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

Layman

Student

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.

Researcher

Examples

Example1

Example2:

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