# nLab connection on a bundle

### Context

#### $\infty$-Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

## Theorems

#### Differential cohomology

differential cohomology

# Contents

## Idea

A connection on a bundle $P \to X$ – a principal bundle or an associated bundle like a vector bundle – is a rule that identifies fibers of the bundle along paths in the base space $X$.

There are several different but equivalent formalizations of this idea:

The usual textbook convention is to say just connection for the distribution of horizontal subspaces, and the objects of the other three approaches one calls more specifically covariant derivative, connection $1$-form and parallel transport.

In the remainder of this Idea-section we discuss a bit more how to understand connections in terms of parallel transport.

Given a smooth bundle $P \to X$, for instance a $G$-principal bundle or a vector bundle, a connection on $P$ is a prescription to associate with each path

$\gamma : x \to y$

in $X$ (which is a morphism in the path groupoid $\mathbf{P}_1(X)$) a morphism $tra(\gamma)$ between the fibers of $P$ over these points

$\array{ P_x &\stackrel{tra(\gamma)}{\to}& P_y \\ x &\stackrel{\gamma}{\to}& y }$

such that

• this assignment respects the structure on the fibers $P_x$ (for instance is $G$-equivariant in the case that $P$ is a $G$-bundle or that is linear in the case that $P$ is a vector bundle);

• this assignment is smooth in a suitable sense;

• this assignment is functorial in that for all pairs $x \stackrel{\gamma}{\to} y$, $y \stackrel{\gamma'}{\to} z$ of composable paths in $X$ we have

$\array{ P_x &\stackrel{tra(\gamma)}{\to}& P_y &\stackrel{tra(\gamma')}{\to}& P_z \\ x &\stackrel{\gamma}{\to}& y &\stackrel{\gamma'}{\to}& z } \;\;\; = \;\;\; \array{ P_x &\stackrel{tra(\gamma' \circ \gamma)}{\to}& P_z \\ x &\stackrel{\gamma'\circ \gamma}{\to}& z }$

In other words, a connection on $P$ is a functor

$tra : \mathbf{P}_1(X) \to At''(P)$

from the path groupoid of $X$ to the Atiyah Lie groupoid of $P$ that is smooth in a suitable sense and splits the Atiyah sequence in that $\mathbf{P}_1(X) \stackrel{tra}{\to} At''(X) \to \mathbf{P}_1(X)$ (see the notation at Atiyah Lie groupoid).

Terminology

The functor $tra$ is called the parallel transport of the connection. This terminology comes from looking at the orbits of all points in $P$ under $tra$ (i.e. from looking at the category of elements of $tra$): these trace out paths in $P$ sitting over paths in $X$ and one thinks of the image of a point $p \in P_x$ under $tra(\gamma)$ as the result of propagating $p$ parallel to these curves in $P$.

Flat connections

It may happen that the assignment $tra : \gamma \mapsto tra(\gamma)$ only depends on the homotopy class of the path $\gamma$ relative to its endpoints $x, y$. In other words: that $tra$ factors through the functor $P_1(X) \to \Pi_1(X)$ from the path groupoid to the fundamental groupoid of $X$. In that case the connection is called a flat connection.

### More concrete picture

By Lie differentiation the functor $tra$, i.e. by looking at what it does to very short pieces of paths, one obtains from it a splitting of the Atiyah Lie algebroid sequence, which is a morphism

$\nabla : T X \to at(P)$

of vector bundles. Locally on $X$ – meaning: when everything is pulled back to a cover $Y \to X$ of $X$ – this is a $Lie(G)$-valued 1-form $A \in \Omega^1(Y, Lie(G))$ with certain special properties.

In particular, since every $G$-principal bundle canonically trivializes when pulled back to its own total space $P$, a connection in this case gives rise to a 1-form $A \in \Omega^1(P)$ satisfying two conditions. This data is called an Ehresmann connection.

