View Single Post


On Wed, 7 Jul 2004, Urs Schreiber wrote: > Well, at least I think so. Since we have free afternoon today I'll go back > to my hotel room and try to write this up in detail. Ok, what I wrote before is essentially correct, but very intransparent, as you may have noticed. ;-) There is a much cleaner way to think about this: So consider on loop space the nonabelian 2-form gauge covariant exterior derivative nabla $= d + \oint_A (B) ,$ where I use the notation as in Hofmann's paper http://www.arxiv.org/abs/hep-th/0207017, but, as I have emphasized before, with the difference that I have A-holonomies $U_A(\sigma_1,\sigma_2)$ on both sides of the insertion, so that explicitly nabla = $\int_0^{2\pi}d\sigma E^{\dagger \mu}(\sigma) (\partial_\mu(\sigma)$ + $U_A(0,\sigma) B_{\mu\nu}X^{\prime \nu}(\sigma) U_A(\sigma,0)$ ) where $E^{\dagger}$ is the operator of exterior multiplication by the respective differential form on loop space. Now, the nice thing about the loop space perspective is that here we are dealing simply with an ordinary connection. The worldsheet is a line in loop space and this line couples to the loop space 1-form $\oint_A (B)$ in just the usual way. This means that gauge transformation of this connection will follow the usual rules. We can work them out and see what they imply for the target space theory, instead of just trying to guess the latter. The gauge transformation on the connection $\oint_A (B)$ is obtained as usual by specifying any group-valued function U on loop space and setting $$\oint_A (B)\mapstoU \circ \oint_A (B) \circ U^\dagger + U (dU^\dagger) .$$ Now first of all let's look at _global_ gauge transformations, i.e. those for which $U(dU^\dagger) = 0,$ which is the case when U does not depend on the embedding coordinates of the string/loop. Given any such U, a little calculation demonstrates the following: Given _any_ group-valued function V on the loop which coincides with U at the arbitrary basepoint $$V = V(\sigma)$$ V(0) = U we have the identity $$U \circ \oint_A (B) \circ U^\dagger = \oint_{A'} (B')$$ where $$A' = V A V^\dagger + V (d V^{dagger})B' = V B V^\dagger .$$ This is the first of the gauge invariances of a 2-form connection, namely essentially the ordinary gauge transformation of the 1-form connection A together with the obvious action on B. So this demonstrates that _global_ gauge transformations on loop space are equivalent to ordinary gauge transformations in target space. I should emphasize again that this crucially depends on that second $U_A$ factor in my definition of the loop space connection, which is derived from boundary state deformation theory. Without that factor the above does not seem to have a meaningful analogue. The next step is the more interesting one. There should be a further gauge transformation associated to the cohomology equivalence classes of the 2-form B, roughly. For instance as stated in Lahiri's papers, e.g. http://www.arxiv.org/abs/hep-th/0109220, one expects there to be a gauge invariance of the form $$B \mapsto B + d_A AA \mapsto A .$$ As far as I can see this is the invariance that is not respected by the action which is given in http://www.arxiv.org/abs/hep-th/0206130 . I am a little confused about the status of the proposal for this second gauge transformation. Why is this expected to be the correct form for nonabelian 2-forms? Is this a definition, a derived result, or a guess? The reason I am asking this question is that I believe to have evidence that this gauge transformation needs some correction terms. In order to demonstrate this I'll just consider a local gauge transforation on loop space, where I believe to know the correct form of the 2-form gauge $_connection_,$ and simply see what the resulting effect on that connection is. So consider a group-valued 1-form $\Lambda$ on loop space, and consider to first order the gauge transformation $$U = 1 + i \oint_A (\Lambda) + \cdots ,$$ in direct analogy with the respective expression for target space theories. Computing the transformation of the connection $\oint_A (B)$ with respect to this transformation gives, by the above formula, something like $$\oint_A (B)\mapsto\oint_A (B + d_A B)[/itex] + $i\oint_A (\Lambda,B - d_A A) - i \oint_A (B + d_A A,\Lambda)$$ Interestingly, the first line is precisely the expected gauge transformation on B. But, due to the action of d on the [itex]U_A$ holonomies and due to the commutation with $\oint (\Lambda),$ there appear correction terms in the second line. As these involve path-ordered integrals of two quantities over the loop, they do not have any analogy on target space at all! (I think.) By construction, the above is a valid gauge transformation on loop space. Unless I am confused this shows that we won't easily find any target space action in terms of a _local_ field theory of point particles which respects this invariance. On the other hand, it is of course possible to compute the gauge curvature $$(d + \oint_A (B))^2$$ on loop space, take it's Hodge-square, trace over it and integrte it over loop space to obtain a YM-like theory whose parameter space is loop space. I'd expect, but haven't checked, that this action, which manifestly has the full loop space gauge symmetry, flows to some of the target space actions which have been written down before.