nLab
infinity-Lie algebroid-valued differential 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

For 𝔤\mathfrak{g} an ∞-Lie algebra (or more generally ∞-Lie algebroid), the \infty-groupoid of 𝔤\mathfrak{g}-valued forms is the ∞-groupoid whose

This naturally refines to a non-concrete ∞-Lie groupoid whose UU-parameterized smooth families of objects are ∞-Lie algebroid-valued differential forms on ZZ.

A cocycle with coefficients in this is a connection on an ∞-bundle.

For an introduction see the section ∞-Lie algebra valued forms at ∞-Chern-Weil theory introduction.

Definition

For XX a smooth manifold and 𝔤\mathfrak{g} an ∞-Lie algebra or more generally an ∞-Lie algebroid, a \infty-Lie algebroid valued differential form on XX is a morphism of dg-algebras

Ω (X)W(𝔤):A \Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A

from the Weil algebra of 𝔤\mathfrak{g} to the de Rham complex of XX. Dually this is a morphism of ∞-Lie algebroids

A:TXinn(𝔤) A : T X \to inn(\mathfrak{g})

from the tangent Lie algebroid to the inner automorphism ∞-Lie algebra.

Its curvature is the composite of morphisms of graded vector spaces

Ω (X)AW(𝔤)F () 1𝔤 *:F A. \Omega^\bullet(X) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{F_{(-)}}{\leftarrow} \wedge^1 \mathfrak{g}^* : F_{A} \,.

Precisely if the curvatures vanish does the morphism factor through the Chevalley-Eilenberg algebra W(𝔤)CE(𝔤)W(\mathfrak{g}) \to CE(\mathfrak{g}).

(F A=0)( CE(𝔤) A flat Ω (X) A W(𝔤)) (F_A = 0) \;\;\Leftrightarrow \;\; \left( \array{ && CE(\mathfrak{g}) \\ & {}^{\mathllap{\exists A_{flat}}}\swarrow & \uparrow \\ \Omega^\bullet(X) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) } \right)

in which case we call AA flat.

The curvature characteristic forms of AA are the composite

Ω (X)AW(𝔤)F ()inv(𝔤):F A, \Omega^\bullet(X) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{\langle F_{(-)} \rangle}{\leftarrow} inv(\mathfrak{g}) : \langle F_A\rangle \,,

where inv(𝔤)W(𝔤)inv(\mathfrak{g}) \to W(\mathfrak{g}) is the inclusion of the invariant polynomials.

Definition

For UU a smooth manifold, the \infty-groupoid of 𝔤\mathfrak{g}-valued forms is the Kan complex

exp(𝔤) conn(U):[k]{Ω (U×Δ k)AW(𝔤)vΓ(TΔ k):ι vF A=0} \exp(\mathfrak{g})_{conn}(U) : [k] \mapsto \left\{ \Omega^\bullet(U \times \Delta^k) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \;\; | \;\; \forall v \in \Gamma(T \Delta^k) : \iota_v F_A = 0 \right\}

whose k-morphisms are 𝔤\mathfrak{g}-valued forms AA on U×Δ kU \times \Delta^k with sitting instants, and with the property that their curvature vanishes on vertical vectors.

The canonical morphism

exp(𝔤) connexp(𝔤) \exp(\mathfrak{g})_{conn} \to \exp(\mathfrak{g})

to the untruncated Lie integration of 𝔤\mathfrak{g} is given by restriction of AA to vertical differential forms (see below).

Remark

Here we are thinking of U×Δ kUU \times \Delta^k \to U as a trivial bundle.

The first Ehresmann condition will be identified with the conditions on lifts \nabla in ∞-anafunctors

exp(𝔤) conn C(U) g exp(𝔤) X \array{ && \exp(\mathfrak{g})_{conn} \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow \\ C(U) &\stackrel{g}{\to}& \exp(\mathfrak{g}) \\ \downarrow^{\mathrlap{\simeq}} \\ X }

that define connections on ∞-bundles. More on this in the Properties-section below.