If instead $P = E$ is a vector bundle, then the two conditions satisfies by $A$ imply that it defines a linear map

$\nabla : \Gamma(E) \to \Omega^1(X) \otimes \Gamma(E)$

from the space $\Gamma(E)$ of section of $E$ that satisfies the properties of a covariant derivative.

If again the connection is flat, then this is manifestly the datum of a Lie infinity-algebroid representation of the tangent Lie algebroid $T X$ of $X$ on $E$: it defines the action Lie algebroid which is the Lie version of the Lie groupoid classified by the parallel transport functor.

### More abstract picture

Recall from the discussion at $G$-principal bundle that the $G$-bundle $P \to X$ is encoded in a a suitable morphism

$X \to \mathbf{B}G$

(namely a morphism in the right (infinity,1)-category which may be represented by an anafunctor).

It turns out that similarly suitable morphisms

$\mathbf{P}_1(X) \to \mathbf{B}G$

encode in one step the $G$-bundle together with its parallel transport functor.

This is described in great detail in the reference by Schreiber–Waldorf below.

(…am running out of time… )

## Definition

Let $G$ be a Lie group. We recall briefly the following discussion of $G$-principal bundles. For an in-depth discussion see Smooth∞Grpd.

Write

$\mathbf{B}G : U \mapsto ( Hom_{Diff}(U,G) \stackrel{\to}{\to} *)$

for the functor that sends a Cartesian space $U$ to the delooping groupoid of the group of $G$-valued smooth functions on $U$: the groupoid with a single object and the group $Hom_{Diff}(U,G)$ of maps as its set of morphisms.

This is a groupoid-valued sheaf on the site CartSp${}_{smooth}$ and in fact is a (2,1)-sheaf/stack.

For $X$ a paracompact smooth manifold, we may also regard it as a (2,1)-sheaf on CartSp in an evident way.

###### Observation

The groupoid $G Bund(X)$ of $G$-principal bundles on $X$ is equivalent to the hom-groupoid

$\mathbf{H}(X,\mathbf{B}G) \simeq G Bund(X)$

taken in the (2,1)-topos of (2,1)-sheaves on CartSp${}_{smooth}$.

A detailed discussion of this is at Smooth∞Grpd in the section on Lie groups.

Now write $\mathfrak{g}$ for the Lie algebra of $\mathfrak{g}$. Then consider the functor

$\mathbf{B} G_{conn} : U \mapsto [\mathbf{P}_1(U),\mathbf{B}G] = \left\{ A \stackrel{g}{\to} (g^{-1} A g + g^{-1} d g) | A \in \Omega^1(U,\mathfrak{g})\,, g \in C^\infty(U,G) \right\}$

that sends a Cartesian space $U$ to the groupoid of Lie-algebra valued 1-forms over $U$.

There is an evident morphism of (2,1)-sheaves

$\mathbf{B}G_{conn} \to \mathbf{B}G$

that forgets the 1-forms on each object $U$.

###### Definition

(connection)

A connection on a smooth $G$-principal bundle $g : X \to \mathbf{B}G$ is a lift $\nabla$ to $\mathbf{B}G_{conn}$

$\array{ && \mathbf{B}G_{conn} \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G } \,.$

The groupoid of $G$-principal bundles with connection on $X$ is

$G Bund_\nabla(X) := Hom(X,\mathbf{B}G_{conn}) \,.$

Explicitly, a morphism $g : X \to \mathbf{B}G$ is a nonabelian Cech cohomology cocycle on $X$ with values in $G$:

1. a choice of good open cover $\{U_i \to X\}$ of $X$;

2. a collection of smooth functions $(g_{i j} \in C^\infty(U_i \cap U_j), G)$

such that on $U_i \cap U_j \cap U_k$ the equation

• $g_{i j} g_{j k} = g_{i k}$

holds.

