nLab line 2-bundle

Idea

Given a 2-ring $R$, a line 2-bundle or 2-line bundle is a 2-module bundle whose typical fiber is a 2-line over $R$.

Definition

Recalling general 2-vector bundles

Let $R$ be a commutative ring, or more generally an E-∞ ring. By the discussion at 2-vector space consider the 2-category

$$2 Vect_R \simeq Alg_R$$

equivalent to that whose objects are associative algebras (or generally algebras) $A$ over $R$, (being placeholders for the 2-vector space $A Mod$ which is the category of modules over $A$) whose 1-morphisms are bimodules between these algebras (inducing linear functors between the corresponding 2-vector spaces = categories of modules) and whose 2-morphisms are homomorphisms between those.

Under Isbell duality and by the discussion at Modules – as generalized vector bundles we may think of this 2-category as being that of (generalized) 2-vector bundles over a space called $Spec R$.

Line 2-bundles

The 2-category $2 Vect_R \simeq Alg_R$ is canonically a monoidal 2-category. An object in $2 Vect_R$ is a line if it is an invertible object with respect to this tensor product, hence if it is an Azumaya algebra. In terms of the above this means that it represents a 2-vector bundle over $Spec R$ which is a line 2-bundle.

The full inclusion

$$\mathbf{Br}(R) \hookrightarrow 2 Vect_R \simeq Alg_R$$

of the maximal 2-groupoid on the line 2-bundles over $Spec R$ is a braided 3-group, the Picard 3-group of $Spec R$. See Relation to Brauer group below for more.

Properties

Relation to Brauer group and Picard group

The braided 3-group $\mathbf{Br}(R)$ of line 2-bundles over $Spec R$ has as homotopy groups

• $\pi_0(\mathbf{Br}(R))$ the Brauer group of $R$;

• $\pi_1(\mathbf{Br}(R))$ the Picard group of $R$, hence the group of ordinary line bundles over $R$;

• $\pi_2(\mathbf{Br}(R))$ the group of units of $R$.

Examples

Super line 2-bundles and twisted K-theory

See at super line 2-bundle.

Related concepts

• line bundle

• holomorphic line 2-bundle, algebraic line 2-bundle

• line n-bundle

• line n-bundle with connection

moduli spaces of line n-bundles with connection on $n$-dimensional $X$

$n$	Calabi-Yau n-fold	line n-bundle	moduli of line n-bundles	moduli of flat/degree-0 n-bundles	Artin-Mazur formal group of deformation moduli of line n-bundles	complex oriented cohomology theory	modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$		unit in structure sheaf	multiplicative group/group of units		formal multiplicative group	complex K-theory
$n = 1$	elliptic curve	line bundle	Picard group/Picard scheme	Jacobian	formal Picard group	elliptic cohomology	3d Chern-Simons theory/WZW model
$n = 2$	K3 surface	line 2-bundle	Brauer group	intermediate Jacobian	formal Brauer group	K3 cohomology
$n = 3$	Calabi-Yau 3-fold	line 3-bundle		intermediate Jacobian		CY3 cohomology	7d Chern-Simons theory/M5-brane
$n$				intermediate Jacobian

References

Line 2-bundles in supergeometry as a model for the B-field and orientifolds are discussed (even if not quite explicitly in the language of higher bundles) in

• Daniel Freed, Lectures on twisted K-theory and orientifolds (pdf)

based on

• Peter Donovan, Max Karoubi, Graded Brauer groups and $K$-theory with local coefficients, Publications Mathématiques de l’IHÉS, 38 (1970), p. 5-25 (numdam)

and

• C. T. C. Wall, Graded Brauer groups, J. Reine Angew. Math. 213 (1963/1964), 187-199.

and developing constructions in

• Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics, AMS (2011) (arXiv:0906.0795, slides)

More on super line 2-bundles is secretly in

• Daniel Freed, Michael Hopkins, Constantin Teleman, Loop groups and twisted K-theory I (arXiv:0711.1906)

The above higher supergeometric story is made explicit in

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, A higher stacky perspective on Chern-Simons theory