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 word that literally means "wheat sheaf."

Definitions

Gerbes on a topological space

A gerbe on a topological space is a stack of groupoids over that is locally non-empty and transitive.
A canonical example is the gerbe of principal bundles with a fixed structure group : the section category over an open set is the category of principal -bundles on with isomorphism as morphisms. As principal bundles glue together, these groupoids form a stack. The trivial bundle 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. Given a site a -gerbe is a category fibered in groupoids such that
  1. There exists a refinement of such that for every object the associated fibered category is non-empty
  2. For every any two objects in the fibered category are locally isomorphic
Note that for a site with a final object, a category fibered in groupoids admits a local section if and only if, in which case automatically satisfies the first axiom.

Motivation for gerbes on a site

One of the main motivations for considering gerbes on a site is to consider the following naive question: if the Cech cohomology group for a suitable covering of a space gives the isomorphism classes of principal -bundles over, what does the iterated cohomology functor represent? Meaning, we are gluing together the groups via some 1-cocycle. Gerbes are a technical response for this question: they give geometric representations of elements in the higher cohomology group. It is expected this intuition should hold for higher gerbes.

Cohomological classification

One of the main theorems concerning gerbes is their cohomological classification whenever they have automorphism groups given by a fixed sheaf of abelian groups, called a band. For a gerbe on a site, an object, and an object, the automorphism group of a gerbe is defined as the automorphism group. Notice this is well defined whenever the automorphism group is always the same. Given a covering, there is an associated class
representing the isomorphism class of the gerbe banded by.
For example, in topology, many examples of gerbes can be constructed by considering gerbes banded by the group. As the classifying space is the second Eilenberg–Maclane space for the integers, a bundle gerbe banded by on a topological space is constructed from a homotopy class of maps in
,
which is exactly the third singular homology group. It has been found that all gerbes representing torsion cohomology classes in are represented by a bundle of finite dimensional algebras for a fixed complex vector space. In addition, the non-torsion classes are represented as infinite-dimensional principal bundles of the projective group of unitary operators on a fixed infinite dimensional separable Hilbert space. Note this is well defined because all separable Hilbert spaces are isomorphic to the space of square-summable sequences.
The homotopy-theoretic interpretation of gerbes comes from looking at the homotopy fiber square
analogous to how a line bundle comes from the homotopy fiber square
where, giving as the group of isomorphism classes of line bundles on.

Examples

C*-algebras

There are natural examples of Gerbes that arise from studying the algebra of compactly supported complex valued functions on a paracompact space pg 3. Given a cover of there is the Cech groupoid defined as
with source and target maps given by the inclusions
and the space of composable arrows is just
Then a degree 2 cohomology class is just a map
We can then form a non-commutative C*-algebra, which is associated to the set of compact supported complex valued functions of the space
It has a non-commutative product given by
where the cohomology class twists the multiplication of the standard -algebra product.

Algebraic geometry

Let be a variety over an algebraically closed field, an algebraic group, for example. Recall that a G-torsor over is an algebraic space with an action of and a map, such that locally on is a direct product. A G-gerbe over M may be defined in a similar way. It is an Artin stack with a map, such that locally on M is a direct product. Here denotes the classifying stack of, i.e. a quotient of a point by a trivial -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 of and are the same, but in each point is equipped with a stabilizer group isomorphic to.

From two-term complexes of coherent sheaves

Every two-term complex of coherent sheaves
on a scheme has a canonical sheaf of groupoids associated to it, where on an open subset there is a two-term complex of -modules
giving a groupoid. It has objects given by elements and a morphism is given by an element such that
In order for this stack to be a gerbe, the cohomology sheaf 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. In addition, two-term spectra from the stabilization of the derived category of comodules of Hopf-algebroids with flat over give additional models of gerbes that are non-strict.

Moduli stack of stable bundles on a curve

Consider a smooth projective curve over of genus. Let be the moduli stack of stable vector bundles on of rank and degree. 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 of vector bundles. For any stable vector bundle the automorphism group consists only of scalar multiplications, so each point in a moduli stack has a stabilizer isomorphic to. It turns out that the map is indeed a -gerbe in the sense above. It is a trivial gerbe if and only if and are coprime.

Root stacks

Another class of gerbes can be found using the construction of root stacks. Informally, the -th root stack of a line bundle over a scheme is a space representing the -th root of and is denoted
pg 52
The -th root stack of has the property
as gerbes. It is constructed as the stack
sending an -scheme to the category whose objects are line bundles of the form
and morphisms are commutative diagrams compatible with the isomorphisms. This gerbe is banded by the algebraic group of roots of unity, where on a cover it acts on a point by cyclically permuting the factors of in. Geometrically, these stacks are formed as the fiber product of stacks
where the vertical map of comes from the Kummer sequence
This is because is the moduli space of line bundles, so the line bundle corresponds to an object of the category .
Root stacks with sections
There is another related construction of root stacks with sections. Given the data above, let be a section. Then the -th root stack of the pair is defined as the lax 2-functor
sending an -scheme to the category whose objects are line bundles of the form
and 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. Locally, we may assume and the line bundle is trivial, hence any section is equivalent to taking an element. Then, the stack is given by the stack quotient
with
If then this gives an infinitesimal extension of.

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.

Introductory articles

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Gerbes in topology

Twisted K-theory

Applications in string theory