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November 24, 2004

Categorified Points: Strings as Morphisms

Posted by urs

Over at sci.math.research there are some comments on categorification and its relation to string theory.

Of course categories, being the language in which god wrote math, show up all over the place once people start to seriously do some mathematical physics, and hence here and there in string theory, but apart from their ‘general purpose applications’ the notion of categorization has a rather special nature, since it always amounts to identifying some well known things as ‘points’ and generalizing them to linearly extended things: morphisms, arrows - strings.

The most obvious example is the categorification of the configuraton space of a single particle: This is just some sort of path space over the original config space (where the paths are morphisms between the point objects in the obvious way) which can generically be identified with the configuration space of a single string. In fact, such a space is known as a 2-space.

If our particle was a superparticle its configuration space was something like the exterior bundle over the original configuration space and similarly we expect the config space of the superstring to be the exterior bundle over a 2-space, which it is. The superparticle may be governed by supersymmetric quantum mechanics of the well-known form e Wde Wψ=0 =e W d e W ψ and lifting that to path space indeed produces SCFTs describing string propagation in various backgrounds encoded in W.

I should take the time to formalize that a little better, but it seems indeed that one can justly say that strings are categorized point particles, in a sense.

My recent occupation with higher gauge theory has added considerably to my impression that this is much more than a general abstract nonsensical play of words. If strings on stacks of 5-branes really are to nonabelian gerbes as points are to fiber bundles and given the fact that these nonabelian gerbes are essentially categorified fiber bundles precisely along the space2-space line of reasoning sketched above, this shows that there is nontrivial string physics made visible by the strings-as-categorized-points point of view which is otherwise not or not as clearly tractable - it seems to me.

Posted at November 24, 2004 7:36 PM UTC

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Category theory and physics

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Posted by: Urs on November 29, 2004 9:12 PM | Permalink | Reply to this
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