Gerbes

Prerequisites: Fiber Bundles, Cohomology. Full path (10 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

If the Guardian asked me to explain a gerbe, I’d cheat a little and explain a gerbe with connection, which is actually easier. I’d say it’s a well-behaved recipe that’ll tell you a time of day — by giving a position of a clock hand — if you specify a way of sticking a balloon in a given space. The rules for what counts as “well-behaved” would take a few pictures to explain, but the first rule is that as you move the balloon around in a smooth sort of way, the time of day changes in a smooth sort of way.John Baez

Suppose you are a painter. Your job is to go to the newly built conference building and

- 1) paint all the doorknobs

- 2) paint all the handrails

- 3) paint all the walls.

But it tuns out that the architect used lots of different materials all over the place. You find that you need different amounts of paint to color these items at different places.

So you sit down and first make a table which lists

- on the left all the doorknobs

- on the right for each doorknob the amount of color needed to paint it (a small amount, right, but suppose there are many many doorknob).

Maybe you recall from your highschool days that such a table is also sometimes called a function: it maps doorknobs to the amount of color coloring them.

What they don’t tell you in high school these days is that

- a function is also called a 0-bundle with connection

- a function is also called a (-1)-gerbe with connection.

So you already know what a (−1) (-1)-gerbe is! It’s just a very strange name for an assignment of milliliters to doorknobs.

Next, you try to sit down and make a table that has

- on the left all the handrails

- on the right the amount of color needed for them.

But now you run into trouble: the material of the handrails turns out to change every few meters. (It’s very modern architecture ). So instead of making a table which maps entire handrails to milliliters, you make a long, long table which maps

- each meter of handrail

- to the amount of color needed for it.

That takes a while. After pages of pages have been filled, this table is finished.

It doesn’t tell you immediately how much color is needed for painting a given handrail. But using the table it is easy to determine this amount:

for every piece of handrail, you chop it up (mentally) into 1-meter parts, look up the color needed for each of them seperatey and add all this up.

It happens that people have invented a funny name for this procedure: this is called

- a bundle with connection

or

- a 0-gerbe with connection.

These are just words. They mean exactly: a procedure for determining how to color handrails, and how to do it piecewise.

At this point, the attentive painter may alreay be able to go all the way up to inventing the concept of gerbes and even 2-gerbes himselves.

For all other painters out there, here is what a gerbe with connection would be:

next the walls need to be painted. They are made of even more variations of material than the handraisl! Grudgingly, you pull out pen and paper and make a huge list which

- has on the left an entry for each square meter of wall in the building

- and next to it on the right the estimated amount of color needed for that,

Cretaing that takes while. When done, you have a procedure that allows to compute the amount of color needed for an arbitrary wall:

for each square meter of it, look up the corresponding number of milliliters in your table, add that all up.

This procedure is what is called

- a 2-bundle with connection

or else

- a gerbe with connection.Urs Schreiber

Concrete

As we've seen, a connection describes how a point particle transforms when you carry it along a path:

          f
x--------->-------y     a path f from the point x to the point y:
                                we write this as f: x → y

Now we need a gadget that'll describe how a path transforms when you carry it along a path of paths:

           f
  --------->-------
 /        ||       \
x         ||F       y   a path-of-paths F from the path f to the path g:
 \        \/       /            we write this as F: f => g
  --------->-------
             g

To do this, we need to boost our level of thinking a notch, working not with "G-torsors" and "principal G-bundles" but instead with "G-2-torsors" and "G-gerbes". Here's how it goes:

We start by picking an abelian group G and a manifold X.

Then we pick a "G-gerbe" over X, say P.

What's that? It's a thing that assigns to each point x of X a "G-2-torsor", say P(x).

What's that? Well, it's a thing where if you pick two points in it, you get a G-torsor describing their difference!

Get it? This is the beginning of a story that goes on forever:

Two points in a G-torsor determine an element of G; two points in a G-2-torsor determine a G-torsor; two points in a G-3-torsor determine a G-2-torsor;http://www.math.ucr.edu/home/baez/week210.html

• http://www.math.ucr.edu/home/baez/week210.html
• What is a Gerbe? by Nigel Hitchin; see also the comment about it here and the discussion here.

Abstract

We describe abelian $U(1)$-gerbes, the case most used in physics, by local data in the same way that principal bundles are described by transition functions. Choose a good open cover $\{U_i\}$ of a manifold $X$ and write $U_{ij}=U_i\cap U_j$, $U_{ijk}=U_i\cap U_j\cap U_k$.

