Three ways to loop flatness

July 22, 2004

Posted by Urs Schreiber

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Today I was contacted by Luiz Agostinho Ferreira who kindly called my attention to the paper

Orlando Alvarez & Luiz. A. Ferreira & J. Sánchez Guillén : A New Approach to Integrable Theories in any Dimension (1998)

that regrettably I was not aware of before but which discusses matters that are relevant for my recent work hep-th/0407122 on strings in nonabelian 2-form backgrounds.

The above paper is motivated from generalizing the method of Lax pairs from 1+1 to higher dimensions. Integrable field theories in 1+1 dimensions admit the construction of a certain connection on 1+1d spacetime which is flat iff the equations of motion hold, so that all holonomies of that connection around, say, $S^{1}$ space, are preserved in time and hence are conserved charges.

The idea is to generalize this procedure to $d = 1 + p$ dimensional field theories by constructing $p$ -holonomies for $p > 1$ which associate group elements with a $p$ -dimensional manifold, so that when the relevant $p$ -curvature vanishes these are again homotopy invariants of these surfaces.

For $p = 2$ this amounts to the study of connections on loop space, which is precisely what I was concerned with in the above mentioned paper (as discussed here before: I, II, III).

Ferreira et al. work on parameterized but based loop space, where all the loops have a given point in target space at parameter $σ = 0,2 π$ in common. This differs a little from the setup needed for string theory, where there is no requirement about this point. So the based loop space in that paper is a subspace of the full loop space relevant in string theory. But as far as I can see this does not restrict any of the constructions and statements made by Ferreira et al., they all generalize easily to unbased loop space.

That said, the first thing to observe is that the general form of the loop space connection used by Ferreira et al. (see their equation (5.1)) is precisely that which dropped out from the boundary state deformation which I used (my equation (1.2) and (3.15)), namely (in the notation adapted from Hofman’s paper but modified as in my equation (3.12))

(1)

d^{(A) (B)} = d + i \oint_{A} (B) .

Here $d$ is the exterior derivative on loop space $A$ is a 1-form and $B$ a 2-form, both taking values in a nonablelian Lie-algebra, on target space and the integral is that over the pull-back of $B$ over the loop with $A$ -holonomies used to parallel tranport everything to the point $σ = 0$ .

I argued in my paper that local gauge transformation on loop space translate into sensible 2-level gauge transformations on target space only when $B = - F_{A}$ , in which case it turns out that the above connection is flat, which again makes sense since it implies that toroidal worldsheets don’t see the nonabelian background.

But $B = - F_{A}$ is not the only condition which implies flatness of the connection on loop space. Alvarez, Ferreira and Guillén don’t need to mind consistency of any boundary state deformations, of course, and they find two further sufficient conditions for flatness.

With the above notation there is a simple calculus for dealing with differential forms on loop space which are multi-path-ordered integrals (studied intensively by Chen) over the loops. Using this one finds for the curvature

(2)

{(d^{(A) (B)})}^{2} = {(d + i \oint_{A} (B))}^{2} = \oint_{A} (d_{A} B) + \oint_{A} (F_{A}, B) - \oint_{A} (B, F_{A}) - \oint_{A} (B) \oint_{A} (B) .

Only the first term is understandable from the purely target space perspective, since this is just the ‘pullback’ (on one index) of the $B$ -field strength to the loop up to these $A$ -holonomies and integration.

The other terms have no direct target space analog. In particular, the field strength $F_{A}$ of the $A$ -field shows up all the time, due to the action of the exterior derivative on the $A$ -holonomies.

Alvarez, Ferreira & Guillé demand that

(3)

F_{A} = 0

in order for these terms to vanish.

This is natural from the point of view of integrable systems, but not so natural when we want to think of $F_{A}$ really as the field strength of a physical background gauge field, like in string theory, instead of as a auxiliary connection coming from a Lax-pair method. The authors show that several known integrable systems have ‘Lax pairs’ which satisfy this condition.

That it would help to have $F_{A} = 0$ was also discussed by Christiaan Hofman in the above mentioned paper, who also discussed constraining it to zero.

(For the connection with $B = - F_{A}$ that I was discussing, however, we can have $F_{A} \neq 0$ and there is full cancellation of all the bothersome terms.)

