More by Gustavsson on Non-Abelian Strings

January 2, 2006

More by Gustavsson on Non-Abelian Strings

Posted by Urs Schreiber

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A while ago I had discussed a preprint by Andreas Gustavsson dealing with ‘nonabelian strings’. Now he has a followup:

Andreas Gustavsson
The non-Abelian tensor multiplet in loop space
hep-th/0512341 .

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One goal of the new preprint is to relate nonabelian 1-forms on loop space to superconformal field theories in six dimensions by trying to fit these loop space 1-forms into a supersymmetry multiplet.

Andreas Gustavsson begins by following the remark which I made in reaction to his previous proposal. (See in the previous entry the paragraph introduced by “I am wondering if we cannot slightly simplify this construction as follows:”).

Namely, from any $\mathrm{Lie}(G)$ -valued 2-form on a space $M$ we get a 1-form on the loop space $L M$ with values in the loop algebra (affine algebra at level 0) of $\mathrm{Lie}(G)$ in the obvious way: if $\lambda_a(\sigma)$ is a basis of the loop algebra $L \mathrm{Lie}(G)$ with Lie bracket

(1)

[\lambda_a(\sigma),\lambda_{a'}(\sigma')] = C_{ab}{}^c \delta(\sigma-\sigma) \lambda_c(\sigma)

then from the 2-form $B \in \Omega^2(M,\mathrm{Lie}(G))$ on $M$ we get, in components, the 1-form

(2)

\Omega^1(L M, L \mathrm{Lie}(G)) \ni A = \gamma \mapsto \int_\gamma d\sigma\; B^a_{\mu\nu}(\gamma(\sigma)) \gamma^{\prime \mu}(\sigma)\, \mathbf{d}\gamma^\nu(\sigma) \; \lambda_a(\sigma)

where $\gamma \in L M$ is a loop.

(This differs in spirit from other constructions which I have discussed here at length. As opposed to these, the above construction is not functorial in that it does not lift from loops to paths (at least not without further effort). On the other hand, it evades the constraints found otherwise.)

Andreas Gustavsson promotes, in a similar spirit, the entire content of a supersymmetry tensor multiplet of fields from target space to loop space. For instance given a scalar $\phi = \phi^a t_a$ with values in $\mathrm{Lie}(G)$ on target space we can imagine cooking up something like a scalar on loop space simply by writing, locally and in components,

(3)

\int_\gamma ds\; \gamma^{\prime \mu}(\sigma) \phi^a(\gamma(\sigma)) \lambda_a(\sigma)

with values in $L \mathrm{Lie}(G)$ . Doing this with an entire susy multiplet it is not hard to see that the result is a collection of fields on loop space which still transforms under a supersymmetry algebra. This is discussed in section 3 of the paper. The nice thing about this is that by its very construction the susy multiplet on loop space reduces to that on target space when ‘compactified on a circle’.

The purpose of section 4 of the paper is not entirely clear to me. The main result is the formula for the loop space curvature of the loop space 1-form given above. I had given this formula in the second but last equation of said previous entry. The bulk of section 4 is occupied by a discussion revolving around a peculiar handling of the delta-distribution on loops, which I cannot quite follow. In particular, it is not clear to me why equation (4) is suddenly replaced by equation (42). Comments are welcome.

Posted at January 2, 2006 12:31 PM UTC

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Re: More by Gustavsson on Non-Abelian Strings

It seems to me that what Gustavsson is doing is similar in spirit to my approach. In particular, the relation between A and B,

A_u[C] = \int ds B_uv(C(s)) C’^v(s)

implies an assumption about locality, which seems to be absent in your own writings (please correct me if I am wrong). This is essentially the relation that I have proposed, except that his formula is too restrictive. What I realized in one of our discussions about a year ago is that A must be allowed to depend on the vector C’(s) in a more complicated way, which is encoded in the lattice theory by having two independent fluctuating variables per plaquette. But if we replace above

B_uv(C(s)) => B_uv(C(s),C’(s)),

or perhaps even B_uv(C(s),C’(s),C”(s)), then I believe that this is the right way to go. Note that this generalization does not ruin the local character of B.

Posted by: Thomas Larsson on January 5, 2006 11:36 AM | Permalink | Reply to this

Re: More by Gustavsson on Non-Abelian Strings

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This is essentially the relation that I have proposed

It is similar. For instance in that it shares the property of lacking horizontal invertibility. I have thought a while about it and I think I can construct a weak monoidal category which plays the role of the ‘gauge 2-group’ for Andreas Gustavsson’s approach and would hence allow to define it globally. But I don’t see how to include horizontal inverses and turn this into a weak 2-group.

Of course it does not have to be one for this to be useful. But with the technology that I am in command of I’d need at least adjunctions in the ‘gauge monoid’ in order to produce a global picture with gluing cocycles.

(please correct me if I am wrong).

Depends on what precisely you mean. I think I can guess what you have in mind. I am not sure if I would call this ‘non-locality’.

