Formal Lie groupoids
On Lie groups
For a Lie group, the Maurer-Cartan form on is a canonical Lie-algebra valued 1-form on . 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 are related by a unique group element such that . If and are infinitesimally close points, defining a tangent vector, then is an element of the Lie algebra of . So restricted to infinitesimally close points is a -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 at may be identified with an equivalence class of smooth function with . The tangent vectors through the origin are canonically identified with the Lie algebra of . By left-translating a path through back to the origin it represents a Lie algebra element. This map
of tangent vectors to Lie algebra elements is the Maurer-Cartan form.
If we write for the identity function on , then is the identity function on the tangent vectors of . With this the Maurer-Cartan form may be written
If is a matrix Lie group, then 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 there is a unique group element such that : therefor for three points we have
i.e. . This is what analytically becomes the statement of vanishing curvature.
If is a smooth manifold and a smooth function with values in , we have the pullback form
of the Maurer-Cartan form on . Using the above notation, writing simply for this is
Now is no longer (necessarily) the identity map as was when we wrote above, but the form of this equation shows why it can be useful to think of itself in terms of the identity map .
The Maurer-Cartan form crucially appears in the formula for the gauge transformation of Lie-algebra valued 1-forms.
For a smooth function and a Lie-algebra valued form, the condition that is flat with respect to is that it satisfies the differential equation
(where denotes the right multiplication action of on itself). This is such that if happens to be a matrix Lie group it is equivalent to
We call the unique solution of this differential equation that satisfies the parallel transport of and write it .
Now for a function, the gauge transformed parallel transport is
This solves a differential equation as above, but for a different 1-form . The relation is
or equivalently, with adopted notation
On smooth -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 with delooping there is canonically an smooth ∞-groupoid as described here. Morphisms correspond to flat -valued differential forms on .
This fits into a double (∞,1)-pullback diagram
in this diagram is the -Maurer-Cartan form on . For 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 a cohesive (∞,1)-topos and an ∞-group object, one may think of
as the Maurer-Cartan form on ∞-group objects
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 in cohesion, then the shape of the Maurer-Cartan form plays the role of the Chern character on -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.