Domenico Fiorenza Flat infinity-connections

Contents

Idea

If a connection $\nabla$ on a principal $G$-bundle $E\to X$ is locally represented by the 1-form $\omega$ with values in the Lie algebra $\mathfrak{g}$, then the connection is flat if and only if the curvature 2-form $F_\nabla=d\omega+\frac{1}{2}[\omega,\omega]$ vanishes, that is, if $\omega$ 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 $\omega$ to an infinitesimal 1-simplex in $X$, then the restriction of $\omega$ is clearly a solution to the Maurer-Cartan equation. One can say that $\nabla$ 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 $F_\nabla$ 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 $\mathfrak{g}$ in degree -1. More precisely this amounts to consider the 2-Lie algebra $inn(\mathfrak{g})$ given by the cone of the identity $\mathfrak{g}\to\mathfrak{g}$.

The Maurer-Cartan equation for $inn(\mathfrak{g})$ coincides with the original one on the 1-simplex (since only degree 1 elements are 1-forms with values in $\mathfrak{g}$. But on the 2-simplex we would have, in addition to these elements, also 2-forms with coefficients in $\mathfrak{g}$, and the Maurer-Cartan equation on the 2-simplex takes the form $d\omega+\frac{1}{2}[\omega,\omega]-F=0$. So the original equation telling that $\nabla$ had curvature $F_\nabla$ is now equivalent to say that $\nabla+F_\nabla$ 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 $n$-flat connection on an $n$-bundle) can be seen as a flat connection on a suitable $\infty$-bundle.

The Chevally-Eilenberg algebra and curvatures

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

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

where $G$ is the Lie group that integrates $\mathfrak{g}$.

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

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

An arbitrary (i.e. non necessarily flat) $\mathfrak{g}$-connection on $X$ can still be seen as a morphism of graded commutative algebras $CE(\mathfrak{g})\to \Omega^*(X)$, 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

then the map

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 $dim X$ 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 $\mathfrak{g}$ is a 1-Lie algebra. Moving to $n$-Lie algebras nontrivial higher obstructions appear.

The canonical vertical flat connection

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

Differentiating this, we get a Lie algebroid morphism

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

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

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

is called the Weil algebra of $\mathfrak{g}$. 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 $(\infty,1)$ categories between (higher) Lie algebras and dg-commutative algebras. Therefore to the fibration sequence

it corresponds the cofibration sequence

where $inv(\mathfrak{g})=CE(\Sigma\mathfrak{g})$ is the dg-algebra of invariant polynomials on $\mathfrak{g}$. 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

is homotopy equivalent to the zero map. Since $W(\mathfrak{g})$ is weakly equivalent to $0$, this is surely so, that is, up homotopy, the looked for lifting exists. This is nothing but the classical statement that any $G$-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 $X$ 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) $\mathfrak{a}$-valued connection over $X$ is a dg-commutative algebra morphism

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

where $\mathbf{B}A$ is the $\infty$-Lie groupoid integrating $\mathfrak{a}$. Curvatures of the connection are the datum of $tra_\omega$ on higher morphisms in $\Pi(X)$. 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 $2$ or higher. Note that if $\mathfrak{a}$ is an $n$-Lie algebroid, then one has curvature forms only in degrees from $2$ to $n$. 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 $\omega:CE(\mathfrak{a})\to \Omega^*(X)$.)

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

Preconnections

The notion of $\mathfrak{g}$-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

and, more in general, $k$-preconnection a dg-algebra morphism $CE(\mathfrak{a})\to \Omega^*(X)/\Omega^{\gt k}(X)$.

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

where $\mathcal{P}_k(X)$ is the sub-$\infty$-groupoid of $\Pi(X)$ whose $j$-morphisms are smooth maps $\Delta^j\to X$ which kill all differential forms on $X$ of degree greater than $k$.

