What is your favourite example of Gerbes?
I would like to know Where do we find Gerbes in "nature"?
The examples could vary from String theory to Galois theory. For example my favourite examples of Gerbes are Nori fundamental gerbe, and that Tannakian categories could be described, in cohomological terms, by the gerbe of its fibre functors. What is yours?
At least when the group $G$ is discrete, and when the base is a topological space (as opposed to e.g. a scheme), I would like to advertise the fact that:
a $G$-gerbe is the same thing as a fibration whose fiber is $BG$ (the latter is the classifying space of $G$, also known as $K(G,1)$).
As an example, by taking $G=\mathbb Z$, we learn that an $S^1$-bundle is the same thing as a $\mathbb Z$-gerbe.
So the Hopf fibration $S^1 \to S^3 \to S^2$ is a $\mathbb Z$-gerbe over $S^2$.
You can get a lot of examples by dimension shifting. Namely, consider any exact sequence of groups $$1\to K\to G \to H\to 1 \; .$$ Fix a $H$-torsor $T$. The stack $\mathcal G_T$ of liftings of the structure group of $T$ to $G$ is clearly a gerbe (the objects are pairs $(T',\alpha)$, where $T'$ is a $G$-torsor and $\alpha : T'\times^G H \simeq T$ is an isomorphism). A more compact description as a quotient stack : $\mathcal G_T=[T/G]$. The band of $\mathcal G_T$ is of course (the band of) $K$.
Special instances of this construction :
the gerbe of $n$-th roots of an invertible sheaf (sometimes called Chern gerbe modulo $n$),
from Brauer theory : to any $\operatorname{PSL}_n$-torsor you can associate a $\mathbb G_m$-gerbe. This is probably my favourite example.
The following is a fancy way of saying that every elliptic curve contains a copy of $\mathbb Z/2$ in its automorphism group:
The moduli stack of elliptic curves is a $\mathbb Z/2$-gerbe over some other Deligne-Mumford stack (which doesn't seem to have a name).
My favorite one, in the sense that I am trying to really understand it for many years, is the determinantal gerbe of a locally linearly compact vector space, as described by Kapranov in "Semiinfinite symmetric powers".
I am making this Community Wiki since I don't really understand well this fascinating thing, so if anybody could make this answer better it would be great.
For a field $k$ and a topological vector space $V$ with certain property (linear local compactness, to be more precise, see below), he associates the $\textit{determinantal gerbe}$ $\operatorname{Det}(V)$. This is a $k^*$-gerbe. Its objects are $\textit{determinantal theories}$ for $V$. A determinantal theory $\Delta$ assigns to each open subspace $U$ of $V$ a 1-dimensional space $\Delta(U)$, and to a pair of open subspaces $U_1\subseteq U_2$ with finite-dimensional quotient, an isomorphism $$ \Delta_{U_1,U_2}:\Delta(U_1)\otimes\det(U_2/U_1)\to\Delta(U_2), $$ where $\det$ of a finite-dimensional vector space is its highest exterior power.
These must satisfy a coherence condition $$ \Delta(U_2,U_3)\circ\left(\Delta(U_1,U_2)\otimes\det(U_3/U_2)\right)=\Delta(U_1,U_3)\circ\det(U_1,U_2,U_3) $$ for $U_1\subseteq U_2\subseteq U_3$, where $\det(U_1,U_2,U_3):\det(U_2/U_1)\otimes\det(U_3/U_2)\to\det(U_3/U_1)$ is the canonical map. Then $k^*$ acts on isomorphisms of determinantal theories and turns the groupoid of such isomorphisms into a $k^*$-gerbe.
This gerbe seems to be related to very important constructions, none of them I really understand well. As Kapranov and several other authors explain, it goes back to Tate's thesis, who used it to describe residues of differentials in terms of traces of operators on adelic spaces. Notable further works after that, to mention just the most striking ones, include "The Infinite Wedge Representation and the Reciprocity Law for Algebraic Curves" by Arbarello, de Concini and Kac (in "Theta Functions, Bowdoin 1987", PSPM 49 (1989), Part 1, 171-190), "Central extensions and reciprocity laws" by Brylinski (Cah. Topol. Géom. Différ. Catégor., 38 (1997), 193-215), "Infinite-Dimensional Vector Bundles in Algebraic Geometry: An Introduction" by Drinfeld (in "The Unity of Mathematics: In Honor of the Ninetieth Birthday of I.M. Gelfand", Birkhäuser Progress in Mathematics series 244 (2007), 263-304)
Appendix - definition of linear local compactness.
