### More by Gustavsson on Non-Abelian Strings

#### Posted by urs

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 .

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 $LM$ 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

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

where $\gamma \in LM$ 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 $\varphi ={\varphi}^{a}{t}_{a}$ 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,

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.

## 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.