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Insights Why Higher Category Theory in Physics? - Comments

  1. Jan 4, 2017 #1

    Urs Schreiber

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  2. jcsd
  3. Jan 4, 2017 #2
    Just a few days ago, for the first time, a paper by Patricia Ritter gave me an explanation that I could understand, for the relevance of n-categories to objects like branes - the categorical identities express the equivalence of different ways of doing certain integrals over a volume, e.g. where in effect you might first integrate in the x direction, then along the xy plane, then throughout the xyz volume; but you might have done all that for a different order of x,y,z... the result needs to be the same for all orderings, and that leads to the categorical formulation of higher gauge theory.

    I want to emphasize, that's not exactly what she says, that's me trying to dumb it down to the simplest way of saying it. But am I even approximately correct in this interpretation?
     
  4. Jan 5, 2017 #3

    Urs Schreiber

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    Yes, that's one good way of thinking about it. This is the motivation from "higher parallel transport".

    Like so: the structure of a group (an ordinary group) is exactly what one needs in order to label the edges in a lattice gauge theory: the group product and associativity give that edge labels may be composed, and inverses gives that going back and forth along the same edge picks up no curvature. Of course this is not restricted to the lattice. In general, group-valued gauge fields are exactly the right data to have consistent Wilson line observables

    Now a 2-group (categorical group) is, similarly, exactly the data needed to consistencly label edges AND plaquettes in a consistent way (with possibly different labels for each). For instance associativity now includes a 2-dimensional codition which says that with four plaquettes arranged in a square, then first composing horizontally and then vertically is the same (in fact: is gauge equivalent to) first composing vertically and then horizontally.

    Again this is not restricted to the lattice. Generally, 2-group valued gauge fields are exactly what one needs for consistent Wilson surfaces.
     
    Last edited: Jan 5, 2017
  5. Jan 5, 2017 #4
    Fascinating topic Urs!
     
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