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.

Definition in synthetic differential geometry

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.

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

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

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

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

Properties

Curvature

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

$$d \theta + \frac{1}{2}[\theta \wedge \theta] = 0$$

This is known as the Maurer-Cartan equation.

Synthetically this is just a restatement of the fact that for $x,y \in G$ there is a unique group element such that $y = x \cdot g$: therefore 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.

Pullback

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

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

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 $d g : T G \to T G$.

Gauge transformations

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 $A$ 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 $u = P \exp\big(\int_0^{(-)} A\big)$.

Now for $g \colon \mathbb{R} \to G$ 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

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

In coordinates under the exponential map

Any choice of linear basis $\mathfrak{g} \;\simeq\; \mathbb{R}\big\langle (T_i)_{i \in I} \big\rangle$ for the Lie algebra $\mathfrak{g}$ of the given Lie group $G$ induces linear coordinate functions $(x^i)_{i \in I}$ on the Cartesian space underlying $\mathfrak{g}$, with points parameterized as

$$x^i T_i \,\in\, \mathfrak{g}$$

(where we use the Einstein summation convention throughout). In terms of these coordinates, the pullback $(e^i)_{i \in I}$ of the Maurer-Cartan form along the exponential map $exp \,\colon\, \mathfrak{g} \longrightarrow G$ at any point $X = x^i T_i$ is (by Schur 1891 (36), Helgason 2001 Thm. 7.4, Meinrenken 2013 Thm. C.2)

$$e^i \;=\; \mathrm{d}x^i \left( \frac {1 - exp(- ad X)} {ad X} (\partial_k) \right) \mathrm{d}x^k$$

where (by Helgason p. 36, Meinrenken p. 99)

$$\frac{1 - exp(-A)}{ A } \;\; \coloneqq \;\; \sum_{n=0}^\infty \tfrac{1}{(n+1)!} (-A)^n$$

(for $A \colon \mathfrak{g} \to \mathfrak{g}$ any linear endomorphism), so that:

$$e^i \;=\; \mathrm{d}x^i \left( \textstyle{ \sum_{n = 0}^\infty } \tfrac{1}{(n+1)!} ( - ad X )^n (\partial_k) \right) \mathrm{d}x^k \,.$$

Unwinding this with

$$(ad X) \;=\; \big( x^j f^\bullet_{j \bullet} \big) \,,$$

(where $f^i_{j k} \in \mathbb{R}$ denote the structure constants in the given basis, such that $[X_j, X_k] = f^i_{j k} X_i$) we get:

$$e^i \;=\; \mathrm{d}x^i - \tfrac{1}{2} f^i_{j k}x^j \mathrm{d}x^k + \tfrac{1}{6} f^i_{j k'} f^{k'}_{k l} x^j x^k \mathrm{d}x^l + \cdots \,.$$

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 $G \in Smooth\infty Grpd$ 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.

On cohesive and stable homotopy types

Definition

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

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

as the Maurer-Cartan form on ∞-group objects

$$\theta_G \;\colon\; G \stackrel{}{\longrightarrow} \flat_{dR}\mathbf{B}G \,.$$

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.

Properties

Relation to the Chern character

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.

Related concepts

• invariant differential form

References

General

Named after

• Ludwig Maurer: Über allgemeinere Invarianten-Systeme, Münchener Berichte 18 (1888) 103-150

and

• Élie Cartan, Sur la structure des groupes infinis de transformation, Annales scientifiques de l’É.N.S. 3e série, tome 21 (1904) 153-206 [numdam:ASENS_1904_3_21__153_0]

Further early discussion:

• Friedrich Schur, around (16) in: Zur Theorie der endlichen Transformationsgruppen, Math. Ann. 38 (1891) 263–286 [doi:10.1007/BF01199254]

Introductions in the broader context of Cartan geometry:

• Richard W. Sharpe, §3 in: An introduction to Cartan Geometries, Proceedings of the 21st Winter School “Geometry and Physics”, Circolo Matematico di Palermo (2002) 61-75 [eudml:220395, dml:701688]

• Benjamin McKay, pp. 3 of: An introduction to Cartan geometries, Proceedings of the 21st Winter School “Geometry and Physics”, Palermo (2002) [arXiv:2302.14457, eudml:220395]

• Raphaël Alexandre, Elisha Falbel, §3.1.1 in: Introduction to Cartan geometry (2023) [pdf, pdf]

See also:

• Wikipedia, Maurer-Cartan form

Discussion in synthetic differential geometry:

• Anders Kock, Ex. 3.7.2 & Cor. 6.7.2 in: Synthetic geometry of manifolds, Cambridge Tracts in Mathematics 180 (2010) [doi:10.1017/CBO9780511691690, pdf]

Discussion via Lie algebroids and Lie groupoids:

• Anthony D. Blaom, A characterisation of smooth maps into a homogeneous space, SIGMA 18 (2022) 029 [arXiv:1702.02717, doi:10.3842/SIGMA.2022.029]

Expression of the Maurer-Cartan form in linear coordinates on the Lie algebra $\mathfrak{g}$ after pullback along the exponential map $exp \colon \mathfrak{g} \longrightarrow G$:

• Sigurdur Helgason, §I.8 in: Differential geometry, Lie groups and symmetric spaces, Graduate Studies in Mathematics 34 (2001) [ams:gsm-34]

• Eckhard Meinrenken, Theorem C.2 in Appendix C of: Clifford algebras and Lie groups, Ergebn. Mathem. & Grenzgeb., Springer (2013) [doi:10.1007/978-3-642-36216-3]

On coset spaces

Discussion of MC forms in the generality on coset spaces (homogeneous spaces):

• Wikipedia, Maurer-Cartan form – On a homogeneous space

In application to first-order formulation of (super-)gravity:

• Leonardo Castellani, L. J. Romans, Nicholas P. Warner, Symmetries of coset spaces and Kaluza-Klein supergravity, Annals of Physics 157 2 (1984) 394-407 [doi:10.1016/0003-4916(84)90066-6, spire:193940]

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, §I.6.6 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, ch I.6: pdf

• Leonardo Castellani, On G/H geometry and its use in M-theory compactifications, Annals Phys. 287 (2001) 1-13 [arXiv:hep-th/9912277, doi:10.1006/aphy.2000.6097]