Gerbe

In mathematics, a gerbe (/ɜːrb/; French: [ʒɛʁb]) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud (Giraud 1971) 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 [1]:318 is a stack of groupoids over which is locally non-empty (each point has an open neighbourhood over which the section category of the gerbe is not empty) and transitive (for any two objects and of for any open set , there is an open covering of such that the restrictions of and to each are connected by at least one morphism).

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 (thus the category is a groupoid). As principal bundles glue together (satisfy the descent condition), 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 [2][3]:129 is a category fibered in groupoids such that

  1. There exists a refinement[4] 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 is a -gerbe admits a local section, meaning satisfies the first axiom, if .

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 one 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 ,[5][2] 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[6] 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 which arise from studying the algebra of compactly supported complex valued functions on a paracompact space [7]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 (in étale topology or fppf topology) 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 (in étale or fppf topology) is a direct product .[8] 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 (projectivity if a graded Hopf-algebroid is used). 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 which 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.[9] 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

[10]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 (considered as a point of the moduli space).

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[10][11]

sending an -scheme to the category whose objects 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.[11]:4 Given an affine scheme , all line bundles are trivial, hence and any section is equivalent to taking an element . Then, the stack is given by the stack quotient

[11]:9

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

  • and -gerbes: Jean-Luc Brylinski's approach

History

Gerbes first appeared in the context of algebraic geometry. They were subsequently developed in a more traditional geometric framework by Brylinski (Brylinski 1993). 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

  1. Basic bundle theory and K-cohomology invariants. Husemöller, Dale. Berlin: Springer. 2008. ISBN 978-3-540-74956-1. OCLC 233973513.{{cite book}}: CS1 maint: others (link)
  2. "Section 8.11 (06NY): Gerbes—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-27.
  3. Giraud, J. (Jean) (1971). Cohomologie non abélienne. Berlin: Springer-Verlag. ISBN 3-540-05307-7. OCLC 186709.
  4. "Section 7.8 (00VS): Families of morphisms with fixed target—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-27.
  5. "Section 21.11 (0CJZ): Second cohomology and gerbes—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-27.
  6. Karoubi, Max (2010-12-12). "Twisted bundles and twisted K-theory". arXiv:1012.2512 [math.KT].
  7. Block, Jonathan; Daenzer, Calder (2009-01-09). "Mukai duality for gerbes with connection". arXiv:0803.1529 [math.QA].
  8. Edidin, Dan; Hassett, Brendan; Kresch, Andrew; Vistoli, Angelo (2001). "Brauer groups and quotient stacks". American Journal of Mathematics. 123 (4): 761–777. arXiv:math/9905049. doi:10.1353/ajm.2001.0024. S2CID 16541492.
  9. Hoffman, Norbert (2010). "Moduli stacks of vector bundles on curves and the King–Schofield rationality proof". Cohomological and Geometric Approaches to Rationality Problems: 133–148. arXiv:math/0511660. doi:10.1007/978-0-8176-4934-0_5. ISBN 978-0-8176-4933-3. S2CID 5467668.
  10. Abramovich, Dan; Graber, Tom; Vistoli, Angelo (2008-04-13). "Gromov-Witten theory of Deligne-Mumford stacks". arXiv:math/0603151.
  11. Cadman, Charles (2007). "Using stacks to impose tangency conditions on curves" (PDF). Amer. J. Math. 129 (2): 405–427. arXiv:math/0312349. doi:10.1353/ajm.2007.0007. S2CID 10323243.

Introductory articles

Gerbes in topology

Twisted K-theory

Applications in string theory

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