Properties

Curvature characteristics

Proposition

For Aexp(𝔤) conn(U,[k])A \in \exp(\mathfrak{g})_{conn}(U,[k]) a 𝔤\mathfrak{g}-valued form on U×Δ kU \times \Delta^k and for W(𝔤)\langle - \rangle \in W(\mathfrak{g}) any invariant polynomial, the corresponding curvature characteristic form F AΩ (U×Δ k)\langle F_A \rangle \in \Omega^\bullet(U \times \Delta^k) descends down to UU.

Proof

It is sufficient to show that for all vΓ(TΔ k)v \in \Gamma(T \Delta^k) we have

  1. ι vF A=0\iota_v \langle F_A \rangle = 0;

  2. vF A=0\mathcal{L}_v \langle F_A \rangle = 0.

The first condition is evidently satisfied if already ι vF A=0\iota_v F_A = 0. The second condition follows with Cartan calculus and using that d dRF A=0d_{dR} \langle F_A\rangle = 0:

vF A=dι vF A+ι vdF A=0. \mathcal{L}_v \langle F_A \rangle = d \iota_v \langle F_A \rangle + \iota_v d \langle F_A \rangle = 0 \,.
Remark

For a general \infty-Lie algebra 𝔤\mathfrak{g} the curvature forms F AF_A themselves are not closed, hence requiring them to have no component along the simplex does not imply that they descend. This is different for abelian \infty-Lie algebras: for them the curvature forms themselves are already closed, and hence are themselves already curvature characteristics that do descent.

It is useful to organize the 𝔤\mathfrak{g}-valued form AA, together with its restriction A vertA_{vert} to vertical differential forms and with its curvature characteristic forms in the commuting diagram

Ω (U×Δ k) vert A vert CE(𝔤) gaugetransformation Ω (U×Δ k) A W(𝔤) 𝔤valuedform Ω (U) F A inv(𝔤) curvaturecharacteristicforms \array{ \Omega^\bullet(U \times \Delta^k)_{vert} &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) &&& gauge\;transformation \\ \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) &&& \mathfrak{g}-valued\;form \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{\langle F_A\rangle}{\leftarrow}& inv(\mathfrak{g}) &&& curvature\;characteristic\;forms }

in dgAlg.

The commutativity of this diagram is implied by ι vF A=0\iota_v F_A = 0.

Definition

Write exp(𝔤) CW(U)\exp(\mathfrak{g})_{CW}(U) for the \infty-groupoid of 𝔤\mathfrak{g}-valued forms fitting into such diagrams.

exp(𝔤) CW(U):[k]{Ω (U×Δ k) vert A vert CE(𝔤) Ω (U×Δ k) A W(𝔤) Ω (U) F A inv(𝔤)}. \exp(\mathfrak{g})_{CW}(U) : [k] \mapsto \left\{ \array{ \Omega^\bullet(U \times \Delta^k)_{vert} &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{\langle F_A\rangle}{\leftarrow}& inv(\mathfrak{g}) } \right\} \,.
Remark

If we just consider the top horizontal morphism in this diagram we obtain the object

exp(𝔤)(U):[k]{Ω (U×Δ k) vert A vert CE(𝔤)} \exp(\mathfrak{g})(U) : [k] \mapsto \left\{ \array{ \Omega^\bullet(U \times \Delta^k)_{vert} &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) } \right\}

discussed in detail at Lie integration. If we consider the top square of the diagram we obtain the object

exp(𝔤) diff(U):[k]{Ω (U×Δ k) vert A vert CE(𝔤) Ω (U×Δ k) A W(𝔤)}. \exp(\mathfrak{g})_{diff}(U) : [k] \mapsto \left\{ \array{ \Omega^\bullet(U \times \Delta^k)_{vert} &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{g}) } \right\} \,.

This forms a resolution of exp(𝔤)\exp(\mathfrak{g}) and may be thought of as the \infty-groupoid of pseudo-connections.

