Maurer-Cartan form



\infty-Lie theory

∞-Lie theory (higher geometry)


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On Lie groups


For GG a Lie group, the Maurer-Cartan form on GG is a canonical Lie-algebra valued 1-form on GG. One can generalize also to the Maurer-Cartan form on a principal bundle.

Definition in synthetic differential geometry

Speaking in terms of synthetic differential geometry the Maurer-Cartan form has the following definition:

any two points x,yGx,y \in G are related by a unique group element θ(x,y)\theta(x,y) such that y=xθ(x,y)y = x \cdot \theta(x,y). If xx and yy are infinitesimally close points, defining a tangent vector, then θ(x,y)\theta(x,y) is an element of the Lie algebra of GG. So θ\theta restricted to infinitesimally close points is a 𝔤\mathfrak{g}-valued 1-form, and this is the Maurer-Cartan form.

Analytic definition

In terms of analysis there is a direct analogue of this definition: a tangent vector on GG at gGg \in G may be identified with an equivalence class of smooth function γ:[0,1]G\gamma : [0,1] \to G with γ(0)=g\gamma(0) = g. The tangent vectors through the origin x=ex = e are canonically identified with the Lie algebra of GG. By left-translating a path through gg back to the origin g 1γ:[0,1]Gg 1()Gg^{-1}\gamma : [0,1] \to G \stackrel{g^{-1} \cdot(-)}{\to} G it represents a Lie algebra element. This map

θ:=g * 1:[γ][g 1γ] \theta := g^{-1}_* : [\gamma] \mapsto [g^{-1} \gamma]

of tangent vectors to Lie algebra elements is the Maurer-Cartan form.

If we write g:GGg : G \to G for the identity function on GG, then dg:TGTGd g : T G \to T G is the identity function on the tangent vectors of GG. With this the Maurer-Cartan form may be written

g * 1dg:TGT eG=𝔤. g^{-1}_* d g : T G \to T_e G = \mathfrak{g} \,.

If GG is a matrix Lie group, then g * 1g^{-1}_* is literally just left-multiplication of matrices and therefore the Maurer-Cartan form is often written just

θ=g 1dg. \theta = g^{-1} d g \,.



The Maurer-Cartan form is a Lie-algebra valued form with vanishing curvature.

dθ+12[θθ]=0 d \theta + \frac{1}{2}[\theta \wedge \theta] = 0

This is known as the Maurer-Cartan equation.

Synthetically this is just a restatement of the fact that for x,yGx,y \in G there is a unique group element such that y=xgy = x \cdot g: therefore for three points x,y,zx,y,z we have

y θ(x,y) θ(y,z) x θ(x,z) z \array{ && y \\ & {}^{\mathllap{\theta}(x,y)}\nearrow && \searrow^{\mathrlap{\theta}(y,z)} \\ x &&\stackrel{\theta(x,z)}{\to}&& z }

i.e. θ(x,y)θ(y,z)=θ(x,z)\theta(x,y) \theta(y,z) = \theta(x,z). This is what analytically becomes the statement of vanishing curvature.


If XX is a smooth manifold and h:XGh : X \to G a smooth function with values in GG, we have the pullback form

h *θΩ 1(X,𝔤) h^* \theta \in \Omega^1(X,\mathfrak{g})

of the Maurer-Cartan form on XX. Using the above notation, writing simply h 1h^{-1} for h * 1h^{-1}_* this is

h *θ=h 1dh. h^* \theta = h^{-1} d h \,.

Now dh:TXTGd h : T X \to T G is no longer (necessarily) the identity map as gg was when we wrote θ=g 1dg\theta = g^{-1} d g above, but the form of this equation shows why it can be useful to think of θ\theta itself in terms of the identity map dg:TGTGd g : T G \to T G.

Gauge transformations

The Maurer-Cartan form crucially appears in the formula for the gauge transformation of Lie-algebra valued 1-forms.

For u:Gu : \mathbb{R} \to G a smooth function and AΩ 1(,𝔤)A \in \Omega^1(\mathbb{R}, \mathfrak{g}) a Lie-algebra valued form, the condition that uu is flat with respect to uu is that it satisfies the differential equation

du=(R u) *A d u = -(R_u)_* \circ A

(where RR denotes the right multiplication action of GG on itself). This is such that if GG happens to be a matrix Lie group it is equivalent to

(d+A)u=0. (d + A) u = 0 \,.

We call the unique solution uu of this differential equation that satisfies u(0)=eu(0) = e the parallel transport of AA and write it u=Pexp( 0 ()A)u = P \exp(\int_0^{(-)} A).

Now for g:Gg : \mathbb{R} \to G a function, the gauge transformed parallel transport is

g 1Pexp( 0 ()A)g. g^{-1} P \exp(\int_0^{(-)} A) g \,.

This solves a differential equation as above, but for a different 1-form AA'. The relation is

A=Ad g 1A+g *θ A' = Ad_{g^{-1}} A + g^* \theta

or equivalently, with adopted notation

A=g 1Ag+g 1dg. A' = g^{-1}A g + g^{-1} d g \,.

On smooth \infty-groups

The theory of Lie groups embeds into the more general context of smooth ∞-groupoids. In this context the Maurer-Cartan form has an (even) more general abstract definition that does not even presuppose the notion of differential form as such:

for every smooth ∞-group GSmoothGrpdG \in Smooth\infty Grpd with delooping BG\mathbf{B}G there is canonically an smooth ∞-groupoid dRBG\mathbf{\flat}_{dR} \mathbf{B}G as described here. Morphisms X dRBGX\to \mathbf{\flat}_{dR}\mathbf{B}G correspond to flat 𝔤\mathfrak{g}-valued differential forms on GG.

This fits into a double (∞,1)-pullback diagram

G * θ dRBG BG * BG. \array{ G &\to& * \\ {}^{\mathllap{\theta}}\downarrow && \downarrow \\ \mathbf{\flat}_{dR} \mathbf{B}G &\to& \mathbf{\flat} \mathbf{B}G \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}G } \,.

The morphism

θ:G dRBG \theta : G \to \mathbf{\flat}_{dR}\mathbf{B}G

in this diagram is the \infty-Maurer-Cartan form on GG. For GG an ordinary Lie group, this reduces to the above definition. This statement and its proof is spelled out here.

On cohesive and stable homotopy types


Therefore generally for H\mathbf{H} a cohesive (∞,1)-topos and GHG \in \mathbf{H} an ∞-group object, one may think of

θfib(B) \theta \coloneqq fib(\flat \to \mathbf{B})

as the Maurer-Cartan form on ∞-group objects

θ G:G dRBG. \theta_G \;\colon\; G \stackrel{}{\longrightarrow} \flat_{dR}\mathbf{B}G \,.

This is discussed at cohesive infinity-topos – structures in the section Maurer-Cartan forms and curvature characteristics.

This includes then for instance Maurer-Cartan forms in higher supergeometry as discussed at Super Gerbes.


Relation to the Chern character

Given a stable homotopy type E^\hat E in cohesion, then the shape of the Maurer-Cartan form plays the role of the Chern character on EΠ(E^)E \coloneqq \Pi(\hat E)-cohomology.

See at Chern character for more on this, and see at differential cohomology diagram.


The synthetic view on the Maurer-Cartan form is discussed in

The synthetic Maurer-Cartan form itself appears in example 3.7.2. The synthetic vanishing of its curvature is corollary 6.7.2.

Last revised on November 24, 2020 at 02:25:26. See the history of this page for a list of all contributions to it.