nLab
connection on a 2-bundle

Context

\infty-Chern-Weil theory

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

BG conn X g BG \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 XX is topologically trivial, then the connections on it are identified with Lie 2-algebra valued differential forms on XX.

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

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

Let 𝔤 0\mathfrak{g}_0 and 𝔤 1\mathfrak{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

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

with these generators.

We thus have

d CE(𝔤)t a=12C a bct bt cr a ib 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 CE(𝔤)b i=α aj it ab jr abct at bt 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}\{C^a{}_{b c}\} (the bracket on 𝔤 0\mathfrak{g}_0) and {r a i}\{r^i_a\} (the differential 𝔤 1𝔤 0\mathfrak{g}_1 \to \mathfrak{g}_0) and {alph i aj}\{\alph^i{}_{a j}\} (the action of 𝔤 0\mathfrak{g}_0 on 𝔤 1\mathfrak{g}_1) and {r abc}\{r_{a b c}\} (the “Jacobiator” for the bracket on 𝔤 0\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(𝔤)) 2=0. (d_{CE(\mathfrak{g})})^2 = 0 \,.

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

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

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

This is given by a 1-form

AΩ 1(U,𝔤 0) A \in \Omega^1(U, \mathfrak{g}_0)

and a 2-form

BΩ 2(U,𝔤 1). B \in \Omega^2(U, \mathfrak{g}_1) \,.

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

β a=d dRA a+12C a bcA bA c+r aiB 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 dRB i+α i ajA aB j+t abcA aA bA 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 XX with respect to a given open cover {U iX}\{U_i \to X\}, following (FSS, SchreiberCohesive)

(…)

Examples

References

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

expanding on

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

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

Applications to physics

Nonabelian 2-connections appear for instance as orientifold B-fields in type II string theory, as differential string structure in heterotic string theory, and as fields in non-abelian 7-dimensional Chern-Simons theory. See at these pages for references.

An appearance in 4-dimensional Yang-Mills theory and 4d TQFT is reported in

Revised on June 26, 2014 22:30:06 by David Roberts (137.195.217.101)