Gerbe

A gerbe on a topological space ${\displaystyle S}$[1]^: 318 is a stack ${\displaystyle {\mathcal {X}}}$ of groupoids over ${\displaystyle S}$ which is locally non-empty (each point ${\displaystyle p\in S}$ has an open neighbourhood ${\displaystyle U_{p}}$ over which the section category ${\displaystyle {\mathcal {X}}(U_{p})}$ of the gerbe is not empty) and transitive (for any two objects ${\displaystyle a}$ and ${\displaystyle b}$ of ${\displaystyle {\mathcal {X}}(U)}$ for any open set ${\displaystyle U}$, there is an open covering ${\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}$ of ${\displaystyle U}$ such that the restrictions of ${\displaystyle a}$ and ${\displaystyle b}$ to each ${\displaystyle U_{i}}$ are connected by at least one morphism).

A canonical example is the gerbe ${\displaystyle BH}$ of principal bundles with a fixed structure group ${\displaystyle H}$: the section category over an open set ${\displaystyle U}$ is the category of principal ${\displaystyle H}$-bundles on ${\displaystyle U}$ 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 ${\displaystyle X\times H\to X}$ 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.

The most general definition of gerbes are defined over a site. Given a site ${\displaystyle {\mathcal {C}}}$ a ${\displaystyle {\mathcal {C}}}$-gerbe ${\displaystyle G}$[2][3]^: 129 is a category fibered in groupoids ${\displaystyle G\to {\mathcal {C}}}$ such that

1. There exists a refinement[4] ${\displaystyle {\mathcal {C}}'}$ of ${\displaystyle {\mathcal {C}}}$ such that for every object ${\displaystyle S\in {\text{Ob}}({\mathcal {C}}')}$ the associated fibered category ${\displaystyle G_{S}}$ is non-empty
2. For every ${\displaystyle S\in {\text{Ob}}({\mathcal {C}})}$ any two objects in the fibered category ${\displaystyle G_{S}}$ are locally isomorphic

Note that for a site ${\displaystyle {\mathcal {C}}}$ with a final object ${\displaystyle e}$, a category fibered in groupoids ${\displaystyle G\to {\mathcal {C}}}$ is a ${\displaystyle {\mathcal {C}}}$-gerbe admits a local section, meaning satisfies the first axiom, if ${\displaystyle {\text{Ob}}(G_{e})\neq \varnothing }$.

One of the main motivations for considering gerbes on a site is to consider the following naive question: if the Cech cohomology group ${\displaystyle H^{1}({\mathcal {U}},G)}$ for a suitable covering ${\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}$ of a space ${\displaystyle X}$ gives the isomorphism classes of principal ${\displaystyle G}$-bundles over ${\displaystyle X}$, what does the iterated cohomology functor ${\displaystyle H^{1}(-,H^{1}(-,G))}$ represent? Meaning, we are gluing together the groups${\displaystyle H^{1}(U_{i},G)}$ via some one cocycle. Gerbes are a technical response for this question: they give geometric representations of elements in the higher cohomology group ${\displaystyle H^{2}({\mathcal {U}},G)}$. It is expected this intuition should hold for higher gerbes.

There are natural examples of Gerbes which arise from studying the algebra of compactly supported complex valued functions on a paracompact space ${\displaystyle X}$[7]^{pg 3}. Given a cover ${\displaystyle {\mathcal {U}}=\{U_{i}\}}$ of ${\displaystyle X}$ there is the Cech groupoid defined as

${\displaystyle {\mathcal {G}}=\left\{\coprod _{i,j}U_{ij}\rightrightarrows \coprod U_{i}\right\}}$

with source and target maps given by the inclusions

${\displaystyle {\begin{aligned}s:U_{ij}\hookrightarrow U_{j}\\t:U_{ij}\hookrightarrow U_{i}\end{aligned}}}$

and the space of composable arrows is just

${\displaystyle \coprod _{i,j,k}U_{ijk}}$

Then a degree 2 cohomology class ${\displaystyle \sigma \in H^{2}(X;U(1))}$ is just a map

${\displaystyle \sigma$ :\coprod U_{ijk}\to U(1)}

We can then form a non-commutative C*-algebra ${\displaystyle C_{c}({\mathcal {G}}(\sigma ))}$ which is associated to the set of compact supported complex valued functions of the space

${\displaystyle {\mathcal {G}}_{1}=\coprod _{i,j}U_{ij}}$

It has a non-commutative product given by

${\displaystyle a*b(x,i,k):=\sum _{j}a(x,i,j)b(x,j,k)\sigma (x,i,j,k)}$

where the cohomology class ${\displaystyle \sigma }$ twists the multiplication of the standard ${\displaystyle C^{*}}$-algebra product.

