Domenico Fiorenza Flat infinity-connections

Contents

Contents

Idea

If a connection βˆ‡\nabla on a principal GG-bundle Eβ†’XE\to X is locally represented by the 1-form Ο‰\omega with values in the Lie algebra 𝔀\mathfrak{g}, then the connection is flat if and only if the curvature 2-form F βˆ‡=dΟ‰+12[Ο‰,Ο‰]F_\nabla=d\omega+\frac{1}{2}[\omega,\omega] vanishes, that is, if Ο‰\omega is a solution of the Maurer-Cartan equation.

Notice that one can see curvature only on 2-dimensional paths (since it is a 2-form): if one restricts Ο‰\omega to an infinitesimal 1-simplex in XX, then the restriction of Ο‰\omega is clearly a solution to the Maurer-Cartan equation. One can say that βˆ‡\nabla is 1-flat.

Then, moving to an infinitesimal 2-simplex one sees that the connection is (generally) not 2-flat: holonomy along two sides of the 2-simplex is not the same thing as holonomy along the third side. Not the same, but in a very precise way: the curvature F βˆ‡F_\nabla exactly measures the gap to go from a horn of the 2-simplex to the third edge. This is very 2-categorical, and suggests one can cure the lack of flatness of the original connection by adding a copy of 𝔀\mathfrak{g} in degree -1. More precisely this amounts to consider the 2-Lie algebra inn(𝔀)inn(\mathfrak{g}) given by the cone of the identity 𝔀→𝔀\mathfrak{g}\to\mathfrak{g}.

The Maurer-Cartan equation for inn(𝔀)inn(\mathfrak{g}) coincides with the original one on the 1-simplex (since only degree 1 elements are 1-forms with values in 𝔀\mathfrak{g}. But on the 2-simplex we would have, in addition to these elements, also 2-forms with coefficients in 𝔀\mathfrak{g}, and the Maurer-Cartan equation on the 2-simplex takes the form dΟ‰+12[Ο‰,Ο‰]βˆ’F=0d\omega+\frac{1}{2}[\omega,\omega]-F=0. So the original equation telling that βˆ‡\nabla had curvature F βˆ‡F_\nabla is now equivalent to say that βˆ‡+F βˆ‡\nabla+F_\nabla is flat.

In other words, what seemed a non-flat connection was so since we were not seeing the 2-bundle, but only a 1-bundle approximation. And on a 1-bundle one can only clearly see up to 1-simplices, where the original connection was actually flat. Once curvature has come in, we can repeat the argument: now we have a 2-flat connection and can test it on the 3-simplex. If it has 3-curvature, that will presumibly be because we are not seeing the 3-bundle, yet. So I find it natural to conjecture that any connection on a principal bundle (and more generally any nn-flat connection on an nn-bundle) can be seen as a flat connection on a suitable ∞\infty-bundle.

The Chevally-Eilenberg algebra and curvatures

For 𝔀\mathfrak{g} a Lie algebra, a flat 𝔀\mathfrak{g}-valued connection (on a trivial bundle) is a morphism of Lie algebroids

Ο‰:TX→𝔀\omega : T X \to \mathfrak{g}

This can be integrated to a morphism of Lie groupoids, where it becomes

tra Ο‰:Ξ (X)β†’BG tra_\omega : \Pi(X) \to \mathbf{B}G

where GG is the Lie group that integrates 𝔀\mathfrak{g}.

An extremely fruitful point of view on the algebroid morphism above is to look at it in the opposite direction, i.e. as a morphism of (sheaves of) differential graded commutative algebras

(CE(𝔀),d CE)β†’(Ξ© *(X),d dR) (CE(\mathfrak{g}),d_{CE})\to (\Omega^*(X),d_{dR})

where the algebra on the left-hand side is the Chevalley-Eilenberg algebra of 𝔀\mathfrak{g} and Ξ© *(X)\Omega^*(X) is the de Rham algebra of XX.

An arbitrary (i.e. non necessarily flat) 𝔀\mathfrak{g}-connection on XX can still be seen as a morphism of graded commutative algebras CE(𝔀)β†’Ξ© *(X)CE(\mathfrak{g})\to \Omega^*(X), but this will in general not be compatible with the differentials. Obstruction to this compatibillity is precisely the curvature of the connection. See details at curvature of ∞-Lie algebroid valued differential forms in Urs’ personal area.

