Exotic Statistics for Strings in 4d BF Theory

[marcus]

Baez et al revised the title of their March 2006 paper to say "strings" instead of "loops" and reposted it

http://arxiv.org/abs/gr-qc/0603085
Exotic Statistics for Strings in 4d BF Theory
John C. Baez, Derek K. Wise, Alissa S. Crans
41 pages, many figures. New version has minor corrections and clarifications, and some added references
Subj-class: General Relativity and Quantum Cosmology; Geometric Topology

"After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the 'loop braid group'. This group has a set of generators that switch two strings just as one would normally switch point particles, but also a set of generators that switch two strings by passing one through the other. The first set generates a copy of the symmetric group, while the second generates a copy of the braid group. Thanks to recent work of Xiao-Song Lin, we can give a presentation of the whole loop braid group, which turns out to be isomorphic to the 'braid permutation group' of Fenn, Rimanyi and Rourke. In the context 4d BF theory this group naturally acts on the moduli space of flat G-bundles on the complement of a collection of unlinked unknotted circles in R^3. When G is unimodular, this gives a unitary representation of the loop braid group. We also discuss 'quandle field theory', in which the gauge group G is replaced by a quandle."

 

[marcus]

Woit flags two Freidels

today 11 May Peter woit blogged links to 2 recent Freidel papers
I was happy because Freidel is for me what Witten might be for some other people, which is to say a weathervane on the roof that one observes from time to time to see which way he points.

Peter said:
"In the category of interesting-looking work that I haven’t yet had time to read carefully and think about, here are two papers on an approach to studying pure Yang-Mills theory, by Freidel and Freidel and collaborators."

The "Freidel" link was
http://arxiv.org/abs/hep-th/0604185
On pure Yang-Mills theory in 3+1 dimensions: Hamiltonian, vacuum and gauge invariant variables
Laurent Freidel
35 pages
"In this work we discuss an analytic approach towards the solution of pure Yang-Mills theory in 3+1 dimensional spacetime which strongly suggests that the recent strategy already applied to pure Yang-Mills theory in 2+1 can be extended to 3+1 dimensions..."

Peter's "Freidel and collaborators" link was
http://arxiv.org/abs/hep-th/0604184
Towards a solution of pure Yang-Mills theory in 3+1 dimensions
Laurent Freidel, Robert G. Leigh, Djordje Minic
12 pages
"We discuss an analytic approach towards the solution of pure Yang-Mills theory in 3+1 dimensional spacetime. ... we point out unexpected parallels between pure Yang-Mills theory in 2+1 and 3+1 dimensions. The most important parallel shows up in the analysis of the ground state wave-functional especially in view of the numerical similarity of the existing large N lattice simulations of the spectra of 2+1 and 3+1 Yang Mills theories."

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Amusingly (at least for me) even though they are working in 4D the researchers often refer to "string tension". Probably there is some obvious explanation for this which escapes me at the moment.
================

I was glad to see that Peter (who does not seem especially interested in F's spinfoam work) took note of Freidel's papers when the topic came into range of his field theory interest.

 

[selfAdjoint]

Amusingly (at least for me) even though they are working in 4D the researchers often refer to "string tension". Probably there is some obvious explanation for this which escapes me at the moment.

Marcus, "string tension" is a common phenomenological reference in QCD. Note that among other non-perturbative results the papers are pursuing is QCD confinement, and this is often modeled as a string with quarks at each end. That was the fons et origo of SST but has nothing else to do with it.

These are very important papers - you ask what will be cited in the future, I'll bet it will be them. Note that down the road they seem to have made a successful start on is Clay Prize number one, and a million bucks.

 

[marcus]

... and this is often modeled as a string with quarks at each end.

this is what I suspected they were talking about! not actual "stringy" string tension.

You put it is clear historical context:

That was the fons et origo of SST but has nothing else to do with it.

I am delighted with this fons et origo.

I will, on your suggestion, add the Freidel paper to the nominations

 

[marcus]

These are very important papers - you ask what will be cited in the future, I'll bet it will be them. Note that down the road they seem to have made a successful start on is Clay Prize number one, and a million bucks.

more information about that

http://www.claymath.org/millennium/Yang-Mills_Theory/

statement of problem:

"Yang-Mills Existence and Mass Gap

Prove that for any compact simple gauge group G,
quantum Yang-Mills theory of R⁴ exists
and has a mass gap Delta > 0."

 

[arivero]

this is what I suspected they were talking about! not actual "stringy" string tension.

You can speak of the real "stringy" string tension in any dimension, as far as you do not try to quantise it.

At the end, the issue of fundamental strings is that they propose a quantisation method that when superimposed with Lorentz invariance and some other general considerations happens to be consistent only for a concrete dimension (D=26 for "bosonic" string).

(actually I can not understand why they ask for lorenz invariance and then they go across these complicated compactifications of space; I suppose they are still lorentz invariant for short distances, but then they proceed to claim that at short distances space time structure is very different than usual...)

 

[marcus]

...(actually I can not understand why they ask for lorenz invariance and then they go across these complicated compactifications of space; I suppose they are still lorentz invariant for short distances, but then they proceed to claim that at short distances space time structure is very different than usual...)

Alfonso the Wise was asking about something like this and concluded that God should have had someone practical advising him. But not to worry. Remember to pray to Kepler on the 15th, the patron saint of new ideas when you need them.