So with $F_{A} = 0$ two of the terms in the above expression for the curvature vanish. Alvarez, Ferreira & Guillén now also set what one might call ‘target space curvature of $B$ ’ equal to zero:

(4)

d_{A} B = 0 .

With this requirement, all that remains is the exterior product of the 1-form $\oint_{A} (B)$ with itself. If this were an abelian 1-form, then its exterior square would vanish automatically. So that’s one condition on $B$ (together with $F_{A} = 0 = d_{A} B$ ) which would give a flat connection (their equation (3.16)).

The second choice discussed by these authors is obtained by demanding not only $d_{A} B = 0$ , but even stronger that $B$ is covariantly constant (their equation (3.18)). Then, as they show and as can easily be seen, $(\oint_{A} (B))^{2} = 0$ follows automatically.

So this are two sets of conditions on $A$ and $B$ which yield a flat metric on loop space without having $B = - F_{A}$ but with instead having $F_{A} = 0$ and constancy constraints on $B$ .

It is annyoing that I wasn’t aware of this work before and I will need to cite it in a revised version of my paper. But I also think that the insights in both papers are to a good part complementary. The form of the loop space connection in both cases is found to be the same, but different conditions on the vanishing of its curvature arise for applications in integrable systems as opposed to applications for the configuration space of relativistic superstring.

(Please note that I have only read this paper today and not all of it, so I have to apologize for any inaccuracies in the above summary. I’ll appreciate corrections and comments. More on how to use the String Coffee Table can be found here.)

Posted at July 22, 2004 5:46 PM UTC

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10 Comments & 0 Trackbacks

Re: Three ways to loop flatness

Ha! I mentioned that paper to you in my
mail of 9th July. ;-)

Posted by: Amitabha on July 23, 2004 8:41 AM | Permalink | Reply to this

Re: Three ways to loop flatness

Yes, I know. I did look at it, but very quickly only. I had gotten the wrong impression that these integrable systems are not what I was looking for. A mistake. But, ok, that’s life.

Anyway, thanks for mentioning this and other references!

Posted by: Urs Schreiber on July 23, 2004 9:08 AM | Permalink | PGP Sig | Reply to this

Even more flat connections on loop space

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As I mentioned, the easiest way to see that the loop space connection $d + \oint_{A} (B)$ with $B = - F_{A}$ is flat is to realize that it is gauge equivalent to the trivial connection $d$ under conjugation with the open Wilson line of $A$ around the loop.

This immediately suggests to apply such gauge transformations on loop space to the flat connections studied by Ferreira et al. This would yield even more flat connections.

I think it is important to note that gauge equivalent connections on loop space have nontrivial effect on the physics of bounded surfaces. In particular, as I argued in my paper, the physics of open strings is not invariant under all gauge transformation on loop space.

I believe that the reason for this is nicely elucidated by the SCFT deformation technique which I am using. One lesson from this is that a pure unitary tranformation of the generator algebra for the closed string translates into nontrivial deformations of boundary SCFTs when the closed string is cut open to produce the open string.

A simple well known example for this is Hashimoto’s boundary state deformation for the abelian $A$ -field background. Another is the more recent paper by Maeda et al.

Posted by: Urs Schreiber on July 23, 2004 12:12 PM | Permalink | PGP Sig | Reply to this

Re: Even more flat connections on loop space

I spent some time thinking about higher gauge theory on the lattice and multi-dimensional integrability fifteen years ago, when the term gerbe was not yet invented and branes were frowned upon by string theorists:

T A Larsson, “p-cell gauge theories, manifold space and multi-dimensional integrability”, Mod Phys Lett A 5 (1990) 255-264

Inspired by a discussion about gerbes on spr, I later wrote a review of this model with an important correction, so don’t bother to look at the original paper.

The model is a straightforward generalization of lattice gauge theory. Recall that in LGT one puts matrices, i.e. 2-index quantities, on links; an index is associated to each endpoint. LGT has a gauge symmetry at each vertex, and one constructs gauge-covariant Wilson lines by gluing adjascent links together, contracting indices at common endpoints.

My 2-LGT consists of putting 4-index quantities on plaquettes, associating an index to each edge. This model has a gauge symmetry at each edge, and gauge-covariant Wilson surfaces are obtained by gluing adjascent plaquettes together, contracting indices at common edges. An interesting feature is that the zero-3-curvature condition is the Yang-Baxter equation, which gives a connection to lattice integrability.