We have an evident sequence of morphisms

exp(𝔤) conn genuineconnections exp(𝔤) CW pseudoconnectionwithglobalcurvaturecharacteristics exp(𝔤) diff pseudoconnections exp(𝔤) barebundles, \array{ \exp(\mathfrak{g})_{conn} &&& genuine\;connections \\ \downarrow \\ \exp(\mathfrak{g})_{CW} &&& pseudo-connection\;with\;global\;curvature\;characteristics \\ \downarrow \\ \exp(\mathfrak{g})_{diff} &&& pseudo-connections \\ \downarrow^{\mathrlap{\simeq}} \\ \exp(\mathfrak{g}) &&& bare bundles } \,,

where we label the objects by the structures they classify, as discussed at ∞-Chern-Weil theory.

Here the botton morphism is a weak equivalence and the others are monomorphisms.

Notice that in full ∞-Chern-Weil theory the fundamental object of interest is really exp(𝔤) diff\exp(\mathfrak{g})_{diff} – the object of pseudo-connections. The other objects only serve the purpose of picking particularly nice representatives:

the object exp(𝔤) CW\exp(\mathfrak{g})_{CW} may contain pseudo-connections, those for which at least the associated circle n-bundles with connection given by the \infty-Chern Weil homomorphism are genuine connections.

This distinction is important: over objects XX \in ∞LieGrpd that are not smooth manifolds but for instance orbifolds, the genuine connections for higher Lie algebras do not exhaust all nonabelian differential cocycles. This just means that not every differential class in this case does have a nice representative in the usual sense.

1-Morphisms: integration of infinitesimal gauge transformations

The 1-morphisms in exp(𝔤)(U)\exp(\mathfrak{g})(U) may be thought of as gauge transformations between 𝔤\mathfrak{g}-valued forms. We unwind what these look like concretely.

Definition

Given a 1-morphism in exp(𝔤)(X)\exp(\mathfrak{g})(X), represented by 𝔤\mathfrak{g}-valued forms

Ω (U×Δ 1)W(𝔤):A \Omega^\bullet(U \times \Delta^1) \leftarrow W(\mathfrak{g}) : A

consider the unique decomposition

A=A U+(A vert:=λdt), A = A_U + ( A_{vert} := \lambda \wedge d t) \; \; \,,

with A UA_U the horizonal differential form component and t:Δ 1=[0,1]t : \Delta^1 = [0,1] \to \mathbb{R} the canonical coordinate.

We call λ\lambda the gauge parameter . This is a function on Δ 1\Delta^1 with values in 0-forms on UU for 𝔤\mathfrak{g} an ordinary Lie algebra, plus 1-forms on UU for 𝔤\mathfrak{g} a Lie 2-algebra, plus 2-forms for a Lie 3-algebra, and so forth.

We describe now how this enccodes a gauge transformation

A 0(s=1)λA U(s=1). A_0(s=1) \stackrel{\lambda}{\to} A_U(s = 1) \,.
Observation

By the nature of the Weil algebra we have

ddsA U=d Uλ+[λA]+[λAA]++ι sF A, \frac{d}{d s} A_U = d_U \lambda + [\lambda \wedge A] + [\lambda \wedge A \wedge A] + \cdots + \iota_s F_A \,,

where the sum is over all higher brackets of the ∞-Lie algebra 𝔤\mathfrak{g}.

In Cartan calculus for 𝔤\mathfrak{g} an ordinary one writes the corresponding second Ehremsnn condition ι sF A=0\iota_{\partial_s} F_A = 0 equivalently

sA=ad λA. \mathcal{L}_{\partial_s} A = ad_\lambda A \,.
Definition

Define the covariant derivative of the gauge parameter to be

λ:=dλ+[Aλ]+[AAλ]+. \nabla \lambda := d \lambda + [A \wedge \lambda] + [A \wedge A \wedge \lambda] + \cdots \,.

