Maurer-Cartan form

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.

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 $\mathfrak{g}$-valued 1-form, and this is the Maurer-Cartan form.

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

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

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 $d g : T G \to T G$ is the identity function on the tangent vectors of $G$. With this the Maurer-Cartan form may be written

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(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

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

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 $g : \mathbb{R} \to G$ a function, the *gauge transformed* parallel transport is

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

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

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

or equivalently, with adopted notation

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

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

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

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

The morphism

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

in this diagram is the $\infty$-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.

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.

Revised on January 9, 2013 13:08:14
by Urs Schreiber
(89.204.153.84)