In particular, ordinary $\mathfrak{g}$-connections of differential geometry are 1-preconnections. Other examples of 1-preconnections can be found in the literature on “abelian gerbes with connection but without curving”, where $\infty$-functors $\mathcal{P}_1(X) \to \mathbf{B}G$, for $G$ the 2-group $\mathbf{B}U(1)$ are considered. Also the truncated versions, i.e., involving the path n-groupoids $t_{\leq n}\mathcal{P}_k(X)$ are often met in existing literature. Note that $t_{\leq n}\mathcal{P}_k(X)\simeq \Pi_n(X)$ for $k\gt n$.

Maurer-Cartan equations

Let us go back to the original problem, and describe the integration problem in terms of Maurer-Cartan functors. If
$\mathfrak{g}$ is a Lie algebra, and $G$ is a Lie group integrating it, then a $\mathfrak{g}$ valued connection on a trivial principal $G$-bundle on a manifold $X$ is a 1-form $\omega$ on $X$ with values in $\mathfrak{g}$, i.e. a degree 1 element in the dgla $\Omega^\bullet(X)\otimes\mathfrak{g}$, where $\Omega^\bullet(X)$ is the de Rham algebra of $X$. Integrating $\omega$ gives a functor

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

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

This is very much a 0-1 situation: either we are blocked on $n=1$ or we have vanishing curvature. To cook something intersting with nontrivial curvature, let us move from Lie algebras to $n$-Lie algebras, which will be convenient to think as $L_\infty$-algebras concentrated in degrees $-(n-1),\dots,-1,0$. Getzler shows in his work on Lie theory for nilpotent $L_\infty$ algebras, that the Maurer-Cartan construction sketched above verbatim generalizes from Lie algebras to $n$-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 $\mathfrak{g}=\mathfrak{g}^{-1}\oplus\mathfrak{g}^0$, where $\mathfrak{g}^{i}$ is the subspace of degree $i$ elements. Since the higher brackets

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

Now, consider a manifold $X$ and the $L_\infty-algebra \mathfrak{g}\otimes \Omega^\bullet(X)$ (it is an $L_\infty$-algebra concentrated in degrees $[-1,\dim(X)]$). A flat connection $\omega$ on a trivial principal $G$-bundle on $X$ (where $G$ is the 2-group integrating $\mathfrak{g}$) is a solution of the Maurer-Cartan equation

on $\mathfrak{g}\otimes \Omega^\bullet(X)$. The 2-bracket and the 3-barcket come from the 2- and 3-bracket in $\mathfrak{g}$; the differential has a contribution from the 1-bracket of $\mathfrak{g}$ and one from the de Rham differential $d_\Omega$. A particularly simple and interesting case is when $\mathfrak{g}$ is the Lie 2-algebra $inn(\mathfrak{h})$, i.e., the Lie 2-algebra coming from the crossed module $\mathfrak{h}\xrightarrow{id}\mathfrak{h}\xrightarrow{ad}Der(\mathfrak{h})$, for a Lie algebra $\mathfrak{h}$. In this case, $\mathfrak{g}^0=\mathfrak{h}$, $\mathfrak{g}^{-1}=\mathfrak{h}[1]$, the 3-bracket vanishes, the 2-brackets are just the 2-brackets of $\mathfrak{h}$, and the differential is (up to a sign) the identity of $\mathfrak{h}$. A flat $inn(\mathfrak{h})$-connection on $X$ has the form $\omega=A+F$, with $A$ a 1-form on $X$ with coefficients in $\mathfrak{g}^0=\mathfrak{h}$, and $F$ a 2-form on $X$ with coefficients in $\mathfrak{g}^{-1}=\mathfrak{h}[1]$ (and so with some abuse we can think of $F$ as a 2-form with coefficients in $\mathfrak{h}$; 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 $\omega$ has the form

(recall that the 3-bracket of $inn(\mathfrak{h})$ vanishes). The $\Omega^\bullet(X)$-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,

i.e., $F$ is the curvature of $A$, and and

i.e., the Bianchi identity.

Flat 3-connections and Chern-Simons terms

(to be continued)

Gauge and homotopy equivalence of connections

(to be continued)