Scandinavian but not Abelian

July 9, 2004

Posted by urs

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Stepping out of the propeller plane on Karlstad airport, I found myself surrounded by pine forests and in an atmosphere quite unlike that on larger airports – but what I did not find was my luggage.

Apart from the obvious inconveniences this meant that a couple of papers on non-abelian 2-form fields which I had brought with me were spending the night in Copenhagen, instead of attending the conference ‘NCG and rep theory in math-phys’ with me.

Not that there weren’t plenty of other things to think about, like Schweigert’s talk on how modular tensor categories and Frobenius algebras know about open strings, as well as many very mathematical talks with categories here and functors there

but after I had given my talk on Loop space methods in string theory it turned out that several people were interested in nonabelian 2-form gauge theories, and on my way back to the hotel I had a very interesting conversation with Martin Cederwall about precisely the lost hep-th/0206130, hep-th/0207017, hep-th/0312112 which I had intended to pull out of my hat on precisely such an occasion.

But maybe I was lucky after all, because when on the next day at lunch I talked about gauge invariances in 2-form theories with Jens Fjelstad, we had to reproduce the essential formulas by ourselves on a sheet of scrap paper, instead of just looking them up, and somehow this triggered the right neurons for me, and after a nap that evening I got up and saw the light.

[Update 07/15/04: The issue discussed below can now be found discussed in hep-th/0407122.]

The point is that the 2-form on target space gives rise to an ordinary 1-form connection on loop space, of course, and that I think that I know precisely how this 1-form connection looks like, because I can derive it from boundary state deformations.

In a somewhat schematical and loose fashion we can write

(1)

\nabla = d + \oint_{A} (B),

following the notation in Hofman’s paper, but including a second factor of the $A$ -holonomy, as I have discussed before.

Using this connection and the ordinary formula for its gauge transformations, one can check that global gauge transformations on loop space correspond to the ordinary 1-form gauge transformations

(2)

A \mapsto U A U^{†} + U ({dU}^{†})

(3)

B \mapsto U B U^{†}

on target space, while local gauge transformations on loop space give rise to

(4)

A \mapsto A

(5)

B \mapsto B + d_{A} λ

up to some correction terms which don’t have a target space analogue. I have given a little more detailed discussion of this on sci.physics.strings.

As with any riddle, after having written this down it looks pretty obvious, but at least I haven’t seen this clearly before.

The question now seems to be: Can we even expect to be able to write down a theory of point particles that is local in target space and respects the above gauge symmetries. What happens to the correction terms?

Rather I’d suspect something like an OSFT which has the true loop space 2-form gauge invariance, but whose level truncated effective field theory breaks some of it. But I don’t know.

When I mentioned to Martic Cederwall that we should maybe consider YM on loop space using the field strength $(d + \oint_{A} (B))^{2}$ he remarked that this would be a theory local in loop space, while ordinary OSFT is non-local in loop space (because the 3 ‘loops’ (or rather open intervals) involved in an interaction are not small deformations of each other and hence do not correspond to nearby points in loop space).

Well, so I don’t know what all this means. But as far as I can see nobody else does either, at least nobody understands it completely. Amitabha Lahiri kindly made me aware of a couple of paper he has on attempts to construct field theories with some reasonable 2-form gauge invariances. I will try to have a look at these papers and see if the Lagrangians considered there might be understood in terms of the loop space connection

(6)

d + \oint_{A} (B) = ℰ^{† (μ, σ)} (\partial_{(μ, σ)} + U_{A} (0, σ) B_{μ ν} X^{' ν} U_{A} (σ,0)) .

Posted at July 9, 2004 8:28 AM UTC

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Re: Scandinavian but not Abelian

Hi -

In a private email (before noticing the new SCT entry), I said…

I haven’t mastered the loop space formulation, but the idea seems really natural to me. I am sure that you are correct and people will take notice of this and your deformation stuff. An obvious question, can the process be iterated? What would be the loop space of loop space? :) Would you get non-abelian 3-form stuff? :)

You responded…

Good question. My advisor asked me the same question today at lunch. My answer
was that, yes, I think this iterates. Naively at least I can imagine the loop
space over loop space, for instance (and I seem to recall John Baez having
mentioned something like that before). This would be “torus space”, the space
of all maps from the torus into target space. It should be relevant for the
supermembrane, which indeed couples to the 3-form that appears in 11d
supergravity. There should then also be nonabelian 3-forms and so on.

