∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
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 : therefore 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
Any choice of linear basis for the Lie algebra of the given Lie group induces linear coordinate functions on the Cartesian space underlying , with points parameterized as
(where we use the Einstein summation convention throughout). In terms of these coordinates, the pullback of the Maurer-Cartan form along the exponential map at any point is (by Schur 1891 (36), Helgason 2001 Thm. 7.4, Meinrenken 2013 Thm. C.2)
where (by Helgason p. 36, Meinrenken p. 99)
(for any linear endomorphism), so that:
Unwinding this with
(where denote the structure constants in the given basis, such that ) we get:
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
The morphism
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.
Therefore generally for a cohesive (∞,1)-topos and 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 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.
Named after
and
Further early discussion:
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:
Discussion in synthetic differential geometry:
Discussion via Lie algebroids and Lie groupoids:
Expression of the Maurer-Cartan form in linear coordinates on the Lie algebra after pullback along the exponential map :
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]
Discussion of MC forms in the generality on coset spaces (homogeneous spaces):
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]
Last revised on September 11, 2024 at 17:37:22. See the history of this page for a list of all contributions to it.