A linearly compact vector space over a field $k$ is a Hausdorff topological vector space $V$ that has a base of neighborhoods of $0$ consisting of vector subspaces and closed affine subspaces have finite intersection property. It is linearly locally compact if it has a $0$-neighborhood base consisting of linearly compact subspaces.
A typical example from deformation theory : fix $i:X_0\to X$ of first order thickening defined by a square zero ideal $\mathcal I$, and let $\mathcal E_0$ be a locally free sheaf of finite rank on $X_0$. Then the stack of deformations of ${\mathcal E_0}$ to $X$ is a gerbe on $X_0$ banded by $\mathcal I\otimes \operatorname{End(}\mathcal E_0)$.
It is neutral if and only if the deformation problem is unobstructed, and in this case the stack of deformations is isomorphic, after choice of a lifting, to $B\left( \mathcal I\otimes \operatorname{End}(\mathcal E_0\right))$.
Let $\mathscr{C}$ be a Tannakian category over $Spec(k)$. Then $FIB(\mathscr{C})$ is an affine gerbe over $Spec(k)$ in the fpqc topology. [1]
Let us consider Tannakian categories $\mathscr{C}$ over a field k, and for which there exists a $K$-valued fibre functor $x$, which values in some finite separable field extension $K$ of k. In this case, the corresponding gerbe $\mathscr{G} = FIB(\mathscr{C})$ of fibre functors on $\mathscr{C}$ is a gerbe over $Spec(k)$ in the étale topology. [2]
[1]: Catégories Tannakiennes, P Deligne, in the Grothendieck Festschrift II, Progr. Math. 87, Birkhäuser, Boston, 1990, pp. 111-195.
[2]: Tannakian categories, L Breen, in Motives, Proceedings of Symposia in Pure Mathematics, 55, Providence, R.I.: American Mathematical Society, 1994.
The root gerbes $${^r}\sqrt{\mathscr{L}/X}$$ associated to a line bundle on a scheme (or stack) $X$
If $\mathscr{M}$ is the moduli stack of mathematical objects $X$ of some specified kind such that $\mathrm{Aut}(X)$ always (naturally) contains a group (object) $G\subseteq\mathrm{Aut}(X)$, then $\mathscr{M}$ will be a $G$-gerbe over some other stack $\mathscr{M}^{\mathrm{rig},G}$. Examples of this include
The following is very nicely described by Malikov in Section 3 of "An Introduction to Algebras of Chiral Differential Operators" ("Perspectives in Lie Theory", pages 73 - 124).
For a smooth algebraic variety $X$ there is a gerbe $\mathscr{PL}_X$ of Picard-Lie algebroids over $X$. Objects of the groupoid $\mathscr{PL}_X(U)$ of its sections over an affine open $U=\operatorname{Spec}(A)$ are Lie $A$-algebroids, i. e. Lie algebras equipped with an action on $A$ by derivations and with an $A$-module structure satisfying certain conditions; "Picard" means that another piece of the structure is an isomorphism of the kernel of the action with $A$.
The lien of $\mathscr{PL}_X$ is the sheaf complex $d:\Omega^1_X\to\Omega^{2,\mathrm{cl}}_X$ where $d$ is the de Rham differential. Roughly, this means two things:
This gerbe has a global section (action of $\mathcal O_X\rtimes\mathcal T_X$ on $\mathcal O_X$) but is nevertheless very interesting, providing a prototype for building de Rham complexes on loop spaces.
Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of topological spaces on $X$.
Then, the map $U\mapsto \pi_1(\mathcal{F}(U))$ for $U\subseteq X$ open is a gerbe over $X$.
I learned this example from André Henriques‘s answer to one of my questions.
Please feel free to edit this to make it more clear and/or to give references.