But - wait - we should be discussing this at the String Coffee Table. It could
need some activity. :-)

Ok! Ok! :)

Another obvious question…

Is it possible to generalize “loop” space to “string” space that includes both open and closed strings? If you could construct a “string space of string space”, then you could talk about general maps from seemingly general 2d manifolds (branes?) into target space.

Just curious, but would even the loop space of loop space admit things more general than tori? I can almost picture a Klein bottle among other things, e.g. closed strings that twist around.

[snip of some stuff about Pohlmeyer invariants I don’t think you want me reproducing in public ;)]

I also said (in regard to the lousy communication skills of most string theorists)

I know! I see you being assimilated! :) Your notes are becoming less and less comprehensible ;)

to which you replied

Ok. So we should start a project: Rephrase everything that I think can be said
about strings in loop space formulation in a way that is understandable for
non-experts in strings. I think there is a very good chance that this is
possible. Many aspects of string theory look surprisingly natural in loop
space formulation. For instance isn’t it kind of remarkable that the concept
of the spacefilling D9 brane translates in loop space language just to the
constant 0-form on loop space? What are called ‘boundary states’ in string
theory are really just the constant 0-form acted on by some unitary operators
on loop space.

This sounds like a good idea :) Maybe we can work on this together and put something on the arxives. Then again, knowing how notoriously slow I am at writing things up, you might want to go ahead on your own. I’d be happy to at least make suggestions :) We started to do this for your string theory seminar, where I was able to write down a pretty sleek coordinate independent version of the Polyakov action. Doing so made the relation to BF-YM theory kind of obvious to me. Knowing that there are probably infinite many ways to write down an action that reduced to Nambu-Goto “on shell” caused me to lose interest :)

This loop space idea is pretty neat though. Of course, I would suggest presenting the discrete version first, which would be much simpler :) Contraction “integrals” of continuum indices become summations, which more closely resembles the usual contraction of indices.

Just a thought…

I also said

One of the things that inspired me to learn some elementary string theory was a statement you made (several times in fact) in response to complaints that string theory does not predict anything. You said that the fact that string theory does not predict anything should not be considered necessarily a bad thing because Newton’s gravitational theory doesn’t predict anything either. Not without specifying initial conditions. Finding the right vacua in string theory is like trying to find the right initial configuration of planets in the solar system. Once this is done, you can make many predictions about the subsequent orbits of the planets. This made a lot of sense to me.

You replied

Nice to hear. I get kind of frustrated hearing [people] repeat [their] claim ‘no predictions’ no matter what. In precisely the same sense field theory as such does not make any predictions. I mean, just givben the statement: “The world is described by field theoy.” does not allow you to predict anything. You instead need to pick a particular Lagrangian. String theory has the advantage that all these Lagrangians are not just there to be picked but are part of a larger framework which in principle should tell you what t choose. So in principle string theory is more predictive than field theory.

I continued…

On the other hand, I am not quite convinced that the situation is as easy as that. Before Newton could write down his gravitational theory, he first had to invent his calculus. Once you have the calculus, you need some basic physical laws, like F = dp/dt. Once you have the physical laws, THEN you can construct a physical model, e.g. the solar system.

to which you replied

I am not sure about your destinction between laws and models. But one should
certainly agree with what everybody is saying, namely that the full picture of
string theory has not yet emerged.

In this case of Newton’s gravitational theory, the “law” is the law of gravitation. The “model” is the specific placement of planets. The model is governed by the law. I don’t know how this analogy quite translates for strings.

I continued

I get the feeling that string theory has not even passed the “calculus” phase yet, i.e. the “calculus of strings” is still being developed.

To which you replied

BTW, I think Martinec used to call conformal field theory the “calculus of
string”.

