Gerbe Explained

In mathematics, a gerbe (; pronounced as /fr/) 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

^[1] is a stack of groupoids over that 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]} 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

1. For every

any two objects in the fibered category are locally isomorphicNote 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 }_(x). Notice this is well defined whenever the automorphism group is always the same. Given a covering , there is an associated class

c(\underline{L})\inH^{3(X,\underline{L})}

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

[X,B^2(U(1))]=[X,K(Z,3)]

,

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

\begin{matrix} l{X}&\to&*\\ \downarrow&&\downarrow\\ S&\xrightarrow{f}&B^{2U(1) \end{matrix}}

analogous to how a line bundle comes from the homotopy fiber square

\begin{matrix} L&\to&*\\ \downarrow&&\downarrow\\ S&\xrightarrow{f}&BU(1) \end{matrix}

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

^{[7] pg 3}. Given a cover of there is the Cech groupoid defined as

l{G}=\left\{\coprod_i,jU_ij\rightrightarrows\coprodU_i\right\}

with source and target maps given by the inclusions

\begin{align} s:U_ij\hookrightarrowU_j\\ t:U_ij\hookrightarrowU_{i \end{align}}

and the space of composable arrows is just

\coprod_i,j,kU_ijk

Then a degree 2 cohomology class is just a map

\sigma:\coprodU_ijk\toU(1)

We can then form a non-commutative C*-algebra , which is associated to the set of compact supported complex valued functions of the space

l{G}₁=\coprod_i,jU_ij

It has a non-commutative product given by

a*b(x,i,k):=\sum_ja(x,i,j)b(x,j,k)\sigma(x,i,j,k)

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

l{E}^\bullet=[l{E}^-1\xrightarrow{d}l{E}^0]

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

l{E}^-1(U)\xrightarrow{d}l{E}^0(U)

giving a groupoid. It has objects given by elements and a morphism is given by an element such that

dy+x=x'

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 that are non-strict.

Moduli stack of stable bundles on a 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

\sqrt[r]{L/S}.

^{[10] pg 52}

The -th root stack of has the property

otimes^{r\sqrt[{r}]{L/S}}\congL

as gerbes. It is constructed as the stack

\sqrt[r]{L/S}:(\operatorname{Sch}/S)^op\to\operatorname{Grpd}

sending an -scheme to the category whose objects are line bundles of the form

\left\{ (M\toT,\alpha_M):\alpha_M:M ^⊗ \xrightarrow{\sim}L x _{ST \right\}}

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

\begin{matrix} X x
	BGm

BG_m&\to&BG_m\\ \downarrow&&\downarrow\\ X&\to&BG_{m \end{matrix}}

where the vertical map of comes from the Kummer sequence

1\xrightarrow{}\mu_r\xrightarrow{}G_m\xrightarrow{( ⋅ )^r}G_m\xrightarrow{}1

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^[11]

\sqrt[r]{(L,s)/S}:(\operatorname{Sch}/S)^op\to\operatorname{Grpd}

sending an -scheme to the category whose objects are line bundles of the form

\left\{ (M\toT,\alpha_M,t): \begin{align} &\alpha_M:M ^⊗ \xrightarrow{\sim}L x _ST\\ &t\in\Gamma(T,M)

	⊗ r
\\ &\alpha
	M(t

)=s \end{align} \right\}

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

\sqrt[r]{(L,s)/S}=[Spec(B)/\mu_r]

with

B=

	A[x]
	xr-s

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:

• Azumaya algebras
• Deformations of infinitesimal thickenings
• Twisted forms of projective varieties
• Fiber functors for motives

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 . 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

• Twisted sheaf
• Azumaya algebra
• Twisted K-theory
• Algebraic stack
• Bundle gerbe
• String group

References

• .
• .

External links

Introductory articles

• Constructions with Bundle Gerbes - Stuart Johnson
• An Introduction to Gerbes on Orbifolds, Ernesto Lupercio, Bernado Uribe.
• What is a Gerbe?, by Nigel Hitchin in Notices of the AMS
• Bundle gerbes, Michael Murray.
• Web site: Moerdijk . Ieke. Ieke Moerdijk . Introduction to the Language of Stacks and Gerbes . 2007-05-20 .

Gerbes in topology

• Homotopy theory of presheaves of simplicial groupoids, Zhi-Ming Luo

Twisted K-theory

• Twisted K-theory and K-theory of bundle gerbes
• Twisted Bundles and Twisted K-Theory - Karoubi

Applications in string theory

• Stable Singularities in String Theory - contains examples of gerbes in appendix using the Brauer group
• Branes on Group Manifolds, Gluon Condensates, and twisted K-theory
• Lectures on Special Lagrangian Submanifolds - Very down-to earth introduction with applications to Mirror symmetry
• The basic gerbe over a compact simple Lie group - Gives techniques for describing groups such as the String group as a gerbe

Notes and References

1. Book: Basic bundle theory and K-cohomology invariants. 2008. Springer. Husemöller, Dale.. 978-3-540-74956-1. Berlin. 233973513.
2. Web site: Section 8.11 (06NY): Gerbes—The Stacks project. 2020-10-27. stacks.math.columbia.edu.
3. Book: Giraud, J. (Jean). Cohomologie non abélienne.. 1971. Springer-Verlag. 3-540-05307-7. Berlin. 186709.
4. Web site: Section 7.8 (00VS): Families of morphisms with fixed target—The Stacks project. 2020-10-27. stacks.math.columbia.edu.
5. Web site: Section 21.11 (0CJZ): Second cohomology and gerbes—The Stacks project. 2020-10-27. stacks.math.columbia.edu.
6. Karoubi. Max. 2010-12-12. Twisted bundles and twisted K-theory. math.KT. 1012.2512.
7. Block . Jonathan . Daenzer . Calder . 2009-01-09 . Mukai duality for gerbes with connection . math.QA . 0803.1529 .
8. Dan . Edidin . Brendan . Hassett. Andrew . Kresch . Angelo . Vistoli . Brauer groups and quotient stacks . . 2001 . 123 . 4 . 761–777 . math/9905049 . 10.1353/ajm.2001.0024. 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. Progress in Mathematics . 282 . 133–148. 10.1007/978-0-8176-4934-0_5. math/0511660. 978-0-8176-4933-3. 5467668.
10. Abramovich. Dan. Graber. Tom. Vistoli. Angelo. 2008-04-13. Gromov-Witten theory of Deligne-Mumford stacks. math/0603151.
11. Cadman. Charles. 2007. Using stacks to impose tangency conditions on curves. Amer. J. Math.. 129. 2. 405–427. math/0312349. 10.1353/ajm.2007.0007. 10323243.