Let ${\displaystyle M}$ be a variety over an algebraically closed field ${\displaystyle k}$, ${\displaystyle G}$ an algebraic group, for example ${\displaystyle \mathbb {G} _{m}}$. Recall that a G-torsor over ${\displaystyle M}$ is an algebraic space ${\displaystyle P}$ with an action of ${\displaystyle G}$ and a map ${\displaystyle \pi :P\to M}$, such that locally on ${\displaystyle M}$ (in étale topology or fppf topology) ${\displaystyle \pi }$ is a direct product ${\displaystyle \pi |_{U}:G\times U\to U}$. A G-gerbe over M may be defined in a similar way. It is an Artin stack ${\displaystyle {\mathcal {M}}}$ with a map ${\displaystyle \pi \colon {\mathcal {M}}\to M}$, such that locally on M (in étale or fppf topology) ${\displaystyle \pi }$ is a direct product ${\displaystyle \pi |_{U}\colon \mathrm {B} G\times U\to U}$.[8] Here ${\displaystyle BG}$ denotes the classifying stack of ${\displaystyle G}$, i.e. a quotient ${\displaystyle [*/G]}$ of a point by a trivial ${\displaystyle G}$-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 ${\displaystyle {\mathcal {M}}}$ and ${\displaystyle M}$ are the same, but in ${\displaystyle {\mathcal {M}}}$ each point is equipped with a stabilizer group isomorphic to ${\displaystyle G}$.

Every two-term complex of coherent sheaves

${\displaystyle {\mathcal {E}}^{\bullet }=[{\mathcal {E}}^{-1}\xrightarrow {d} {\mathcal {E}}^{0}]}$

on a scheme ${\displaystyle X\in {\text{Sch}}}$ has a canonical sheaf of groupoids associated to it, where on an open subset ${\displaystyle U\subseteq X}$ there is a two-term complex of ${\displaystyle X(U)}$-modules

${\displaystyle {\mathcal {E}}^{-1}(U)\xrightarrow {d} {\mathcal {E}}^{0}(U)}$

giving a groupoid. It has objects given by elements ${\displaystyle x\in {\mathcal {E}}^{0}(U)}$ and a morphism ${\displaystyle x\to x'}$ is given by an element ${\displaystyle y\in {\mathcal {E}}^{-1}(U)}$ such that

${\displaystyle dy+x=x'}$

In order for this stack to be a gerbe, the cohomology sheaf ${\displaystyle {\mathcal {H}}^{0}({\mathcal {E}})}$ 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 ${\displaystyle (A,\Gamma )}$ with ${\displaystyle \Gamma }$ flat over ${\displaystyle A}$ give additional models of gerbes which are non-strict.

Consider a smooth projective curve ${\displaystyle C}$ over ${\displaystyle k}$ of genus ${\displaystyle g>1}$. Let ${\displaystyle {\mathcal {M}}_{r,d}^{s}}$ be the moduli stack of stable vector bundles on ${\displaystyle C}$ of rank ${\displaystyle r}$ and degree ${\displaystyle d}$. It has a coarse moduli space ${\displaystyle M_{r,d}^{s}}$, 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 ${\displaystyle E}$ the automorphism group ${\displaystyle Aut(E)}$ consists only of scalar multiplications, so each point in a moduli stack has a stabilizer isomorphic to ${\displaystyle \mathbb {G} _{m}}$. It turns out that the map ${\displaystyle {\mathcal {M}}_{r,d}^{s}\to M_{r,d}^{s}}$ is indeed a ${\displaystyle \mathbb {G} _{m}}$-gerbe in the sense above.[9] It is a trivial gerbe if and only if ${\displaystyle r}$ and ${\displaystyle d}$ are coprime.