This model differs significantly from gerbes (I misunderstood that when I chose the title) in one respect: different Wilson surfaces take values in different spaces. If you glue links together you always end up with either zero or two endpoints, so any Wilson line is either a c-number or a two-index matrix. However, when you glue plaquettes together you can end up with a boundary of any length L (a well-defined notion on a lattice). Therefore, the amplitude for such a Wilson surface has L indices.

I still think this old model is quite cool. This two-form generalization of non-abelian gauge theory on the lattice is a well-defined statistical model, and if anyone has the energy it can be treated by standard methods such as Monte Carlo. It may even be useful for studying things like membranes in condensed matter. The thing I like most is its manifest locality; the terms in the action only involves plaquettes around a single cube, which is as local as it gets on thelattice. It seems to me that you can get in trouble with locality if you start from a loop space formulation.

Alas, I later learned that Nepomechie and Orland considered essentially the same model as early as 1984 (references in the eprint). So it seems that essentially the same idea keeps popping up in different guises (2-LGT, gerbes, multi-dimensional integrability, 2-categories, branes, etc.), but a striking breakthrough is still missing. This makes me somewhat pessimistic.

Posted by: Thomas Larsson on July 26, 2004 7:03 AM | Permalink | Reply to this

Re: Even more flat connections on loop space

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My 2-LGT consists of putting 4-index quantities on plaquettes, associating an index to each edge.

I have had a look at your paper. Some points are not yet clear to me:

On p.4 you discuss the need for labels NE and SE. Why does only NE appear in (2.6)? Are the SE terms implicit in the ‘7 more terms’?

Does the expression in (2.6) act on $V^{\otimes n}$ or on $⨂_{ℓ \in δ Σ} V_{ℓ}$ ?

In (2.8) it is understood that $f_{μ}$ acts on $V_{μ}$ , right?

I can see the reasoning behind the construction in section 2. But except for the gauge transformation (2.8) it does not make use of any 1-form holonomy, does it? In this respect it seems to be rather different from loop space and gerbe approaches, right?

Indeed, you write:

This model differs significantly from gerbes […]

Therefore, the amplitude for such a Wilson surface has L indices.

I see. You seem to avoid the issue of ordering the nonabelian surface contributions by keeping track of the orientations of the steps that pull the string over the surface in terms of these indices. This way no 1-form is needed to relate the 2-forms at different points, roughly.

So this seems to be an approach strictly different from that by, for instance, Baez, Girelli & Pfeiffer.

Not that this implies anything. Your (2.7) is arguably a sensible generalization of the Wilson action.

Have you tried to work out its implications, e.g. its equations of motion? (This is not related to your section 4.6, is it?)

It seems to me that you can get in trouble with locality if you start from a loop space formulation.

I would think the nonlocality on target space that may appear is of the same general form as that generally associated with strings.

As far as I can see the 2-group as well as the loop space approach currently do not and perhaps (as indicated by the work by Girelli and Pfeiffer) cannot produce interesting higher generalizations of YM theory. But it does give insights into the general problem of assigning surface holonomy, which allows to study strings in nonabelian 2-form backgrounds.

(From my perspective, the fact that the effective field theory of such strings does not appear to be higher YM looks like an indication that higher YM is maybe not such a good idea after all, but of course that’s a pretty bold conclusion. :-)

So in particular with the argument given in the current entry I think that it is sufficient to have any flat connection on loop space in order to have a well defined surface holonomy of spherical surfaces, even when the second homotopy group of the target is nontrivial. As discussed by Ferreira et al. this alone is sufficient to get interesting physics.

Posted by: Urs Schreiber on July 26, 2004 11:37 AM | Permalink | Reply to this

Re: Even more flat connections on loop space

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On p.4 you discuss the need for labels NE and SE. Why does only NE appear in (2.6)? Are the SE terms implicit in the ‘7 more terms’?

Yes. In 1-LGT, the action consists of two terms, corresponding to the two ways you can do parallel transport around the square. My action consists of terms which do parallel transport of a string segment around a diagonal of an elementary cube, as illustrated in figure 3. Since there are eight directed diagonals, each cube contributes eight terms.