However, if we postcompone with the projection

Ξ© *(X)/Ξ© >1(X) \Omega^*(X)/\Omega^{\gt1}(X)

then the map

(CE(𝔀),d CE)β†’(Ξ© *(X)/Ξ© >1(X),d dR) (CE(\mathfrak{g}),d_{CE})\to (\Omega^*(X)/\Omega^{\gt1}(X),d_{dR})

is a morphism of differential graded commutative algebras. This because curvature is a 2-form. So we can look at curvature as the obstruction to lifting in the following diagram of dg-commutative algebras

This lifting problem can be factorized into dimXdim X lifting problems as follows:

And what we said about curvature can be rephrased as follows: once the first obstruction vanishes, all higher obstruction vanish. This happens because 𝔀\mathfrak{g} is a 1-Lie algebra. Moving to nn-Lie algebras nontrivial higher obstructions appear.

The canonical vertical flat connection

While on the base XX of a principal GG-bundle E→XE\to X there is genrally no flat GG-connection, on each fiber of E→XE\to X there is a canonical one. More precisely, let Π vert(E)\Pi^{vert}(E) be the subgroupoid of Π(E)\Pi(E) consisting of vertical paths (i.e., of paths whose tangent vector at each point is a vertical vector with respect to the projection π:E→X\pi:E\to X). By the very definition of GG-principal bundle, there is a tautological morphism

Ξ  vert(E)β†’BG \Pi^{vert}(E)\to \mathbf{B}G

Differentiating this, we get a Lie algebroid morphism

T vertE→𝔀, T^{vert}E\to \mathfrak{g},

the canonical vertical flat connection on EE. In the dual picture we have a canonical dg-commutative algebra morphism

CE(𝔀)β†’Ξ© vert *(E), CE(\mathfrak{g})\to\Omega^*_{vert}(E),

where the dg-algebra Ξ© vert *(E)\Omega^*_{vert}(E) of vertical differential forms on EE is the quotient of Ξ© *(E)\Omega^*(E) by the sub-dg-algebra of differential form vanishing on vertical multivectors.

inn(𝔀)inn(\mathfrak{g})-connections

A first appearance of higher Lie algebras comes from noticing that the lower corner of the above diagram can be completed to a (homotopy) commutative diagram

where inn(𝔀)inn(\mathfrak{g}) is the 2-Lie algebra given by the cone over the identity of 𝔀\mathfrak{g}. Indeed (by definition of cone) the 2-Lie algebra inn(𝔀)inn(\mathfrak{g}) is contractible, and therefore so is its Chevalley-Eilenberg algebra. It’s worth giving this dg-algebra a name:

W(𝔀)=CE(inn(𝔀)) W(\mathfrak{g})=CE(inn(\mathfrak{g}))

is called the Weil algebra of 𝔀\mathfrak{g}. Our lifting problem has now been translated to a (homotopy) factorization problem:

To shed a bit of light on this problem, recall that the Chevalley-Eilenberg construction induces an opposite equivalence of (∞,1)(\infty,1) categories between (higher) Lie algebras and dg-commutative algebras. Therefore to the fibration sequence

it corresponds the cofibration sequence

where inv(𝔀)=CE(Σ𝔀)inv(\mathfrak{g})=CE(\Sigma\mathfrak{g}) is the dg-algebra of invariant polynomials on 𝔀\mathfrak{g}. This way we see that the diagram of our lifting problem is a part of a larger homotopy commutative diagram of dg-commutative algebras:

and the universal property of push-out tells us that the seeked lifting exists if and only if the composite map

inv(𝔀)⟢W(𝔀)⟢cΞ© *(X) inv(\mathfrak{g})\longrightarrow W(\mathfrak{g})\stackrel{c}{\longrightarrow} \Omega^*(X)

is homotopy equivalent to the zero map. Since W(𝔀)W(\mathfrak{g}) is weakly equivalent to 00, this is surely so, that is, up homotopy, the looked for lifting exists. This is nothing but the classical statement that any GG-connection on a trivial principal bundle is gauge-equivalent to the trivial connection. But seeing things from the abstract homotopy nonsense above clearly tells us which is the right setup if we want to consider curvature, and not only curvature up to homotopy: twisted cohomology.

