01
Feb
08

From the ‘n-Category Café’…

The folks at The n-Category Café wet my appetite today with 2 very interesting posts:

The second one, by Urs, touches in a topic that i hold dear: Knot Theory and QFT — and how these are related to gerbes, and how 2-gerbes are the fundamental geometric objects in Chern-Simons. Urs provides a series of references in a very rational way. Computing the Configuration space (which he called “Space of States”) of certain QFTs has been in my mind quite heavily for quite some time now, including for Chern-Simons. However, the question I have in mind has more to do with Wilson Loops and how their combination is related to Knot Theory; more to the point, how would combination of Wilson Loops in different representations relate to [Spontaneous] Symmetry Breaking. This definitely needs a better explanation than what’s here… but, it’ll have to wait (i’m a bit late to meet some friends!😉 ).

As for the first link, by John Baez, …, what can i say: Apparently Modular Forms show up in every interesting place nowadays! I had a vague idea about their appearance in Fermat’s Last Theorem but, given that they are related to Hecke operators, they also show up in Langland’s Geometrical Conjecture! And this is the part that really gets to me: the connections between Number Theory and Physics (specially Symmetry Breaking — as can be seen via a Higgs Bundle construction) have always seemed completely magical to my eyes.🙂

Anyway, now I’m really late… more on this later.😉

[]’s.


1 Response to “From the ‘n-Category Café’…”


  1. Wednesday, 12 Mar 2008; \11\UTC\UTC\k 11 at 05:46:39 UTC

    Hi,

    have only seen this entry now, by chance. But maybe I can still leave a comment.

    Understanding how the Wilson lines enter into Chern-Simons theory “from first principles”, i.e. just by using the idea that there is a 2-gerbe and that we do natural things to it, is something I really want to better understand eventually.

    As Bruce Bartlett just recently emphasized to me, there is a brief remark in Witten’s original article which seems to point the way to a deeper understanding, but which also, to the best of my knowledge, has never so far been followed up on:

    one of the ingredients that have to be added “by hand” in CS theory is the choice of representation and the trace of the Wilson line we take in that representation. Witten remarked that by the Borel-Weil-Bott theorem, the trace in any irrep of a compact group is itself given by a path integral over paths in a certain auxiliary space. This seems to be a hint that it should be “path integrals all the way to the bottom” for Chern-Simons theory with Wilson lines, in a way that has maybe not been fully exploited. (Or has it? Maybe I am just being ignorant).

    One last remark: you write:

    Configuration space (which he called “Space of States”)

    Hm, not sure if I agree with that terminology. Let’s see:

    we are talking about a QFT of generalized Sigma model type. These theories come with

    – a target space X

    – a parameter space Sigma

    – an n-gerbe P with connection on X .

    From this we get the

    – configuration space = space of maps from Sigma to X

    and the

    – space of states over Sigma = space of sections of the transgression of P to configuration space.

    If dim(Sigma) = n then the transgression of P to configuration space is an ordinary bundle, and its space of sections is the familiar vector (Hilbert, if we do it right) space of states over Sigma.

    For example, for the ordinary electromagnetically charged particle would have

    X = a (Riemannian) space

    Sigma = the point

    P = a line bundle over X

    configuration space = space of maps from the point to X = X

    transgressed P = P

    space of states = space of (L^2) sections of P = indeed the orinary Hilbert space of states that the quantum theory assigns to the point.

    Now for Chern-Simons theory: here we have

    X = a smooth model of BG

    Sigma = a (complex) surface

    P = the canonical Chern-Simons 2-gerbe with connection over BG

    then

    configuration space = smooth maps from the surface into BG = G-bundles with connection on Sigma

    space of states = (holomorphic) sections of the line bundle on this configuration space obtained from the transgression of P = conformal blocks over Sigma = Hilbert space of states that CS theory assigns to Sigma.

    While this makes it sound easier and more well understood than it is currently in practice, that’s the general pattern.


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