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 $g : X \to \mathbf{B}G$ 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

• 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

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

• Urs Schreiber, Konrad Waldorf,

  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

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

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

• Arthur Parzygnat, Two-dimensional algebra in lattice gauge theory, Journal of Mathematical Physics 60, 043506 (2019) (arXiv:1802.01139, doi:10.1063/1.5078532)

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, $L_\infty$-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.

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

• Sergei Gukov, Anton Kapustin, Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories (arXiv:1307.4793)