∞\infty-Lie algebroid and ∞\infty-connections

From the above discussion we learnt two things:

  1. there’s no point in starting with a Lie algebroid over XX rather than with an arbitrary ∞\infty-Lie algebroid: even if one starts with Lie algebroids, higher structures naturally come in.
  2. from the higher categorical point of view every connection is flat (what seems to be non flat is actually a partial datum of something flat).

This motivates the following definition:

A (local) π”ž\mathfrak{a}-valued connection over XX is a dg-commutative algebra morphism

Ο‰:CE(π”ž)β†’Ξ© *(X) \omega:CE(\mathfrak{a})\to \Omega^*(X)

Equivalently, it is an ∞\infty-Lie groupoid morphism

tra Ο‰:Ξ (X)β†’BA, tra_\omega :\Pi(X)\to \mathbf{B}A,

where BA\mathbf{B}A is the ∞\infty-Lie groupoid integrating π”ž\mathfrak{a}. Curvatures of the connection are the datum of tra Ο‰tra_\omega on higher morphisms in Ξ (X)\Pi(X). On the dg-algebra side, they correspond to the datum of curvature forms; since by definition curvature is related to higher morphisms, one has curvature forms in degree 22 or higher. Note that if π”ž\mathfrak{a} is an nn-Lie algebroid, then one has curvature forms only in degrees from 22 to nn. The connection is flat if all of its curvature forms vanish.

(Deatils to be added, here, to show how to express curvatures in term of the dg-algebra morphism Ο‰:CE(π”ž)β†’Ξ© *(X)\omega:CE(\mathfrak{a})\to \Omega^*(X).)

With this definition, what is called a 𝔀\mathfrak{g}-valued connection Ο‰\omega in classic differential geometry (with 𝔀\mathfrak{g} a Lie algebra), is the partial datum for a true inn(𝔀)inn(\mathfrak{g})-connection Ο‰ ∞\omega_\infty. Since inn(𝔀)inn(\mathfrak{g}) is a 2-Lie algebra when 𝔀\mathfrak{g} is a Lie algebra, the connection Ο‰ ∞\omega_\infty has exactly one curvature form concentrated in degree 22. This 22-form is nothing but the usual curvature 2-form of Ο‰\omega from classic differential geometry.

Preconnections

The notion of 𝔀\mathfrak{g}-valued connection is too well rooted in classic differential geometry, to just repalce it with another notion as above and forget about it. So it is convenient and reasonable to keep the classical notion, slightly changing its name in order to stress that it only gives a partial picture. So, let us call preconnection a dg-algebra morphism

CE(π”ž)β†’Ξ© *(X)/Ξ© >1(X) CE(\mathfrak{a})\to \Omega^*(X)/\Omega^{\gt 1}(X)

and, more in general, kk-preconnection a dg-algebra morphism CE(π”ž)β†’Ξ© *(X)/Ξ© >k(X)CE(\mathfrak{a})\to \Omega^*(X)/\Omega^{\gt k}(X).

In the dual (integrated) picture, a kk-preconnection is a morphism of ∞\infty-groupoids

𝒫 k(X)β†’BA, \mathcal{P}_k(X)\to \mathbf{B}A,

where 𝒫 k(X)\mathcal{P}_k(X) is the sub-∞\infty-groupoid of Ξ (X)\Pi(X) whose jj-morphisms are smooth maps Ξ” jβ†’X\Delta^j\to X which kill all differential forms on XX of degree greater than kk.

In particular, ordinary 𝔀\mathfrak{g}-connections of differential geometry are 1-preconnections. Other examples of 1-preconnections can be found in the literature on β€œabelian gerbes with connection but without curving”, where ∞\infty-functors 𝒫 1(X)β†’BG\mathcal{P}_1(X) \to \mathbf{B}G, for GG the 2-group BU(1)\mathbf{B}U(1) are considered. Also the truncated versions, i.e., involving the path n-groupoids t ≀n𝒫 k(X)t_{\leq n}\mathcal{P}_k(X) are often met in existing literature. Note that t ≀n𝒫 k(X)≃Π n(X)t_{\leq n}\mathcal{P}_k(X)\simeq \Pi_n(X) for k>nk\gt n.

