# nLab connection on a 2-bundle

Contents

### Context

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

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

## Theorems

#### Differential cohomology

differential cohomology

# Contents

## Idea

The notion of connection on a 2-bundle generalizes the notion of connection on a bundle from principal bundles to principal 2-bundles / gerbes.

It comes with a notion of 2-dimensional parallel transport.

For an exposition of the concepts here see also at infinity-Chern-Weil theory introduction the section Connections on principal 2-bundles .

## Definition

For $G$ a Lie 2-group, a connection on a $G$-principal 2-bundle coming from a cocycle $g : X \to \mathbf{B}G$ is a lift of the cocycle to the 2-groupoid of Lie 2-algebra valued forms $\mathbf{B}G_{conn}$

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

## Properties

### On trivial 2-bundles

When the underlying principal 2-bundle over a smooth manifold $X$ is topologically trivial, then the connections on it are identified with Lie 2-algebra valued differential forms on $X$.

Recall from the discussion there what such form data looks like.

Let $\mathfrak{g}$ be some Lie 2-algebra. For instance for discussion of connections on $G$-gerbes ($G$ a Lie group) this would be the derivation Lie 2-algebra of the Lie algebra of $G$.

Let $\mathfrak{g}_0$ and $\mathfrak{g}_1$ be the two vector spaces involved and let

$\{t^a\} \,, \;\;\; \{b^i\}$

be a dual basis, respectively. The structure of a Lie 2-algebra is conveniently determined by writing out the most general Chevalley-Eilenberg algebra

$CE(\mathfrak{g}) \in cdgAlg_\mathbb{R}$

with these generators.

We thus have

$d_{CE(\mathfrak{g})} t^a = - \frac{1}{2}C^a{}_{b c} t^b \wedge t^c - r^a{}_i b^i$
$d_{CE(\mathfrak{g})} b^i = -\alpha^i_{a j} t^a \wedge b^j - r^i{}_{a b c} t^a \wedge t^b \wedge t^c \,,$

for collections of structure constants $\{C^a{}_{b c}\}$ (the bracket on $\mathfrak{g}_0$) and $\{r^i_a\}$ (the differential $\mathfrak{g}_1 \to \mathfrak{g}_0$) and $\{\alpha^i{}_{a j}\}$ (the action of $\mathfrak{g}_0$ on $\mathfrak{g}_1$) and $\{r_{a b c}\}$ (the “Jacobiator” for the bracket on $\mathfrak{g}_0$).

These constants are subject to constraints (the weak Jacobi identity and its higher coherence laws) which are precisely equivalent to the condition

$(d_{CE(\mathfrak{g})})^2 = 0 \,.$

Over a test space $U$ a $\mathfrak{g}$-valued form datum is a morphism

$\Omega^\bullet(U) \leftarrow W(\mathfrak{g}) : (A,B)$

from the Weil algebra $W(\mathfrak{g})$.

This is given by a 1-form

$A \in \Omega^1(U, \mathfrak{g}_0)$

and a 2-form

$B \in \Omega^2(U, \mathfrak{g}_1) \,.$

The curvature of this is $(\beta, H)$, where the 2-form component (“fake curvature”) is

$\beta^a = d_{dR} A^a + \frac{1}{2}C^a{}_{b c} A^b \wedge A^c + r^{a}_{i} B^i$

and whose 3-form component is

$H^i = d_{dR} B^i + \alpha^i{}_{a j} A^a \wedge B^j + r^i{}_{a b c} A^a \wedge A^b \wedge A^c \,.$

### Differential Čech cocycle data

We spell out the data of a connection on a 2-bundle over a smooth manifold $X$ with respect to a given open cover $\{U_i \to X\}$, following (FSS, SchreiberCohesive)

(…)

## Examples

### General

Connections on 2-bundles with vanishing 2-form curvature and arbitrary 3-form curvature are defined in terms of their higher parallel transport are discussed in

• Smooth Functors and Differential Forms, Homology, Homotopy Appl., 13(1), 143-203 (2011) (arXiv:0802.0663)

Connections on non-abelian gerbes and their holonomy, Theory and Applications of Categories, Vol. 28, 2013, No. 17, pp 476-540. (TAC, arXiv:0808.1923, web)

expanding on

Much further discussion and illustration and relation to tensor networks is in

Examples of 2-connections with vanishing 2-form curvature obtained from geometric quantization are discusssed in

• Olivier Brahic, On the infinitesimal Gauge Symmetries of closed forms (arXiv)

The cocycle data for 2-connections with coeffcients in automorphism 2-groups but without restrictions on the 2-form curvature have been proposed in

and

• Paolo Aschieri, Luigi Cantini, Branislav Jurco, Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory , Communications in Mathematical Physics Volume 254, Number 2 (2005) 367-400,(arXiv:hep-th/0312154).

A discussion of fully general local 2-connections is in

and the globalization is in

For a discussion of all this in a more comprehensive context see section xy of