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Wikipedia / Gerbe

In mathematics, a gerbe (; ) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analogue of fibre bundles where the fibre is the classifying stack of a group. Gerbes provide a convenient, if highly abstract, language for dealing with many types of deformation questions especially in modern algebraic geometry. In addition, special cases of gerbes have been used more recently in differential topology and differential geometry to give alternative descriptions to certain cohomology classes and additional structures attached to them.

"Gerbe" is a French (and archaic English) word that literally means "wheat sheaf."

Definitions

Gerbes on a topological space

A gerbe on a topological space is a stack of groupoids that is locally non-empty over which the section category of the gerbe is not empty) and transitive are connected by at least one morphism).

of principal bundles with a fixed structure group shows that the local non-emptiness condition is satisfied, and finally as principal bundles are locally trivial, they become isomorphic when restricted to sufficiently small open sets; thus the transitivity condition is satisfied as well.

Gerbes on a site

The most general definition of gerbes are defined over a site such that

  1. There exists a refinement is non-empty

  2. are locally isomorphic

automatically satisfies the first axiom.

Motivation for gerbes on a site

. It is expected this intuition should hold for higher gerbes.

Cohomological classification

,, there is an associated classrepresenting the isomorphism class.

is the second Eilenberg–Maclane is constructed from a homotopy class of maps in,which is exactly the third singular homology. It has been found of the projective group of unitary operators on a fixed infinite dimensional separable Hilbert space.

The homotopy-theoretic interpretation of gerbes comes from looking at the homotopy fiber squareanalogous to how a line bundle comes from the homotopy fiber square.

Examples

C*-algebras

pg 3 there is the Čech groupoid defined aswith source and target maps given by the inclusionsand the space of composable arrows is just is just a mapWe can then form a non-commutative C*-algebra, which is associated to the set of compact supported complex valued functions of the spaceIt has a non-commutative product given by-algebra product.

Algebraic geometry

be a variety over an algebraically closed field an algebraic group. Recall that a G-torsor is an algebraic space (in étale topology or fppf topology. A G-gerbe over M may be defined in a similar way. It is an Artin stack, such that locally on M. denotes the classifying stack-action. There is no need to impose the compatibility with the group structure in that case since it is covered by the definition of a stack. The underlying topological spaces.

From two-term complexes of coherent sheaves

Every two-term complex of coherent sheaves-modules such that must always have a section. This hypothesis implies the category constructed above always has objects. Note this can be applied to the situation of comodules over Hopf-algebroids to construct algebraic models of gerbes over affine or projective stacks (projectivity if a graded Hopf-algebroid is used). In addition, two-term spectra from the stabilization of the derived category give additional models of gerbes that are non-strict.

Moduli stack of stable bundles on a curve

Consider a smooth projective curve be the moduli stack of stable vector bundles. It has a coarse moduli space, which is a quasiprojective variety. These two moduli problems parametrize the same objects, but the stacky version remembers automorphisms-gerbe in the sense above. are coprime.

Root stacks

over a scheme and is denotedpg 52 has the propertyas gerbes. It is constructed as the stack to the category whose objects are line bundles of the form. This gerbe is banded by the algebraic group. Geometrically, these stacks are formed as the fiber product of stacks comes from the Kummer sequence (considered as a point of the moduli space).

Root stacks with sections

is defined as the lax 2-functor to the category whose objects are line bundles of the formand morphisms are given similarly. These stacks can be constructed very explicitly, and are well understood for affine schemes. In fact, these form the affine models for root stacks with sections.. Then, the stack is given by the stack quotientwith.

Examples throughout algebraic geometry

These and more general kinds of gerbes arise in several contexts as both geometric spaces and as formal bookkeeping tools:

Differential geometry

History

Gerbes first appeared in the context of algebraic geometry. They were subsequently developed in a more traditional geometric framework by Brylinski . One can think of gerbes as being a natural step in a hierarchy of mathematical objects providing geometric realizations of integral cohomology classes.

A more specialised notion of gerbe was introduced by Murray and called bundle gerbes. Essentially they are a smooth version of abelian gerbes belonging more to the hierarchy starting with principal bundles than sheaves. Bundle gerbes have been used in gauge theory and also string theory. Current work by others is developing a theory of non-abelian bundle gerbes.

See also

References

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Introductory articles

Gerbes in topology

Twisted K-theory

Applications in string theory