nLab
Maurer-Cartan form

On Lie groups

Idea

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

Synthetic definition

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

any two points $x,y\in G$ are related by a unique group element $\theta (x,y)$ such that $y=x\cdot \theta (x,y)$. If $x$ and $y$ are infinitesimally close points, defining a tangent vector, then $\theta (x,y)$ is an element of the Lie algebra of $G$. So $\theta$ restricted to infinitesimally close points is a $𝔤$-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 $G$ at $g\in G$ may be identified with an equivalence class of smooth function $\gamma :[0,1]\to G$ with $\gamma (0)=g$. The tangent vectors through the origin $x=e$ are canonically identified with the Lie algebra of $G$. By left-translating a path through $g$ back to the origin $g^{-1}\gamma :[0,1]\to G\stackrel{g^{-1}\cdot (-)}{\to }G$ it represents a Lie algebra element. This map

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

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

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

$$g_{*}^{-1}dg:TG\to T_{e}G=𝔤.$$g^{-1}_* d g : T G \to T_e G = \mathfrak{g} \,.

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

$$\theta =g^{-1}dg.$$\theta = g^{-1} d g \,.

Properties

Curvature

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

$$d\theta +[\theta \wedge \theta ]=0$$d \theta + [\theta \wedge \theta] = 0

This is known as the Maurer-Cartan equation.

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

$$\begin{array}{ccc}& & y\\ & {}^{\theta (x,y)}\nearrow & & {\searrow }^{\theta (y,z)}\\ x& & \stackrel{\theta (x,z)}{\to }& & z\end{array}$$\array{ && y \\ & {}^{\mathllap{\theta}(x,y)}\nearrow && \searrow^{\mathrlap{\theta}(y,z)} \\ x &&\stackrel{\theta(x,z)}{\to}&& z }

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

Pullback

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

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

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

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

Now $dh:TX\to TG$ is no longer (necessarily) the identity map as $g$ was when we wrote $\theta =g^{-1}dg$ above, but the form of this equation shows why it can be useful to think of $\theta$ itself in terms of the identity map $dg:TG\to TG$.

Gauge transformations

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

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

$$du=-(R_u{)}_{*}\circ A$$d u = -(R_u)_* \circ A

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

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

We call the unique solution $u$ of this differential equation that satisfies $u(0)=e$ the parallel transport of $A$ and write it $u=P\exp ({\int }_0^{(-)}A)$.

Now for $g:ℝ\to G$ a function, the gauge transformed parallel transport is

$$g^{-1}P\exp ({\int }_0^{(-)}A)g.$$g^{-1} P \exp(\int_0^{(-)} A) g \,.

This solves a differential equation as above, but for a different 1-form $A\prime$. The relation is

$$A\prime ={\mathrm{Ad}}_{g^{-1}}A+g^{*}\theta$$A' = Ad_{g^{-1}} A + g^* \theta

or equivalently, with adopted notation

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

On smooth $\mathrm{\infty }$-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 $G\in \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$ with delooping $BG$ there is canonically an smooth ∞-groupoid ${♭}_{\mathrm{dR}}BG$ as described here. Morphisms $X\to {♭}_{\mathrm{dR}}BG$ correspond to flat $𝔤$-valued differential forms on $G$.

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

$$\begin{array}{ccc}G& \to & *\\ {}^{\theta }\downarrow & & \downarrow \\ {♭}_{\mathrm{dR}}BG& \to & ♭BG\\ \downarrow & & \downarrow \\ *& \to & BG\end{array}.$$\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

$$\theta :G\to {♭}_{\mathrm{dR}}BG$$\theta : G \to \mathbf{\flat}_{dR}\mathbf{B}G

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

Related concepts

• invariant differential form

References

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

• Anders Kock, Synthetic geometry of manifolds (pdf)

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