Neat :)

I continued

On another note, I have actually been working hard on the discrete stuff. I don’t have a lot to show for it though :) It might be a waste of time, but I have been LaTeX’ifying Robb’s 21 postulates for axiomatizing Minkowski space. Then I will LaTeX’ify Zeeman’s postulates and finally Penrose’s. I hope to compare and contrast them. I will then try to distill out the “meaning” of what they are trying to say. I will then try to use this “meaning” to motivate the discrete approaches of Sorkin, D&MH, and our notes. Well, that is the plan anyway :)

to which you replied

Robb has 21 axioms? So many?

Yes. Although all of them rely only on the relations “before” and “after”, it does seem unnecessarily convoluted. By the way, I forgot to mention Mundy in my list. Mundy basically reformulated what Robb did in a much more concise manner using the relation of being “light-like” separated as opposed to Robb’s “time-like” relation. Another motivation for :aTeX’ifying everthing is to help me verify that Mundy actually does reproduce everything Robb does.

Gotta run!

Eric

Posted by: Eric on July 9, 2004 5:35 PM | Permalink | Reply to this

Re: Scandinavian but not Abelian

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Oops!

… sleek coordinate independent version of the Polyakov action.

Of course Polyakov is coordinate independent. I meant that I wrote down a sleek notation that was “coordinate free”. If coordinates do not even appear in the expression, then it is obviously coordinate independent. I prefer “coordinate free” notation whenever possible.

Eric

Posted by: Eric on July 9, 2004 5:44 PM | Permalink | Reply to this

Re: Scandinavian but not Abelian

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Is it possible to generalize ‘loop’ space to ‘string’ space that includes both open and closed strings?

Hm, well, er, in principle, why not? I mean, this beast certainly exists somehow. But I think an important insight is that lots of open string physics can be captured instead much more elegantly by boundary state formalism, where open string physics happens inside closed string inner products whith a closed string state on one side inserted.

Just curious, but would even the loop space of loop space admit things more general than tori? I can almost picture a Klein bottle among other things, e.g. closed strings that twist around.

Depends on how precisely you define loop space. In order for a Klein bottle to appear as a point in the loop space of loop space of target space the loop space of target space has to be that of unoriented loops, i.e. where a single point in loop space corresponds to a given loop or its orientation reverse.

Contraction ‘integrals’ of continuum indices become summations, which more closely resembles the usual contraction of indices.

That would be polygon space. Polygon space is different from a discretized loop space. But of course one could consider it, too. The problem is that it badly breaks conformal invariance on the worldsheet. I have speculated before how one could try to make sense of it anyway. I am not sure yet that there is anything interesting to be found. But maybe there is.

In this case of Newton’s gravitational theory, the ‘law’ is the law of gravitation. The ‘model’ is the specific placement of planets. The model is governed by the law. I don’t know how this analogy quite translates for strings.

So by model you mean a point in phase space, i.e. one solution of the system. Take the IKKT version of strings, as a drastic example. The law is $[A^{μ}, [A_{μ}, A_{ν}]] = 0$ and a ‘model’ in your sense is any set of 10 large matrices $A_{μ}$ that solve this equation.

Posted by: Urs Schreiber on July 9, 2004 6:17 PM | Permalink | PGP Sig | Reply to this

Re: Scandinavian but not Abelian

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BTW, concerning loop space and CFT and the ‘calculus of string’ I should emphasize that loop space formulation and usual CFT language are two sides of the same coin. Usual CFT is working in the Heisenberg pciture with worldsheet-time dependent operators (fields), whereas the loop space context uses canonical Schrödinger picture formulation of the worldsheet field theory. As always, some aspects of a theory are more easily visible in Heisenberg picture, other in Schrödinger pricture. In field theory the Schrödinger picture is usually very awkward. But I think at least on the 2d worldsheet it proves to be a useful point of view for some questions.

Posted by: Urs Schreiber on July 9, 2004 7:21 PM | Permalink | PGP Sig | Reply to this

Re: Scandinavian but not Abelian

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Ok. So we should start a project: Rephrase everything that I think can be said about strings in loop space formulation in a way that is understandable for non-experts in strings.

