∞-Lie theory (higher geometry)
synthetic differential ∞-groupoid?
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
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
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
The Maurer-Cartan form is a Lie-algebra valued form with vanishing curvature.
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
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
of the Maurer-Cartan form on $X$. Using the above notation, writing simply $h^{-1}$ for $h^{-1}_*$ this is
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
(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
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
This solves a differential equation as above, but for a different 1-form $A'$. The relation is
or equivalently, with adopted notation
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
The morphism
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.
Therefore generally for $\mathbf{H}$ a cohesive (∞,1)-topos and $G \in \mathbf{H}$ an ∞-group object, one may think of
as the Maurer-Cartan form on ∞-group objects
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.
Given a stable homotopy type $\hat E$ in cohesion, then the shape of the Maurer-Cartan form plays the role of the Chern character on $E \coloneqq \Pi(\hat E)$-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 December 12, 2017 at 13:48:49. See the history of this page for a list of all contributions to it.