Does the expression in (2.6) act on $V^{\otimes n}$ or on $⨂_{ℓ \in δ Σ} V_{ℓ}$ ?

Huh, this looks akward without MathML support; I know I should install it, but I haven’t gotten around to do it. I’ll stick to old-fashioned Ascii notation and write @ for tensor product [Converted to itex so the rest of us can read it.].

Anyway, (2.6) is supposed to be a cube cut open along three edges, as the ones in figure 3. Each link gives a $V$ , so it acts on $V \otimes V \otimes V$ . The trace in (2.7) contracts the indices living on the three links.

In (2.8) it is understood that $f_{μ}$ acts on $V_{μ}$ , right?

Yes, I should have written ( $f_{μ} \otimes f_{ν}$ ) rather than just $f_{μ} f_{ν}$ .

I can see the reasoning behind the construction in section 2. But except for the gauge transformation (2.8) it does not make use of any 1-form holonomy, does it? In this respect it seems to be rather different from loop space and gerbe approaches, right?

Yes, but you can construct more complicated models which have 1-form connections as well - I think Orland did that.

The gerbe approach certainly uses both 1- and 2-form connections, but in loop space you only have a 2-form (i.e. a 1-form in loop space), haven’t you?

Indeed, you write:

“This model differs significantly from gerbes […]

Therefore, the amplitude for such a Wilson surface has L indices.”

I see. You seem to avoid the issue of ordering the nonabelian surface contributions by keeping track of the orientations of the steps that pull the string over the surface in terms of these indices. This way no 1-form is needed to relate the 2-forms at different points, roughly.

Yes.

So this seems to be an approach strictly different from that by, for instance, Baez, Girelli and Pfeiffer.

Yes, although this wasn’t clear to me when I wrote the eprint, which is why I used the buzzword “gerbe”.

Not that this implies anything. Your (2.7) is arguably a sensible generalization of the Wilson action.

Have you tried to work out its implications, e.g. its equations of motion? (This is not related to your section 4.6, is it?)

Well, (4.6) and (4.7) should be the equations of motion, but the classical continuum limit is very formal, and I am not completely happy with it.

One thing that I both like and dislike is the formally diff-and gauge-covariant curvature in (4.35), $F = dA + [A, s \cdot A]$ . $F = 0$ formally becomes the classical Yang-Baxter equation in section 4.7. There is something right about this, since the quantum Yang-Baxter equation is the zero-curvature condition on the lattice. But this is also a difficulty, because the CYBE is an equation in $End (V \otimes V \otimes V)$ , which seems to break rotational symmetry down to cubic symmetry.

“It seems to me that you can get in trouble with locality if you start from a loop space formulation.”

I would think the nonlocality on target space that may appear is of the same general form as that generally associated with strings.

Yes, and this is something which has bugged me about string theory for the past 20 years. Say that you embed a loop in space (not spacetime) so that it looks like the figure “8”. Then one would expect an interaction at the intersection point, since the two strands are very close to each other. However, this would give an interaction between two points on the loop that are far apart along the loop. This effect is difficult to describe in worldsheet langauge, so people just ignore it AFAIU. But I think that it is important, provided that embedding space has low enough dimension. My guess that the critical dimension is four, since self-avoiding and unconstrained random walks have different fractal dimensions below that.

OTOH, if we start from a theory which has manifest locality in spacetime, like 1-LGT or 2-LGT, we are safe. Spacetime locality cannot be ruined by a reformulation in terms of nonlocal holonomy variables, although loop space locality is probably lost.

As far as I can see the 2-group as well as the loop space approach currently do not and perhaps (as indicated by the work by Girelli and Pfeiffer) cannot produce interesting higher generalizations of YM theory. But it does give insights into the general problem of assigning surface holonomy, which allows to study strings in nonabelian 2-form backgrounds.

I have not studied Pfeiffer’s papers in great depth, but I also got the impression that that line of thought runs into trouble.

My own biggest problem is that the formal continuum limit is awkward. However, maybe one can live without such a limit. A lattice model needs of course a continuum limit to be of interest, but not necessarily a classical one. Maybe one can simply define the full, non-perturbative quantum theory as the critical point of the lattice theory. This assumes, of course, that a second-order phase transition exists, which must be checked by simulations or otherwise.