I am spending a little time thinking about this. The first thing I would suggest is that you stop calling

(1)

d_{K} = d + i_{K}

a “deformation of the exterior derivative.” This is not a deformation of the exterior derivative because the exterior derivative is the transpose of the boundary operator. Instead, $d_{K}$ is some other operator constructed from $d$ and $i_{K}$ , i.e. it is the “square root” of the Lie derivative. Since the motivation is to try to make things more transparent to the non-experts (like myself), I think it is worth some haggling over defining terminology and notation.

Another thing, why is the kinematical configuration space of bosonic strings the space of parameterized strings on target space as opposed to unparameterized strings? Why introduce parameterizations if nothing is going to depend on them?

Another thing, can you define integration of forms on loop space? Stokes’ theorem? This is how you should define the exterior derivative on loop space and not via some abstract algebraic (unmotivated) definition. I believe that Stokes’ theorem on loop space should play a central role and as far as I can see, it hasn’t even been mentioned yet (unless I missed it).

More coming soon…

Eric

Posted by: Eric on July 10, 2004 10:14 PM | Permalink | Reply to this

Re: Scandinavian but not Abelian

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In http://arxiv.org/abs/hep-th/0401175 you say

Let $(ℳ, g)$ be a pseudo-Riemannian manifold, the target space, with metric $g$ , and let $ℒℳ$ be its loop space consisting of smooth embeddings of the circle into $ℳ$ :

(1) $ℒℳ : = C^{∞} (S^{1}, ℳ) .$

To me this seems to only define the point set of $ℒℳ$ and says nothing about its topology. Are two points that are “close” in $ℒℳ$ corresponding to two loops that are “close” in $ℳ$ ? In both instances, how is “close” defined?

How is the topology of loop space defined?

Eric

Posted by: Eric on July 10, 2004 10:51 PM | Permalink | Reply to this

Re: Scandinavian but not Abelian

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Several years ago, I went bonkers over how beautiful loop space methods were and how I thought it was a shame that it didn’t seem to have a place in the “northern” approach to loop quantum gravity. I was rooting for the southern approach because it made use of the “loop derivative.” I did believe and I do believe that loop space methods are too beautiful not to be important for something.

Now here I am full circle approaching it again from a slightly different angle. I thought that for historical purposes, it might be entertaining to take a look at

Loop derivative

Eric

PS: For even more fun, check this out.

Posted by: Eric on July 11, 2004 3:06 AM | Permalink | Reply to this

Re: Scandinavian but not Abelian

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Walking through memory lane, i.e. rereading old s.p.r. posts, I found a url I gave to be no longer valid. Some minor tweaking turned up the current valid url

http://www-dft.ts.infn.it/~ansoldi/RedTape/Curriculum/HTML/Articles.html

In particular, I see something you (Urs) might find interesting (if you haven’t seen it)

String Propagator: A Loop Space Representation

To quote the abstract

The string quantum kernel is normally written as a functional sum over the string coordinates and the world-sheet metrics. As an alternative to this quantum field-inspired approach, we study the closed bosonic string propagation amplitude in the functional space of loop configurations. This functional theory is based entirely on the Jacobi variational formulation of quantum mechanics, without the use of a lattice approximation. The corresponding Feynman path integral is weighed by a string action which is a reparametrization invariant version of the Schild action. We show that this path integral formulation is equivalent to a functional “Schrödinger” equation defined in loop-space. Finally, for a free string, we show that the path integral and the functional wave equation are exactly solvable.

Eric

Posted by: Eric on July 11, 2004 3:19 AM | Permalink | Reply to this

Re: Scandinavian but not Abelian

String Propagator: A Loop Space Representation

I am aware of this paper. In spite of what it seems to indicate in the abstract I could never really relate it to what I am concerned with, though. Maybe my fault.

Concerning the ‘loop derivative’, I have looked at some (possibly not all) the links that you proivded, in particular the simple explanation that John baez gives here.

