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advanced_tools:gerbes

Gerbes

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

Gerbes are generalization of principal bundles.

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

Abstract

The motto in this section is: the higher the level of abstraction, the better.

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

Contributing authors:

Jakob Schwichtenberg Paul Tremper
advanced_tools/gerbes.txt · Last modified: 2018/05/03 09:42 by paul