You are probably thinking of the parallel transport of the 2-form by means of the 1-form along the loop. This looks non-local, but note that it simply implements the semidirect product operation. The 2-connections with values in 2-groups which I have been talking about with John Baez are perfectly local in that they are - unsurprisingly - functorial. That is, for every surface $\Sigma$ which is composed of two smaller surfaces $\Sigma_1$ and $\Sigma_2$ , the parallel transport along $\Sigma$ is just the composition of the parallel transport along $\Sigma_1$ and $\Sigma_2$ , seperately. That’s locality.

In fact, that’s precisely the notion of locality in field theory. The reason why a D-dimensional local field theory can be described by a functor on cobordisms is precisely because functoriality here encodes the physical concept of locality.

Posted by: Urs on January 5, 2006 12:08 PM | Permalink | Reply to this

Re: More by Gustavsson on Non-Abelian Strings

It is similar. For instance in that it shares the property of lacking horizontal invertibility.

Hm. I assume that the horizontal product is the diagram which looks like the figure 8 laying down, and the vertical product looks like a setting-sun diagram without external legs. If so, then I agree that I have no horizontal inverse. In fact, the horizonal product does not really make sense in my approach; it is some kind of singular limit.

The products that do make sense to me are two deformations of the vertical product. The first deformation amounts to sliding the two half-suns (above and below the horizon) relative to each other. In the second, the half-suns are scaled by different amounts, so the result looks like half an avocado with the whole nucleus kept, looked at from the side. Up to smooth deformations, these are the only possibilities to glue two disks to a new one, keeping the “in” and “out” sides intact. Which is natural if you want surfaces to describe parallel transport in path space.

That is, for every surface S which is composed of two smaller surfaces S 1 and S 2 , the parallel transport along S is just the composition of the parallel transport along S 1 and S 2 , seperately. That’s locality.

My 2-product also has this property.

Posted by: Thomas Larsson on January 5, 2006 7:10 PM | Permalink | Reply to this

Re: More by Gustavsson on Non-Abelian Strings

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My 2-product also has this property.

There is a TV commercial here in Germany, where two former classmates meet after many years. One of them pulls a bunch of photographs out of his wallet, throws them on the table: “my house, my car, my boat,…”.

But the other picks his wallet and some even better photographs: “my house, my car, my boat … my bank”.

Sometimes our conversation reminds me of that commercial. :-)

So, to reassure you, all I was saying was that my house, er, I mean ‘my’ surface transport is local, which you seemd to doubt. I know that yours is, too.

And I think, when formulated suitably, your car also has a horizontal product. As I said before, I believe your construction amounts to looking at 2-functors (or double functors) to $\mathrm{Vect}$ . And this has horizontal products.

Posted by: Urs on January 5, 2006 7:36 PM | Permalink | Reply to this

Re: More by Gustavsson on Non-Abelian Strings

Sorry, I did not try to steal credit. As I have mentioned before, similar models were put forward before me by Nepomechie, Orland, Maillet, Nijhoff and maybe others in the 1980s. But since we have two different notions of 2-gauge theory, calling them mine and yours seemed like the simplest way to distinguish between them. I am perfectly happy to talk about 2-gauge theory in the sense of Gustavsson and in the sense of Baez/Pfeiffer instead.

Since this post was about Gustavsson’s work, it seemed on-topic to make a comment on 2-gauge theory in his sense. And my comment was that his B-field does not depend on the most general local data available, which includes not only C(s) but also C’(s), C”(s), etc.

You know that I cannot follow your fancy category-theoretical notation. However, if you claim that some relevant notion of inverse is missing, then I think you are wrong. Recall from the lattice formulation in my (!) e-print that the zero-curvature condition is nothing but the Yang-Baxter equation. So you seem to be saying that a Yang-Baxter matrix lacks a horizontal inverse. What does it mean, and why should we care?

Posted by: Thomas Larsson on January 7, 2006 5:45 AM | Permalink | Reply to this

Re: More by Gustavsson on Non-Abelian Strings

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that his B-field does not depend on the most general local data available

I guess you are right. In general, I believe, local (nonabelian) connection 1-forms on path space (or loop space) that have rep-invariant local surface holonomy are easy to come by. A dime a dozen, if you wish.

What is much more restrictive is the condition that these 1-forms can glue appropriately to a global object.

This is a problem I currently cannot quite resolve for Gustavsson’s loop space 1-form. And it has to do with the lack of horizontal inverses.

But actually, as I said, (weak) horizontal inverses are not necessary to make a local surface connection global - it suffies to have (special) adjunctions. Think of that like a weak inversion weakened even one more step.

I currently don’t see how the structure 2-category which seems to encode Gustavsson’s proposal would admit such adjunctions. But I might not have thought hard enough.

On the other hand, in the category Vect of vector spaces we don’t have horizontal inverses in general, either, but we do have adjunctions. So here it does work! That was one point of what I wrote about TFT from 2-transport. There I worked out how the purely topological part of local 2-functors to Vect glue in a way that reproduces 2D TFT presciption.

Hence, in as far as ‘your’ approach is identifiable with 2-functors (or double functors or whatever) to Vect, I see no problem. It is still true that there are no horizontal inverses (except for 1D vector spaces) but there are adjunctions.

Posted by: Urs on January 7, 2006 1:13 PM | Permalink | Reply to this

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January 2, 2006