✎ Gerbes [The Physics Travel Guide]

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====== Gerbes ======

<tabbox Why is it interesting?> 
<blockquote>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.<cite>http://www.math.ucr.edu/home/baez/week210.html</cite></blockquote>

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
</note>
  
<tabbox Student>

<note tip>Gerbes are generalization of principal [[advanced_tools:fiber_bundles|bundles]].</note>

<blockquote>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;<cite>http://www.math.ucr.edu/home/baez/week210.html</cite></blockquote>

* http://www.math.ucr.edu/home/baez/week210.html
  * [[http://www.ams.org/notices/200302/what-is.pdf|What is a Gerbe?]] by Nigel Hitchin; see also the comment about it [[https://www.theguardian.com/education/2007/aug/21/highereducation.research|here]] and the discussion [[https://golem.ph.utexas.edu/category/2007/08/gerbes_in_the_guardian.html|here]].
 
<tabbox Researcher>

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

--> Example1#

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--> Example2:#

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</tabbox>

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