋯,a p=f aa 1 bω b,a 2,⋯,a p+f aa 2 bω a 1,b,⋯,a p+⋯, \mathcal{L}_{v^a} \omega_{a_1, a_2 \cdots , a_p} = f_{a a_1}{}^b \omega_{b , a_2, \cdots , a_p} + f_{a a_2}^{}^b \omega_{a_1 , b, \cdots , a_p} + \cdots \,,

where {f ab b}\{f_{a b}{}^b\} are the structure constants of 𝔤\mathfrak{g} (2).

Equip this graded vector space Ω •(G,𝔥 *[1]) G\Omega^\bullet(G, \mathfrak{h}^\ast[1])^G with a differential dd by

d:ω↦d dRω−F aι v aω d \colon \omega \mapsto d_{dR}\omega - F^a \iota_{v^a} \omega

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

The resulting dgc-algebra (Ω •(G,𝔤 *[1]) G,d)(\Omega^\bullet(G,\mathfrak{g}^\ast[1])^G, d) is the Cartan model for GG-equivariant de Rham cohomology on XX.

Equivariant de Rham theorem

The point of the above dgc-algebra models is that, under suitable conditions, their cochain cohomology computes the real cohomology of the homotopy type of the homotopy quotient X⫽HX \sslash H, which, as an actual topological space, may be presented by the Borel construction X× GEGX \times_G E G, hence the Borel equivariant de Rham cohomology of XX.

This is the equivariant cohomology-generalization of the plain de Rham theorem:

Proposition

(equivariant de Rham theorem)

Let

GG be a Lie group which is
1. compact;
2. connected;
XX be a smooth GG-manifold (Def. ).

Then the cochain cohomology of (the cochain complex underlying) the Weil model dgc-algebra (11), and hence, by Lemma , also of the Cartan model dgc-algebra (17). is isomorphic to the real cohomology of the homotopy quotient X⫽GX \!\sslash\! G of the action on (the topological space underlying) XX by the (topological group underlying) GG, hence in particular of the Borel construction X× GEG≃X⫽GX \times_G E G \simeq X \!\sslash\! G :

Cartan model cohomology H •((Ω •(X)⊗ℝ[{r a} a]) G,d dR+r aι v a) ≃↓ H •(:exp(−θ aι a):) H •((Ω •(X)⊗W(𝔤)) basic,d dR+d W) ⟶≃ H •(X⫽G,ℝ) Weil model cohomology equivariant real cohomology \array{ \text{Cartan model cohomology} \\ H^\bullet \Big( \Big( \Omega^\bullet\big(X \big) \otimes \mathbb{R}\big[ \{r^a\}_a \big] \Big)^G \,,\, d_{dR} + r^a \iota_{v^a} \Big) \\ {}^{\simeq} \Big\downarrow {}^{ H^\bullet\big( : \exp\big( - \theta^a \iota_a \big) : \big) } \\ H^\bullet \Big( \Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \Big)_{basic} , d_{dR} + d_W \Big) &\underoverset{\simeq}{\;\;\;\;\;\;\;\;\;\;\;\;\;}{\longrightarrow}& H^\bullet \big( X \!\!\sslash\!\! G \,,\, \mathbb{R} \big) \\ \text{Weil model cohomology} && \mathclap{ \text{equivariant real cohomology} } }

(e.g Meinrenken 06, Theorem 6.1)

Cartan’s map

If the G-manifold XX has a free action, hence is the total space X=PX = P of a GG-principal bundle P→BP \to B, then the Cartan map (or Cartan's map or similar) is a quasi-isomorphism from the Cartan model for the equivariant de Rham cohomology of X=PX = P to the ordinary de Rham complex model for the ordinary de Rham cohomology of the base manifold BB

((Ω •(P)⊗ℝ[{r a} a]) G,d dR+r aι v a) ⟶≃ qi (Ω dR •(B),d dR) ω⊗⟨−⟩ ↦ ω hor⊗⟨F ∇⟩ \array{ \bigg( \Big( \Omega^\bullet\big(P \big) \otimes \mathbb{R}\big[ \{r^a\}_a \big] \Big)^G \,,\, d_{dR} + r^a \iota_{v^a} \bigg) & \overset{ \simeq_{qi} }{\longrightarrow} & \Big( \Omega^\bullet_{dR}(B), \, d_{dR} \Big) \\ \omega \otimes \langle - \rangle & \mapsto & \omega_{hor} \otimes \langle F_\nabla\rangle }

given by choosing an Ehresmann connection ∇\nabla on P→BP \to B and inserting its curvature form into the invariant polynomials ⟨−⟩\langle-\rangle (essentially the Chern-Weil homomorphism).

(Guillemin-Sternberg 99, Chapter 5, Albin-Melrose 09, Theorem 11.1)

References

The Cartan model for equivariant de Rham cohomology is originally due to

Henri Cartan, La transgression dans un groupe de Lie et dans un espace fibré principal, Colloque de topologie (espaces fibrés). Bruxelles, 1950

Review:

Nicole Berline, Ezra Getzler, Michèle Vergne, Section 7.1 of: Heat Kernels and Dirac Operators, Grundlehren 298, Springer 2004 (ISBN:9783540200628)
Eckhard Meinrenken, Section 5 of: Group actions on manifolds, Lecture Notes 2003 (pdf, pdf)
Eckhard Meinrenken, Equivariant cohomology and the Cartan model, in: Encyclopedia of Mathematical Physics, Pages 242-250 Academic Press 2006 (pdf, doi:10.1016/B0-12-512666-2/00344-8)
Oliver Goertsches, Leopold Zoller, Equivariant de Rham Cohomology: Theory and Applications, São Paulo J. Math. Sci. (2019) (arXiv:1812.09511, doi:10.1007/s40863-019-00129-4)

Comprehensive textbook account:

Loring Tu, Introductory Lectures on Equivariant Cohomology, Annals of Mathematics Studies 204, AMS 2020 (ISBN:9780691191744)