Posted by: Thomas Larsson on July 27, 2004 8:38 AM | Permalink | Reply to this

Re: Even more flat connections on loop space

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Thomas Larsson wrote:

but in loop space you only have a 2-form (i.e. a 1-form in loop space), haven’t you?

Not quite. There is of course just a 1-form on loop space, but its definition in general involves a 2-form as well as a 1-form on target space.

You can see this discussed in detail in equations (3.12), (3.15) of hep-th/0407122 and in equation (1.4) and (5.1) of hep-th/9710147. The general idea can also be found in the expression at the top of p. 7 in hep-th/0207017.

Roughly, the 1-form on loop space is obtained by integrating the 2-form on target space over the loop. But in the nonabelian case this integral alone would be ill-defined, since it involves addition of elements sitting in different fibers. This is cured by using a 1-form to parallel transport the 2-form to a given point on the loop, before adding up. In symbols:

(1)

loop space connection = \int_{0}^{2 π} W_{A} (0, σ) B_{μ ν} (σ) (\frac{\partial}{\partial σ} X^{ν} (σ)) W_{A} (σ,0) {dX}^{μ} (σ)

Here $W_{A}$ is the holonomy of the 1-form $A$ along the loop.

In the my paper mentioned above I show that this construction need not to be postulated but follows from a boundary state deformation analysis of the string.

But the general connection on loop space is even more general. This can be seen by simply making a loop-space gauge transformation of the expression above. In general the result of such a gauge transformation won’t have this form anymore, but will need several target space 1- and 2-forms for its description.

This is not the case for precisely the connections for which $B = - F_{A}$ , where $F_{A}$ is the field strength of $A$ . Only in this case does hence a gauge transformation on loop space have the interpretation of a 2nd order gauge transformation on target space. $B = - F_{A}$ (and two of its generalizations) is also the only known case for nontrivial $A$ which makes the connection on loop space flat, which is good, because it is related to the existence of unique surface holonomies induced by the loop space holonomy. This I discuss in the newly added section 3.4 of hep-th/0407122

Well, (4.6) and (4.7) should be the equations of motion

How do you see this? These equations are derived in the context of gerbes, right? But, as you say, your approach is different from gerbes.

Say that you embed a loop in space (not spacetime) so that it looks like the figure ‘8’. Then one would expect an interaction at the intersection point, since the two strands are very close to each other. However, this would give an interaction between two points on the loop that are far apart along the loop. This effect is difficult to describe in worldsheet langauge, so people just ignore it AFAIU.

True. String interactions are not directly related to string self-intersections, but only to the topology of the parameter space, so to say. But I wouldn’t say that anything is ignored here. After all, the dynamics of the string is what drops out e.g. from the ‘t Hooft limit or Wilson line dynamics in gauge theory.

I have not studied Pfeiffer’s papers in great depth

I haven’t compared the details, but it seems to me that what they are doing is, morally at least, precisely what you are doing, only that they allow for a non-vanishing 1-form together with their 2-form.

In particular, in section 2.5, in figure 2 and in the text below equation (3.25) they explain how they compute the surface holonomy of a little cube, much as envisioned in your paper.

Note that they claim that consistency of this whole setup requires their equation (2.10), which implies that (2.37) must be the unit, which again implies the condition $B = - F_{A}$ (3.25) mentioned above. I don’t know if this can be circumvented somehow. If not, it would mean that your approach is consistent only for $B = 0$ , because you set $A = 0$ .

But I haven’t worked that out in detail. Maybe you can have a look at it and tell us what you think.

the formal continuum limit is awkward

If the way Girelli and Pfeiffer compute the surface holonomy of a little lattice cube is the same as how you do it, then the continuum limit would be their equation (3.27) (for $A = 0$ ). Actually, i think it might be different. It would be good to understand how different precisely.

Posted by: Urs Schreiber on July 28, 2004 2:23 PM | Permalink | PGP Sig | Reply to this

Re: Even more flat connections on loop space

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I have sat down and looked through the literature again.

What you, Thomas, propose in equation (2.6) of your math-ph/0205017 as the surface holonomy of a cube is precisely the same construction as described by Pfeiffer and Girelli in section 2.5 of their hep-th/0309173 for the case of the tetrahedron and applied to the cube in equation (3.26) except for the fact that they have actions of the edge holonomy in their expression, which are missing in yours.