Cocycle description. A principal $U(1)$-bundle is given by functions $g_{ij}:U_{ij}\to U(1)$ with $g_{ij}g_{jk}g_{ki}=1$ on triple overlaps. A $U(1)$-gerbe is given one level higher, by functions $g_{ijk}:U_{ijk}\to U(1)$ satisfying, on quadruple overlaps, $$g_{jkl}\,g_{ikl}^{-1}\,g_{ijl}\,g_{ijk}^{-1}=1 ,$$ the Čech 2-cocycle condition, and $g_{ijk}$ and $g_{ijk}\,h_{jk}h_{ik}^{-1}h_{ij}$ (for any $h_{ij}:U_{ij}\to U(1)$) describe equivalent gerbes. Equivalently, following Hitchin, a gerbe is a collection of line bundles $L_{ij}$ on $U_{ij}$ with trivializations $g_{ijk}$ of $L_{ij}\otimes L_{jk}\otimes L_{ki}$ on $U_{ijk}$ satisfying the cocycle condition. Murray's bundle gerbes give a cover-independent version: a line bundle over the fiber product $Y\times_XY$ of a surjective submersion $Y\to X$, with an associative product.

Classification. Equivalence classes of gerbes form the sheaf cohomology group $H^2(X,\underline{U(1)})$ with coefficients in the sheaf of smooth $U(1)$-valued functions. The exponential sequence $0\to\mathbb{Z}\to\underline{\mathbb{R}}\to\underline{U(1)}\to0$, together with the fact that the sheaf $\underline{\mathbb{R}}$ of smooth real functions has no higher cohomology, gives $$H^2(X,\underline{U(1)})\cong H^3(X;\mathbb{Z}).$$ The image of a gerbe is its Dixmier-Douady class. This is the analogue, one degree up, of $H^1(X,\underline{U(1)})\cong H^2(X;\mathbb{Z})$, which classifies line bundles by their first Chern class. In homotopy terms, line bundles are classified by maps to $BU(1)\simeq K(\mathbb{Z},2)$ and gerbes by maps to $B^2U(1)\simeq K(\mathbb{Z},3)$; a gerbe is a principal 2-bundle for the 2-group $BU(1)$.

Connections. A connection on a gerbe consists of 1-forms $A_{ij}$ on $U_{ij}$ and 2-forms $B_i$ on $U_i$ with $$A_{ij}+A_{jk}+A_{ki}=-i\,g_{ijk}^{-1}dg_{ijk},\qquad B_j-B_i=dA_{ij}.$$ Then $H=dB_i$ agrees on overlaps and is a global closed 3-form, the curvature, and $H/2\pi$ represents the image of the Dixmier-Douady class in $H^3(X;\mathbb{R})$. A gerbe with connection assigns a holonomy in $U(1)$ to every closed oriented surface $\Sigma\to X$, which reduces to $\exp(i\int_\Sigma B)$ when $\Sigma$ lies in a single $U_i$ and to $\exp(i\int_VH)$ when $\Sigma=\partial V$. Gerbes with connection up to equivalence are classified by smooth Deligne cohomology (differential cohomology), which surjects onto $H^3(X;\mathbb{Z})$ with kernel $\Omega^2(X)/\Omega^2_{\mathbb{Z}}(X)$, the 2-forms modulo closed 2-forms with periods in $2\pi\mathbb{Z}$.

Physics. The Kalb-Ramond field of string theory is a gerbe connection: the string couples to it through the surface holonomy of its worldsheet, as a charged point particle couples to a $U(1)$ connection through its worldline holonomy. The same structure gives the Wess-Zumino term of the WZW model on a compact simple simply connected Lie group $G$; since $H^3(G;\mathbb{Z})\cong\mathbb{Z}$, the level is quantized. In the presence of a nontrivial $H$-flux, D-brane charges are classified by twisted K-theory, with the twist given by the Dixmier-Douady class. More general nonabelian gerbes were introduced by Giraud as stacks of groupoids that are locally nonempty and locally connected.

Why is it interesting?

Gerbes show up when we try to invent a kind of "higher gauge theory" that describes how not just point particles but 1-dimensional objects transform when you move them around. For example, the strings in string theory, or the loops in loop quantum gravity.http://www.math.ucr.edu/home/baez/week210.html

The Euler-Lagrange p-gerbes […] are singled out as being exactly the right coherent refinement of locally defined local Lagrangians that may be integrated over a (p+1)-dimensional spacetime/worldvolume to produce a function, the action functional.https://arxiv.org/abs/1601.05956

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2. Linear Algebra
3. Group Theory
4. Manifold
5. Product Spaces
6. Vector Calculus
7. Differential Forms
8. Fiber Bundles
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10. Cohomology