In this notation we have

  • the general identity

    (1)ddsA U=λ+(F A) s \frac{d}{d s} A_U = \nabla \lambda + (F_A)_s
  • the horizontality or rheonomy constraint or second Ehresmann condition ι sF A=0\iota_{\partial_s} F_A = 0, the differential equation

    (2)ddsA U=λ. \frac{d}{d s} A_U = \nabla \lambda \,.

This is known as the equation for infinitesimal gauge transformations of an \infty-Lie algebra valued form.

Observation

By Lie integration we have that A vertA_{vert} – and hence λ\lambda – defines an element exp(λ)\exp(\lambda) in the ∞-Lie group that integrates 𝔤\mathfrak{g}.

The unique solution A U(s=1)A_U(s = 1) of the above differential equation at s=1s = 1 for the initial values A U(s=0)A_U(s = 0) we may think of as the result of acting on A U(0)A_U(0) with the gauge transformatin exp(λ)\exp(\lambda).

Examples

Lie algebra valued 1-forms

Proposition

(connections on ordinary bundles)

For 𝔤\mathfrak{g} an ordinary Lie algebra with simply connected Lie group GG and for BG conn\mathbf{B}G_{conn} the groupoid of Lie algebra-valued forms we have an isomorphism

τ 1exp(𝔤) conn=BG conn \tau_1 \exp(\mathfrak{g})_{conn} = \mathbf{B}G_{conn}
Proof

To see this, first note that the sheaves of objects on both sides are manifestly isomorphic, both are the sheaf of Ω 1(,𝔤)\Omega^1(-,\mathfrak{g}). For morphisms, observe that for a form Ω (U×Δ 1)leftarowW(𝔤):A\Omega^\bullet(U \times \Delta^1) \leftarow W(\mathfrak{g}) : A which we may decompose into a horizontal and a verical pice as A=A U+lamndadtA = A_U + \lamnda \wedge d t the condition ι tF A=0\iota_{\partial_t} F_A = 0 is equivalent to the differential equation

tA=d Uλ+[λ,A]. \frac{\partial}{\partial t} A = d_U \lambda + [\lambda, A] \,.

For any initial value A(0)A(0) this has the unique solution

A(t)=g(t) 1(A+d U)g(t), A(t) = g(t)^{-1} (A + d_{U}) g(t) \,,

where g:[0,1]Gg : [0,1] \to G is the parallel transport of λ\lambda:

t(g (t) 1(A+d U)g(t)) = g(t) 1(A+d U)λg(t)g(t) 1λ(A+d U)g(t) \begin{aligned} & \frac{\partial}{\partial t} \left( g_(t)^{-1} (A + d_{U}) g(t) \right) \\ = & g(t)^{-1} (A + d_{U}) \lambda g(t) - g(t)^{-1} \lambda (A + d_{U}) g(t) \end{aligned}

(where for ease of notaton we write actions as if GG were a matrix Lie group).

In particular this implies that the endpoints of the path of 𝔤\mathfrak{g}-valued 1-forms are related by the usual cocycle condition in BG conn\mathbf{B}G_{conn}

A(1)=g(1) 1(A+d U)g(1). A(1) = g(1)^{-1} (A + d_U) g(1) \,.

In the same fashion one sees that given 2-cell in exp(𝔤)(U)\exp(\mathfrak{g})(U) and any 1-form on UU at one vertex, there is a unique lift to a 2-cell in exp(𝔤) conn\exp(\mathfrak{g})_{conn}, obtained by parallel transporting the form around. The claim then follows from the previous statement of Lie integration that τ 1exp(𝔤)=BG\tau_1 \exp(\mathfrak{g}) = \mathbf{B}G.

Lie 2-algebra valued forms

Ordinary nn-forms and the de Rham complex

For nn \in \mathbb{N}, n1n \geq 1 we have that b n1b^{n-1}\mathbb{R}-valued differential forms are in natural bijection to ordinary closed differential forms in degree nn

Observation

Notice that under addition of differential forms, exp(b n1) conn\exp(b^{n-1}\mathbb{R})_{conn} is over each UCartSpU \in CartSp an abelian simplicial group.