A lift $\nabla : X \to \mathbf{B}G_{conn}$ of this is in addition

1. a choice of Lie-algebra valued 1-forms $(A_i \in \Omega^1(U_i, \mathfrak{g}))$

such that on $U_i \cap U_j$ the equation

• $A_j = g^{-1} A_i g + g^{-1} d g$

holds, where on the right we have the pullback $g^* \theta$ of the Maurer-Cartan form on $G$ (see there).

## Properties

### Existence of connections

###### Definition

(existence of connections)

Every $G$-principal bundle admits a connection. In other words, the forgetful functor

$Hom(X, \bar \mathbf{B}G_{conn}) \to Hom(X,\mathbf{B}G)$
###### Proof

Choose a partition of unity $(\rho_i \in C^\infty(X,\mathbb{R}))$ subordinate to the good open cover $\{U_i \to X\}$ with respect to which a given cocycle $g : X \to \mathbf{B}G$ is expressed:

• $(x \;not\; in\; U_i) \Rightarrow \rho_i(x) = 0$;

• $\sum_i \rho_i = 1$.

Then set

$A_i := \sum_{i_0} \rho_{i_0}|_{U_{i_0}} (g_{i_0 i}|^{-1}_{U_{i_0}}) d_{dR} (g_{i_0 i}|_{U_{i_0}}) \,.$

By slight abuse of notation we shall write this and similar expressions simply as

$A_i := \sum_{i_0} \rho_{i_0}(g_{i_0 i}^{-1} d_{dR} g_{i_0 i}) \,.$

Using the that $(g_{i j})$ satisfies its cocycle condition, one checks that this satisfies the cocycle condition for the 1-forms:

\begin{aligned} A_j - g_{i j}^{-1} A_i g_{i j} &= \sum_{i_0} \rho_{i_0} ( g_{i_0 j}^{-1} d g_{i_0 j} - ( g_{i_0 i} g_{i j}) ^{-1} (d g_{i_0 i}) g_{i j} ) \\ & = \sum_{i_0} \rho_{i_0} ( g_{i j}^{-1} d g_{i j} ) \\ & = g_{i j}^{-1} d g_{i j} \end{aligned} \,.

## Special cases

### Connections on the tangent bundle

Connections on tangent bundles are also called affine connections, or Levi-Civita connections.

They play a central role for instance on Riemannian manifolds and pseudo-Riemannian manifolds. From the end of the 19th century and the beginning of the 20th centure originates a language to talk about these in terms of Christoffel symbols.

### Connections in physics

In physics connections on bundles model gauge fields.

For more on this see higher category theory and physics.

## Generalizations

### Superconnections

Generalizing the parallel transport definition from ordinary manifolds to supermanifolds yields the notion of superconnection.

### Simons-Sullivan structured bundles

When the notion of connection on a principal bundle is slightly coarsened, i.e. when more connections are regarded as being ismorphic than usual, one arrives at a structure called a Simons-Sullivan structured bundle. This has the special property that for $G = U$ the unitary group, the corresponding Grothendieck group of such bundles is a model for differential K-theory.

### Connections on a principal $\infty$-bundle

higher Atiyah groupoid

higher Atiyah groupoid:standard higher Atiyah groupoidhigher Courant groupoidgroupoid version of quantomorphism n-group
coefficient for cohomology:$\mathbf{B}\mathbb{G}$$\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})$$\mathbf{B} \mathbb{G}_{conn}$
type of fiber ∞-bundle:principal ∞-bundleprincipal ∞-connection without top-degree connection formprincipal ∞-connection

gauge field: models and components

## References

A classical textbook reference is

The formulation of connections in terms of their smooth parallel transport functors is in

based on a series of classical observations. I farily comprehensive commented list of related references is here:

Basic facts about connections, such as the existence proof in terms of Cech cocycles, are collected in the brief lecture note

Revised on July 1, 2014 02:50:48 by Urs Schreiber (89.204.138.13)