In other words, we obtain your equation (2.6) for the 3-curvature from Girelli&Pfeiffer’s (3.26) when we set $g = 1$ throughout in their equation, i.e. when the 1-form connection vanishes.

If this is correct, then the continuum limit of your field strength should be, according to Pfeiffer’s (3.28)

(1)

G = d_{A} B =_{(for A = 0)} dB

i.e. just the ordinary exterior derivative of the 2-form. (This would mean that the curvature which you discuss in your equation (4.48) would be something else. BTW, where does this expression come from?)

Note the discussion at the end of Girelli&Pfeiffer’s section 2.5, which explains the heuristic meaning of the 1-form holonomies which they have in their expression for the curvature.

Now in principle one could set $g = 1$ and get your expression as a special case of Girelli&Pfeiffer’s. But there is a theorem that the combination of the gauge group $G$ of the edges with the group $H$ of the surfaces into a crossed module is a Lie 2-group if (and apparently only if) Pfeiffer’s equation (2.10) holds

(2)

t (h) = g_{2} \cdot g_{1}^{- 1} .

Here $h$ is the group label of a surface and $g_{1}$ and $g_{2}$ that of its source and its target edge. $t$ is the homomorphism which maps $H$ to $G$ . In the simplest case $H = G$ and we can just forget about $t$ .

This equation apparently goes back to John Baez’ proposition 5 in hep-th/0206130, where it is implicit in the condition

(3)

target (h, g) = t (h) g,

where $(h, g)$ is the surface with the source edge labeled by $g$ and the surface labeled by $h$ .

It is obvious that the above condition

(4)

t (h) = g_{2} \cdot g_{1}^{- 1}

is nothing but the finite version of

(5)

B = - F_{A} .

This means that this condition (which from the loop space perspective is special due to a couple of properties) ensures that the 2-form theory one is dealing with is really a 2-group theory.

So I conclude from that that the 2-form YM action which you give is reasonable, but does not give us a 2-form gauge theory involving a 2-Lie group, that would require $B = - F_{A}$ and since $A = 0$ in your approach we would be left with the trivial case $B = 0$ .

But of course one should ask if anything forces us to satisfy the condition of 2-groups! I have seen nowhere any statement why we should want to deal with a 2-group.

- From the loop space and string theory perspective I know that $B = - F_{A}$ is what we should probably want to have, due to some facts which I have mentioned before. But we know from Alvarez, Ferreira & Sánchez Guillén that there are other reasonable (namely flat) connections on loop space which do not have $B = - F_{A}$ . Instead these have $F_{A} = 0$ , which means that locally a gauge transformation will turn this into something along the lines of your idea. They showed that such 2-form theories indeed do describe a couple of integrable models. This proves that there is certainly a physical application for 2-form theories which don’t satisfy $B = - F_{A}$ and hence are not 2-group theories, strictly.

Posted by: Urs Schreiber on July 28, 2004 5:18 PM | Permalink | PGP Sig | Reply to this

Re: Even more flat connections on loop space

Urs wrote:
But of course one should ask if anything
forces us to satisfy the condition of
2-groups! I have seen nowhere any
statement why we should want to deal with a 2-group.
–

There is no absolutely compelling reason. However, if you do have a `non-Abelian string’, you will want to build its world-sheet by adding infinitesimal surfaces. You know that inifinitesimal squares can be added in two ways, either along an edge or at a corner. So if each
little square corresponds to some group element, there should be two composition laws for those elements. Which should commute if the string world-sheet is to be independent of how you have added the little bits of surface. Then you either get an Abelian group (or F=0), or you get a 2-group. Maybe other structures are possible, but I haven’t seen any.

Posted by: Amitabha on July 29, 2004 1:29 PM | Permalink | Reply to this

Re: Even more flat connections on loop space

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Yes, thanks for pointing this out. Thomas Larsson was concerned with writing down 3-curvatures, and to do just that one does not necessarily need a unique surface holonomy. But of course one would probably run into serious problems with a 2-form theory which does not admit a unique notion of surface holonomy. Probably the continuum limit of Thomas Larsson’s theory won’t be well defined when the surface holonomy of a big cube cannot be computed from small cubes.

Posted by: Urs Schreiber on July 29, 2004 1:45 PM | Permalink | PGP Sig | Reply to this

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July 22, 2004