Under the Dold-Kan correspondence Ch +ΞNsAbCh_\bullet^+ \stackrel{\overset{N}{\leftarrow}}{\underset{\Xi}{\to}} sAb we may therefore identify exp(b n1) conn\exp(b^{n-1}\mathbb{R})_{conn} with a presheaf Nexp(b n1) connN \exp(b^{n-1}\mathbb{R})_{conn} of chain complexes.

Proposition

The degreewise fiber integration of differential forms over simplices constitutes a morphism

Δ :Nexp(b n1) conn(C (,)d dRΩ 1()d dRd dRΩ n()). \int_{\Delta^\bullet} : N\exp(b^{n-1}\mathbb{R})_{conn} \to \left( C^\infty(-, \mathbb{R}) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n(-) \right) \,.

that is a weak equivalence.

This is shown at circle n-bundle with connection – from Lie intgeration based on the discussion at ∞-Lie groupoid – Lie-integrated ∞-groups – differential coefficients.

Supergravity fields

What is called an “extended soft group manifold” in the literature on the D'Auria-Fre formulation of supergravity is really precisely a collection of \infty-Lie algebroid valued forms with values in a super \infty-Lie algebra such as the supergravity Lie 3-algebra (for 11-dimensional supergravity). The way curvature and Bianchi identity are read off from “extded soft group manifolds” in this literature is – apart from this difference in terminology – precisely what is described above.

References

The (obvious but conceptually important) observation that Lie algebra-valued 1-forms regarded as morphisms of graded vector spaces Ω (X) 1𝔤 *:A\Omega^\bullet(X) \leftarrow \wedge^1 \mathfrak{g}^* : A are equivalently morphisms of dg-algebras out of the Weil algebra Ω (X)W(𝔤):A\Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A and that one may think of as the identity W(𝔤)W(𝔤):IdW(\mathfrak{g}) \leftarrow W(\mathfrak{g}) : Id as the universal 𝔤\mathfrak{g}-connection appears in early articles for instance highlighted on p. 15 of

  • Franz W. Kamber; Philippe Tondeur, Semisimplicial Weil algebras and characteristic classes for foliated bundles in Čech cohomology , Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, pp. 283–294. Amer. Math. Soc., Providence, R.I., (1975).

following Eli Cartan’s influential work (see Weil algebra for more references).

The (evident) generalization to Weil algebras of ∞-Lie algebras and ∞-Lie algebroids is considered explicitly in

  • Hisham Sati, Urs Schreiber, Jim Staasheff, L L_\infty-algebra valued connections (web)

but – somewhat implicitly – this construction appears earlier, notably in the D'Auria-Fre formulation of supergravity. A collection of such precursors to these notions is collected at

The structure of the formula (2) for infinitesimal gauge transformations of higher forms is widely known in the literature on supergravity and string theory, if maybe not formalized in terms of \infty-Lie algebra theory as we do here. One exception is the remarkable book

In this old book no \infty-Lie algebras are mentioned explicitly, but the dg-algebra computations that are considered are easily seen to be precisely their Chevalley-Eilenberg algebra-incarnations.

The authors use the term extended soft group manifold for what here we identify as an \infty-Lie algebra valued form TXinn(𝔤)T X \to inn(\mathfrak{g}).

With this terminological translation understood, and observing that all their constructions straightforwardly generalize to more general dg-algebras than these authors conisder explicitly, we find that

  • our equation (1) for the possibly shifted gauge transformation is their equation I.3.136;

  • our equation (2) for the genuine gauge transfomation is their equation for horizontal or rheonomic gauge transformations III.3.23 .

In fact their full rheonomy constraint III.3.32 is essentialy the same horizontality constraint, but applied not just to the 1-simplex Δ 1\Delta^1, but to the super simplex Δ 1p\Delta^{1|p}.

Revised on October 10, 2013 02:51:53 by Urs Schreiber (89.204.135.176)