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August 23, 2007

Wilson Loop Defects on the String

Posted by Urs Schreiber

Just heard a very interesting talk by Samuel Monnier on his work

A. Alekseev, S. Monnier
Quantization of Wilson Loops in Wess Zumino Witten models
hep-th/0702174 .

This was a little bit of a déjà-vu to me, since it addressed questions in 2-dimensional conformal field theory, a simplified version of which originally, back then, made me run into higher gauge theory while looking into strings, then contact John Baez, and – the rest is history – mine at least.

The question is simple: given a 2-dimensional quantum field theory coming from a σ\sigma-model which describes the propagation of string-shaped objects in some space, one may wonder if there are any quantum observables of this theory which come, classically, from the holonomy of some connection around the string.

The way I had originally encountered this question was in terms of Pohlmeyer invariants: Klaus Pohlmeyer was interested in understanding if the quantum observable algebra of the (bosonic) string (propagating on a Minkowski target space) could be understood entirely in terms of holonomies of constant gl(n)\mathrm{gl}(n)-connections on target space around the string, for arbitrary nn.

In fact, in his work the question was motivated by an attempt to apply the method of Lax pairs to the bosnic string, which has the feature of an infinite-dimensional integrable system.

While, classically, these holonomy-observables for the string are easily written down, using the standard formulas for path-ordered exponentials, one immediately runs into the problem of how to regularize these observables (i.e. how to normal order them) such as to yield admissible quantum observables.

Pohlmeyer, his collaborators and students had followed a certain iterative approach to solve this problem. The current endpoint of the developments in these directions, as far as I am aware, is

C. Meusburger, K.-H. Rehren
Algebraic quantization of the closed bosonic string math-ph/0202041 ,

which achieves a partial understanding of a possible quantization of the algebra of Pohlmeyer invariants.

For some reason, back then, I had gotten interested in this and was wondering why people didn’t simply re-express the Pohlmeyer invariants in terms of the well-known DDF invariants.

After I didn’t seem to get a satisfactory answer to this question, I made this observation a research article

U. S.
DDF and Pohlmeyer invariants of (super)string hep-th/0403260 .

But, alas, it turned out that this insight had in fact been published twenty years before, but apparently forgotten, as described here.

In any case, I kept thinking about these string observables. In most parts of the 2d CFT community these entities were considered a lot, but to a rather different end than in Pohlmeyer’s original program: their quantum version played the role of generators of boundary states. These are, heuristically, to be thought of as states which describe condensates of closed strings which form a D-brane, i.e. a boundary condition for open strings.

The particular boundary operators that the Pohlmeyer observables corresponded to, coming from the Wilson line observable of a connection on the target space, describe D-branes with Chan-Paton bundles that carry a connection

Super-Pohlmeyer invariants and boundary states for non-abelian gauge fields

Strikingly, the structures appearing when dealing with these super Pohlmeyer observables have a close resemblance to structures which one encounters in the theory 2-connections. Back then I tried to understand this using a certain formalism for Deformations of 2d SCFTs, since that seemd to admit a very natural way to include the data provided by a 2-connection with values in a 2-group.

This is how I originally entered the world of categorified gauge theory. With hindsight, the time at this point wasn’t really ripe to seriously try to address boundary states for D-branes using this relation. And in fact I have so far not come back to this issue, being busy thinking about various other things.

But today I was starkly reminded of these times, when Konnier talked about of what is, essentially, the problem of quantizing the Pohlmeyer invariants, not for the simple case of a string on Minkowski space, but for the Wess-Zumino-Witten model, describing a string propagating on a compact group manifold.

This investigation had been initiated in

Constantin Bachas, Matthias Gaberdiel
Loop Operators and the Kondo Problem

These authors very much emphasize that inserting the Wilson loop operator (the “Pohlmeyer observable”) is an instance of inserting a defect line on the worldsheet.

Alekseev and Monnier succeed in finding a quantum regularization of the Wilson line operator for the string propagating on a group, and they extract a bunch of interesting results from that.

Posted at August 23, 2007 12:28 PM UTC

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Re: Wilson Loop Defects on the String

Interesting read, Urs. Sadly I don’t think I’ll ever understand this stuff till my dying day, but underneath it all lies some nice geometry, that much I can at least see! At least I know who to come to when I have to duel with these monsters :-)

I see the words “defect line” featuring prominently in your posts nowadays. I find it a great paradox that at the heart of all this smooth geometry is… some kind of defect. Spooky! , as Ace Ventura would say.

