(see also Chern-Weil theory, parameterized homotopy theory)
vector bundle, (∞,1)-vector bundle
topological vector bundle, differentiable vector bundle, algebraic vector bundle
direct sum of vector bundles, tensor product of vector bundles, inner product of vector bundles?, dual vector bundle
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
Over a site of complex analytic spaces, where the multiplicative group $\mathbb{G}_m$ classifies non-vanishing holomorphic functions and $\mathbf{B}\mathbb{G}_m$ classifies holomorphic line bundles, then a holomorphic line 2-bundle is a $\mathbb{G}_m$-principal 2-bundle, modulated by maps to $\mathbf{B}^2 \mathbb{G}_m$.
This means that the moduli stack of holomorphic line 2-bundles on a complex analytic space or more generally on a complex analytic ∞-groupoid $X$ is the Brauer stack $\mathbf{Br}(X) \coloneqq [X,\mathbf{B}^2 \mathbb{G}_m]$ (the line 2-bundle itself is the associated ∞-bundle to the $\mathbf{B}\mathbb{G}_m$-principal ∞-bundle which is the homotopy fiber of a given map $X \to \mathbf{B}^2 \mathbb{G}_m$). In particular equivalence classes of holomorphic line 2-bundles form the elements of the bigger Brauer group of $X$ (the Brauer group proper if they are torsion).
Discussion in terms of bundle gerbes includes (Chatterjee 98,Brylinski 00 Mathai-Stevenson 02, section 7).
The Dixmier-Douady class of holomorphic line 2-bundles, hence the higher analog of the first Chern class, is given by the connecting homomorphism on degee 2 of the long exact sequence in cohomology which is induced by the exponential exact sequence in complex analytic geometry:
Holomorphic line 2-bundles appear in the higher degree analogs of twistor transforms. See (Chatterjee 98) and see twistor – References – Application to self-dual 2-forms
Discussion in relation to Beilinson regulators is in
Early discussion in terms of bundle gerbes includes
Discussion with an eye towards of holomorphic twisted K-theory is in
An equivariant example arising from more algebro-geometric origin is in
Discussion connecting explicitly to the holomorphic Brauer group includes
Oren Ben-Bassat, Gerbes and the Holomorphic Brauer Group of Complex Tori, Journal of Noncommutative Geometry, Volume 6, Issue 3 (2012) 407-455 (arXiv:0811.2746)
Edoardo Ballico, Oren Ben-Bassat, Meromorphic Line Bundles and Holomorphic Gerbes, Math. Res. Lett. 18 (2011), 6, 1-14 (arXiv:1101.2216)
See also
The existence of the “basic” 2-line bundle (see at Chern-Simons line 3-bundle) on a complex reductive group (such as $SL(n,\mathbb{C})$) is mentioned in
The actual construction appears in
Last revised on December 8, 2016 at 10:35:38. See the history of this page for a list of all contributions to it.