Maurer-Cartan equations

Let us go back to the original problem, and describe the integration problem in terms of Maurer-Cartan functors. If
𝔀\mathfrak{g} is a Lie algebra, and GG is a Lie group integrating it, then a 𝔀\mathfrak{g} valued connection on a trivial principal GG-bundle on a manifold XX is a 1-form Ο‰\omega on XX with values in 𝔀\mathfrak{g}, i.e. a degree 1 element in the dgla Ξ© β€’(X)βŠ—π”€\Omega^\bullet(X)\otimes\mathfrak{g}, where Ξ© β€’(X)\Omega^\bullet(X) is the de Rham algebra of XX. Integrating Ο‰\omega gives a functor

Hol Ο‰:𝒫 1(X)β†’BG. Hol_\omega:\mathcal{P}_1(X)\to \mathbf{B}G.

This can be seen as an ∞\infty-functor from the nerve of 𝒫 1(X)\mathcal{P}_1(X) to the nerve of BG\mathbf{B}G. It is a remarkable result by Hinich that this latter nerve is homotopy equivalent to the Kan complex {(π”€βŠ—Ξ© Ξ” n β€’)} nβˆˆβ„•\{(\mathfrak{g}\otimes\Omega^\bullet_{\Delta_n})\}_{n\in\mathbb{N}}, where MC denotes the set of solutions of the Maurer-Cartan equation in a dgla. Using this model for N(BG)N(\mathbf{B}G), the functor Hol Ο‰Hol_\omega is straightforwardly described: a path Ξ³:[0,1]β†’X\gamma\colon [0,1]\to X is mapped to the degree 1 element Ξ³ *(Ο‰)\gamma^*(\omega) in π”€βŠ—Ξ© 1 β€’\mathfrak{g}\otimes \Omega^\bullet_1. This element is automatically a Maurer-Cartan element, since dΟ‰+12[Ο‰,Ο‰]d\omega+\frac{1}{2}[\omega,\omega] is a 2-form and so it vanishes when pulled back via Ξ³\gamma. By the very same reason, Ξ³ *(Ο‰)\gamma^*(\omega) is a Maurer-Cartan element in π”€βŠ—Ξ© n β€’\mathfrak{g}\otimes \Omega^\bullet_n for every thin map Ξ³:Ξ” nβ†’X\gamma\colon \Delta_n\to X (by definition of thin map). Hence we have the seeked Hol Ο‰Hol_\omega functor.

Trying to lift Hol Ο‰Hol_\omega to a functor stemming from 𝒫 2(X)\mathcal{P}_2(X) we meet the curvature obstruction: for a generic map Ξ³:Ξ” 2β†’X\gamma:\Delta_2\to X, the element Ξ³ *(Ο‰)\gamma^*(\omega) is not a Maurer-Cartan element in π”€βŠ—Ξ© 2 β€’\mathfrak{g}\otimes \Omega^\bullet_2 unless Ο‰\omega is flat. Once we require Ο‰\omega to be flat, we are done: what we are asking is that \omega is a Maurer-Cartan element in π”€βŠ—Ξ© β€’(X)\mathfrak{g}\otimes \Omega^\bullet(X), and so it will produce Maurer-Cartan elements wherever we pull it back. In particular the functor Hol Ο‰Hol_\omega can be lifted to 𝒫 n(X)\mathcal{P}_n(X) for any nn, and so to a functor from Ξ (X)\Pi(X).

This is very much a 0-1 situation: either we are blocked on n=1n=1 or we have vanishing curvature. To cook something intersting with nontrivial curvature, let us move from Lie algebras to nn-Lie algebras, which will be convenient to think as L ∞L_\infty-algebras concentrated in degrees βˆ’(nβˆ’1),…,βˆ’1,0-(n-1),\dots,-1,0. Getzler shows in his work on Lie theory for nilpotent L ∞L_\infty algebras, that the Maurer-Cartan construction sketched above verbatim generalizes from Lie algebras to nn-Lie algebras (it is possible some version of this result predates Getzler, it seems I’m quite unable to give proper credits for results..). So let us take for instance a 2-Lie algebra 𝔀=𝔀 βˆ’1βŠ•π”€ 0\mathfrak{g}=\mathfrak{g}^{-1}\oplus\mathfrak{g}^0, where 𝔀 i\mathfrak{g}^{i} is the subspace of degree ii elements. Since the higher brackets

[,…,] n:∧ n𝔀→𝔀[2βˆ’n] [,\dots,]_n\colon \wedge^n \mathfrak{g}\to\mathfrak{g}[2-n]

have degree 2βˆ’n2-n, for a 2-Lie algebra only the following brackets survive:

[] 1:𝔀 βˆ’1→𝔀 0 []_1: \mathfrak{g}^{-1} \to \mathfrak{g}^{0}
[] 2:𝔀 βˆ’1βŠ—π”€ 0→𝔀 βˆ’1 []_2: \mathfrak{g}^{-1}\otimes\mathfrak{g}^0 \to \mathfrak{g}^{-1}
[] 2:∧ 2𝔀 0→𝔀 0 []_2: \wedge^2\mathfrak{g}^0 \to \mathfrak{g}^{0}
[] 3:∧ 3𝔀 0→𝔀 βˆ’1 []_3: \wedge^3\mathfrak{g}^{0} \to \mathfrak{g}^{-1}

Now, consider a manifold XX and the L βˆžβˆ’algebraπ”€βŠ—Ξ© β€’(X)L_\infty-algebra \mathfrak{g}\otimes \Omega^\bullet(X) (it is an L ∞L_\infty-algebra concentrated in degrees [βˆ’1,dim(X)][-1,\dim(X)]). A flat connection Ο‰\omega on a trivial principal GG-bundle on XX (where GG is the 2-group integrating 𝔀\mathfrak{g}) is a solution of the Maurer-Cartan equation

dω+12[ω,ω]+16[ω,ω,ω]=0 d\omega+\frac{1}{2}[\omega,\omega]+\frac{1}{6}[\omega,\omega,\omega]=0

on π”€βŠ—Ξ© β€’(X)\mathfrak{g}\otimes \Omega^\bullet(X). The 2-bracket and the 3-barcket come from the 2- and 3-bracket in 𝔀\mathfrak{g}; the differential has a contribution from the 1-bracket of 𝔀\mathfrak{g} and one from the de Rham differential d Ξ©d_\Omega. A particularly simple and interesting case is when 𝔀\mathfrak{g} is the Lie 2-algebra inn(π”₯)inn(\mathfrak{h}), i.e., the Lie 2-algebra coming from the crossed module π”₯β†’idπ”₯β†’adDer(π”₯)\mathfrak{h}\xrightarrow{id}\mathfrak{h}\xrightarrow{ad}Der(\mathfrak{h}), for a Lie algebra π”₯\mathfrak{h}. In this case, 𝔀 0=π”₯\mathfrak{g}^0=\mathfrak{h}, 𝔀 βˆ’1=π”₯[1]\mathfrak{g}^{-1}=\mathfrak{h}[1], the 3-bracket vanishes, the 2-brackets are just the 2-brackets of π”₯\mathfrak{h}, and the differential is (up to a sign) the identity of π”₯\mathfrak{h}. A flat inn(π”₯)inn(\mathfrak{h})-connection on XX has the form Ο‰=A+F\omega=A+F, with AA a 1-form on XX with coefficients in 𝔀 0=π”₯\mathfrak{g}^0=\mathfrak{h}, and FF a 2-form on XX with coefficients in 𝔀 βˆ’1=π”₯[1]\mathfrak{g}^{-1}=\mathfrak{h}[1] (and so with some abuse we can think of FF as a 2-form with coefficients in π”₯\mathfrak{h}; we should avoid doing so, since degrees are relevant, but let us note this in order to make contact with a familiar picture).

Then the Maurer-Cartan equation for Ο‰\omega has the form

d Ωω+d 𝔀ω+12[Ο‰,Ο‰]=0 d_{\Omega}\omega+d_{\mathfrak{g}}\omega+\frac{1}{2}[\omega,\omega]=0

(recall that the 3-bracket of inn(π”₯)inn(\mathfrak{h}) vanishes). The Ξ© β€’(X)\Omega^\bullet(X)-degree makes a set of several equations out of the single equation above (one for each degree). So we actually end up with two equations, one for a 2-form, and one for a 3-form. Namely,

d Ξ© β€’(X)A+12[A,A]=F, d_{\Omega^\bullet(X)} A+ \frac{1}{2}[A,A]=F,

i.e., FF is the curvature of AA, and and

d Ξ© β€’(X)F+[A,F]=0 d_{\Omega^\bullet(X)} F +[A,F] =0

i.e., the Bianchi identity.

Flat 3-connections and Chern-Simons terms

(to be continued)

Gauge and homotopy equivalence of connections

(to be continued)

Discussion goes here.

Revised on March 5, 2010 at 13:55:30 by Domenico Fiorenza