Posted by: Bruce Bartlett on August 23, 2007 6:10 PM | Permalink | Reply to this

Re: Wilson Loop Defects on the String

I see the words “defect line” featuring prominently in your posts nowadays. I find it a great paradox that at the heart of all this smooth geometry is some kind of defect. Spooky! , as Ace Ventura would say.

I see what you mean. But instead of defect line being a defect in the logic of things, they fit in beautifully.

As I tried to indicate here and there – for instance here: Eigenbranes and CatLinAlg and there: Branes, Bi-Branes, 2-Vectors, 2-Linear Maps, what looks like a “defect” to the naked eye, is really a “2-liner transformation” under the nn-categorical microscope.

The best way to get an intuition for what these defect lines are is, I think, to realize that they are pretty much entirey analogous to the data you put on a triangulation line when computing the surface holonomy for a gerbe with connection (as here).

That’s the right way to think of it, I believe. That, and some basic nonsense about Fourier-Moukai. I guess Simon can tell you more about that.

Posted by: Urs Schreiber on August 23, 2007 6:25 PM | Permalink | Reply to this

Re: Wilson Loop Defects on the String

Howcome they’re called defect lines? When I see the word “defect”, my mind thinks of something like a crystal structure which has a tear, a crack in it at some point. I’m disturbed by how the word “defect” apparently crops up in the context of a Lie group, the smoothest thing in mathematics! In fact, I find it so disturbing my mind goes a little crazy and starts to soliloquize:

The eye wink at the hand; yet let that be, Which the eye fears, when it is done, to see.


I read through the links you gave, and I’ve even looked at Bi-branes: Target Space Geometry for World Sheet topological Defects, but the viscosity of my mental molasses is unfortunately extremely high. I was also bamboozled by the fact that the only appearance of the word “defect” in that paper was in the title! Truly, physics is a dark and mysterious art.

Posted by: Bruce Bartlett on August 23, 2007 10:03 PM | Permalink | Reply to this

Re: Wilson Loop Defects on the String

In re: Bruce’s last line, see the review of
Smolin’s book in the Sept 07 AMS Notices.

Posted by: jim stasheff on August 24, 2007 1:17 AM | Permalink | Reply to this

Re: Wilson Loop Defects on the String

Howcome they’re called defect lines?

Because that used to be the standard name for the first incarnation of these operations:

for instance the 2-dimensional Ising model, which is a crude model for a bunch of elementary magnets located on the vertices on a rectangular grid, is one of the standard examples of a 2-dimensional CFT: as you take a continuum limit in which the lattice spacing shrinks to zero while at the same time the temperature scales in a certain way, the result will show scale independence: there will be regions of positive magnetization inside regions of negative magnetization inside those of positive, all the way from infinity to infinity.

In that limit, this model hence gives rise to a 2-dimensional conformal (= scale invariant) field theory.

Now, one can insert a “defect line” into this setup by imagining drawing any line onto the lattice, and forcing the coupling of our little magnets to change sign as this line is crossed.

So this is like two different Ising models connected along a common line.

If one then takes that conformal continuum limit, this line becomes one of those CFT defect lines I mentioned. And that’s, as far as I know, where the name comes from, originally.

So you can think of a defetc line in CFT, roughly, as something like a 2-sideed boundary condition: two different CFTs (of same central charge) may touch along a defect line.

Of particular interest are the “duality defects” which relate two CFTs that are not the same, but equivalent.

For instance to T-dual 2D CFTs should be connectable by a defect line. So you could imagine computing a string correlator on, say, a torus, by treating half of that torus as mapping into some spacetime, and the other half as mapping into the corresponding T-dual spacetimes. The defect line on the torus would then encode that information which tells you how to glues these two theories, which are the same, but different.

the only appearance of the word “defect” in that paper was in the title!

The goal of that paper is to find a differential geometric target space realization of the defect lines which were already discussed in the purely abstract, algebroaic, worldsheet based discussion in Duality and defects in rational conformal field theory.

If you look at the pictures which I had drawn here you find that these defect lines arise as the most natural thing in the world:

there are algebras around (“algebras of open string states”), there are modules for them around (describing boundary conditions, D-branes). So there is nothing more natural than also using the bimodules for these algebras. And indeed, one can discuss rational CFT algebraically including such bimodules.

But, apart from some simple examples like the Ising model defect line I mentioned above, people had rather little idea what these algebriacally defined defect lines corespond to in target space interpretations. That paper I mentioned argues that in as far as D-branes come from gerbes modules (not all of them do, of course), defect lines come from something now called gerbe bimodules.

Posted by: Urs Schreiber on August 24, 2007 8:50 AM | Permalink | Reply to this

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