nLab
line 2-bundle

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cohomology

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Idea

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

Definition

Recalling general 2-vector bundles

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

2Vect RAlg R 2 Vect_R \simeq Alg_R

equivalent to that whose objects are associative algebras (or generally algebras) AA over RR, (being placeholders for the 2-vector space AModA Mod which is the category of modules over AA) 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 SpecRSpec R.

Line 2-bundles

The 2-category 2Vect RAlg R2 Vect_R \simeq Alg_R is canonically a monoidal 2-category. An object in 2Vect R2 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 SpecRSpec R which is a line 2-bundle.

The full inclusion

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

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

Properties

Relation to Brauer group and Picard group

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

Examples

Super line 2-bundles and twisted K-theory

See at super line 2-bundle.

moduli spaces of line n-bundles with connection on nn-dimensional XX

nnCalabi-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=0n = 0unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
n=1n = 1elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
n=2n = 2K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
n=3n = 3Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
nnintermediate 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

based on

  • Peter Donovan, Max Karoubi, Graded Brauer groups and KK-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

Last revised on September 3, 2014 at 12:31:59. See the history of this page for a list of all contributions to it.