Contents

bundles

cohomology

# Contents

## 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.

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

$n$Calabi-Yau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
$n = 1$elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
$n = 2$K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
$n = 3$Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
$n$intermediate Jacobian

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

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

More on super line 2-bundles is secretly in

The above higher supergeometric story is made explicit in