The bigerbes introduced here give a refinement of the notion of
–gerbes,
representing degree four integral cohomology classes of a space. Defined in terms of
bisimplicial line bundles, bigerbes have a symmetry with respect to which they form “bundle
–gerbes”
in two ways; this structure replaces higher associativity conditions. We provide
natural examples, including a Brylinski–McLaughlin bigerbe associated to a principal
–bundle
for a simply connected simple Lie group. This represents the first Pontryagin class of
the bundle, and is the obstruction to the lifting problem on the associated
principal bundle over the loop space to the structure group consisting of
a central extension of the loop group; in particular, trivializations of this
bigerbe for a spin manifold are in bijection with string structures on
the original manifold. Other natural examples represent “decomposable”
–classes
arising as cup products, a universal bigerbe on
involving its based double loop space, and the representation of any
–class
on a space by a bigerbe involving its free double loop space. The generalization to
“multigerbes” of arbitrary degree is also described.
