nLab
curvature characteristic form

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

\infty-Chern-Weil theory


Contents

Idea

A curvature characteristic form is a differential form naturally associated to a Lie algebra-valued 1-form that is a measure for the non-triviality of the curvature of the 1-form.

More generally, there is a notion of curvature characteristic forms of L-∞-algebra-valued differential forms and ∞-Lie algebroid valued differential forms.

Of connection 1-forms

For 𝔤\mathfrak{g} a Lie algebra, ,,,\langle -,-, \cdots, -\rangle an invariant polynomial of nn arguments on the Lie algebra and AΩ 1(P,𝔤)A \in \Omega^1(P,\mathfrak{g}) a Lie-algebra-valued 1-form with curvature 2-form F A=d dRA+[AA]F_A = d_{dR} A + [A \wedge A], the curvature characteristic form of AA with respect to \langle \cdots \rangle is the differential form

F AF AΩ 2n(P). \langle F_A \wedge \cdots \wedge F_A \rangle \in \Omega^{2 n}(P) \,.

This form is always an exact form. The (2n1)(2 n -1)-form trivializing it is called a Chern-Simons form.

Notably if GG is a Lie group with Lie algebra 𝔤\mathfrak{g}, PP is the total space of a GG-principal bundle π:PX\pi : P \to X, and AΩ 1(P,𝔤)A \in \Omega^1(P,\mathfrak{g}) is an Ehresmann connection 1-form on PP then by the very definition of the GG-equivariance of AA and the invariance of \langle \cdots \rangle it follows that the curvature form is invariant under the GG-action on PP and is therefore the pullback along π\pi of a 2n2 n-form P nΩ 2n(X)P_n \in \Omega^{2 n}(X) down on XX. This form is in general no longer exaxt, but is always a closed form and hence represent a class in the de Rham cohomology of XX. This establishes the Weil homomorphism from invariant polynomials to de Rham cohomology

In terms of \infty-Lie algebroids

The above description of curvature characteristic forms may be formulated in terms of ∞-Lie theory as follows.

For PXP \to X a GG-principal bundle write TXT X, TPT P and T vertPT_{vert} P for the tangent Lie algebroid of XX, of PP and the vertical tangent Lie algebroid of PP, respectively. Write inn(𝔤)inn(\mathfrak{g}) for the Lie 2-algebra given by the differential crossed module 𝔤Id𝔤\mathfrak{g}\stackrel{Id}{\to} \mathfrak{g} and finally ib n i\prod_i b^{n_i} \mathbb{R} for the L-∞-algebra with one abelian generator for each generating invariant polynomial of 𝔤\mathfrak{g}

From the discussion at invariant polynomial we have a canonical morphism inn(𝔤) ib n iinn(\mathfrak{g}) \to \prod_i b^{n_i}\mathbb{R} that represents the generating invariant polynomials.

Recall that a morphism of ∞-Lie algebroids

TXb n T X \to b^n \mathbb{R}

is equivalently a closed nn-form on XX. The data of an Ehresmann connection on PP then induces the following diagram of ∞-Lie algebroids

T vertP A vert 𝔤 flatverticalform firstEhresmanncondition TP A inn(𝔤) formontotalspace secondEhresmanncondition TX (P n) ib n i curvaturecharacteristicforms. \array{ T_{vert} P &\stackrel{A_{vert}}{\to}& \mathfrak{g} &&& flat vertical form \\ \downarrow && \downarrow &&& first Ehresmann condition \\ T P &\stackrel{A}{\to}& inn(\mathfrak{g}) &&& form on total space \\ \downarrow && \downarrow &&& second Ehresmann condition \\ T X &\stackrel{(P_n)}{\to}& \prod_i b^{n_i} \mathbb{R} &&& curvature characteristic forms } \,.

Examples

References

Revised on September 24, 2010 00:57:24 by Toby Bartels (98.19.53.0)