nLab
connection on a 2-bundle

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 $g:X\to BG$ is a lift of the cocycle to the 2-groupoid of Lie 2-algebra valued forms $B{G}_{\mathrm{conn}}$

$$\begin{array}{ccc}& & B{G}_{\mathrm{conn}}\\ & {}^{\nabla }\nearrow & \downarrow \\ X& \stackrel{g}{\to }& BG\end{array}$$\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 $𝔤$ 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 ${𝔤}_0$ and ${𝔤}_1$ be the two vector spaces involved and let

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

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

with these generators.

We thus have

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

$$d_{\mathrm{CE}(𝔤)}{b}^i=-{\alpha }_{aj}^i{t}^a\wedge b^j-r_{abc}{t}^a\wedge t^b\wedge t^c,$$d_{CE(\mathfrak{g})} b^i = -\alpha^i_{a j} t^a \wedge b^j - r_{a b c} t^a \wedge t^b \wedge t^c \,,

for collections of structure constants $\{C^a{}_{bc}\}$ (the bracket on ${𝔤}_0$) and $\{r_a^i\}$ (the differential ${𝔤}_1\to mathgfrak{g}_0$) and $\{{alph}^i{}_{aj}\}$ (the action of ${𝔤}_0$ on ${𝔤}_1$) and $\{r_{abc}\}$ (the “Jacobiator” for the bracket on ${𝔤}_0$).

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

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

Over a test space $U$ a $𝔤$-valued form datum is a morphism

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

from the Weil algebra $W(𝔤)$.

This is given by a 1-form

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

and a 2-form

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

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

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

$$H^i=d_{\mathrm{dR}}{B}^i+{\alpha }^i{}_{aj}{A}^a\wedge B^j+t_{abc}{A}^a\wedge A^b\wedge A^c.$$H^i = d_{dR} B^i + \alpha^i{}_{a j} A^a \wedge B^j + t_{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

• A differential string structure (untwisted) is a 2-connection with coefficients in the string 2-group / string Lie 2-algebra.

• The worldvolume theory of the fivebrane is expected to be a 6d (2,0)-supersymmetric QFT containing a self-dual higher gauge theory whose fields are 2-connections (see Self-dual 2-connections there).

• A connection on a twisted vector bundle is naturally a 2-connection.

Related concepts

• connection on a bundle

• connection on a 2-bundle / connection on a gerbe / connection on a bundle gerbe

  • 2-groupoid of Lie 2-algebra valued forms

• connection on a 3-bundle / connection on a bundle 2-gerbe

• connection on an ∞-bundle

• parallel transport, higher parallel transport

• holonomy

References

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

• Urs Schreiber, Konrad Waldorf, Connections on non-abelian gerbes and their holonomy (web)

expanding on

• John Baez, Urs Schreiber, Higher gauge theory (arXiv:math/0511710) (web)

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

• Lawrence Breen, William Messing, Differential Geometry of Gerbes Advances in Mathematics, Volume 198, Issue 2, 20 (2005) (arXiv:math/0106083)

• Lawrence Breen, Differential Geometry of Gerbes and Differential Forms (arXiv:0802.1833)

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

• Hisham Sati, Urs Schreiber, Jim Stasheff, $L_{\mathrm{\infty }}$-connections (arXiv:0801.3480)

and the globalization is in

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech cocycles for differential characteristic classes

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

• Urs Schreiber, differential cohomology in a cohesive topos

See also connection on an infinity-bundle for the general theory.