Another class of gerbes can be found using the construction of root stacks. Informally, the ${\displaystyle r}$-th root stack of a line bundle ${\displaystyle L\to S}$ over a scheme is a space representing the ${\displaystyle r}$-th root of ${\displaystyle L}$ and is denoted

${\displaystyle {\sqrt[{r}]{L/S}}.\,}$[10]^{pg 52}

The ${\displaystyle r}$-th root stack of ${\displaystyle L}$ has the property

${\displaystyle \bigotimes ^{r}{\sqrt[{r}]{L/S}}\cong L}$

as gerbes. It is constructed as the stack

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

sending an ${\displaystyle S}$-scheme ${\displaystyle T\to S}$ to the category whose objects are line bundles of the form

${\displaystyle \left\{(M\to T,\alpha _{M}):\alpha _{M}:M^{\otimes r}\xrightarrow {\sim } L\times _{S}T\right\}}$

and morphisms are commutative diagrams compatible with the isomorphisms ${\displaystyle \alpha _{M}}$. This gerbe is banded by the algebraic group of roots of unity ${\displaystyle \mu _{r}}$, where on a cover ${\displaystyle T\to S}$ it acts on a point ${\displaystyle (M\to T,\alpha _{M})}$ by cyclically permuting the factors of ${\displaystyle M}$ in ${\displaystyle M^{\otimes r}}$. Geometrically, these stacks are formed as the fiber product of stacks

${\displaystyle {\begin{matrix}X\times _{B\mathbb {G} _{m}}B\mathbb {G} _{m}&\to &B\mathbb {G} _{m}\\\downarrow &&\downarrow \\X&\to &B\mathbb {G} _{m}\end{matrix}}}$

where the vertical map of ${\displaystyle B\mathbb {G} _{m}\to B\mathbb {G} _{m}}$ comes from the Kummer sequence

${\displaystyle 1\xrightarrow {} \mu _{r}\xrightarrow {} \mathbb {G} _{m}\xrightarrow {(\cdot )^{r}} \mathbb {G} _{m}\xrightarrow {} 1}$

This is because ${\displaystyle B\mathbb {G} _{m}}$ is the moduli space of line bundles, so the line bundle ${\displaystyle L\to S}$ corresponds to an object of the category ${\displaystyle B\mathbb {G} _{m}(S)}$ (considered as a point of the moduli space).

There is another related construction of root stacks with sections. Given the data above, let ${\displaystyle s:S\to L}$ be a section. Then the ${\displaystyle r}$-th root stack of the pair ${\displaystyle (L\to S,s)}$ is defined as the lax 2-functor[10][11]

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

sending an ${\displaystyle S}$-scheme ${\displaystyle T\to S}$ to the category whose objects line bundles of the form

${\displaystyle \left\{(M\to T,\alpha _{M},t):{\begin{aligned}&\alpha _{M}:M^{\otimes r}\xrightarrow {\sim } L\times _{S}T\\&t\in \Gamma (T,M)\\&\alpha _{M}(t^{\otimes r})=s\end{aligned}}\right\}}$

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.[11]^: 4 Given an affine scheme ${\displaystyle S={\text{Spec}}(A)}$, all line bundles are trivial, hence ${\displaystyle L\cong {\mathcal {O}}_{S}}$ and any section ${\displaystyle s}$ is equivalent to taking an element ${\displaystyle s\in A}$. Then, the stack is given by the stack quotient

${\displaystyle {\sqrt[{r}]{(L,s)/S}}=[{\text{Spec}}(B)/\mu _{r}]}$[11]^: 9

with

${\displaystyle B={\frac {A[x]}{x^{r}-s}}}$

If ${\displaystyle s=0}$ then this gives an infinitesimal extension of ${\displaystyle [{\text{Spec}}(A)/\mu _{r}]}$.

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

• ${\displaystyle H^{3}(X,\mathbb {Z} )}$ and ${\displaystyle {\mathcal {O}}_{X}^{*}}$-gerbes: Jean-Luc Brylinski's approach

• 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.
• Moerdijk, Ieke. "Introduction to the Language of Stacks and Gerbes". Retrieved 2007-05-20.

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

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

• 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