Seems to me that when the space of diffential forms on loop space is taken to be well-behaved enough this derivative exists and is pretty much just a Lie derivative on loop space. From what John Baez says in that message it looks like the reason this object does not exist in ‘northern LQG’ is the fact which we have discussed before in the context of the ‘LQG string’, namely that there we have non-weakly continuous reps and nonseperable Hilber spaces.

Posted by: Urs Schreiber on July 11, 2004 9:46 AM | Permalink | PGP Sig | Reply to this

Re: Scandinavian but not Abelian

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How is the topology of loop space defined?

The best is really to think of loop space as an $∞$ -dimensional manifold with coordinates ${X^{(μ, σ)}}$ . Then the natural topology is just the usual one, where an open set is given by open intervals in each of the coordinates.

Heuristically, two loops are close together if one is obtained from the other by deforming it ever so slightly.

Posted by: Urs Schreiber on July 11, 2004 9:27 AM | Permalink | Reply to this

Re: Scandinavian but not Abelian

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I would suggest is that you stop calling $d_{K} = d + i_{K}$ a ‘deformation of the exterior derivative’

Here ‘deformation of X’ is meant in the sense of ‘a continuous 1-parameter familiy of objects such that for parameter=0 we have the original object and for parameter $\neq$ 0 we have the deformed object’. In this sense this is a deformation, since you really should have $d_{K} = d + iT ι_{K}$ , where $T$ , the string’s tenstion, is the parameter, which, when turned on, deforms this operator away from the original exterior derivative.

why is the kinematical configuration space of bosonic strings the space of parameterized strings on target space as opposed to unparameterized strings? Why introduce parameterizations if nothing is going to depend on them?

Good point. The answer is: Because it is easier. The question is pretty much the same as ‘Why work with coordinates in GR if nothing is going to depend on them anyway?’ We need the coordinates to even write down the formulas which express their irrelevance! :-)

So for the string we have for instance the Polyakov action. It has a redundency, namely conformal invariance. That’s why we get constraints, which express that physical states should not have this redundancy. But the constraints hence must act on a space of states which does have the redundancy. The subspace annihilated by them is that subspace where the redundancy is gone - the physical subspace.

But the ‘deformed’ exterior derivative above is a pretty neat way to deal with this. It is not nilpotent, but it’s nilpotence is restored when restricted to the rep-invariant subspace.

So this is somewhat similar to your construction of chains from path algebras. Paths, on which the ‘boundary operator’ is not nilpotent restrict to chains, on which it is.

Another thing, can you define integration of forms on loop space?

Yes. The Hodge inner product on forms over loop space is essentially (up to a certain switch of sign in the 0-modes) the inner product on the superstring’s Hilbert space. A state in the string’s Hilbert space is a (inhomogeneous) differential form on loop space.

Stokes’ theorem?

In principle, yes. I think I haven’t come across its analogue in the CFT language yet, but it should be there.

Posted by: Urs Schreiber on July 11, 2004 9:21 AM | Permalink | PGP Sig | Reply to this

Re: Scandinavian but not Abelian

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Hi Urs,

I think I am going to put a little more effort into trying to convince you to consider changing terminology about deformed exterior derivatives.

As we discussed here, there are at least three profound ways to “deform” the exterior derivative:

(1)

1 .) d + A

(2)

2 .) d + i_{X}

(3)

3 .) d + d^{†} .

The first one is the “square root” of the curvature, the second is the square root of the Lie derivative, and the third is the square root of the Lapace-Beltrami operator. In each case, we could append a factor $T$ to the second term that we could, in principle, let go to zero and recover the usual exterior derivative. In this sense, we could consider each one to be a deformation of the exterior derivative. However, I would argue that this belies the geometrical meaning behind each one.

Would it be so controversial to think of some name besides “deformed exterior derivative” for $d + i_{X}$ ? The first item is called the “covariant exterior derivative”. I am not too thrilled about this name either, but at least it makes it clear that it is geometrically different than the exterior derivative. The third item is called the Dirac-Kaehler operator, which I am perfectly happy with. I think I recall you referring to $d + i_{X}$ as the susy generator. Is that true? Could we call the second item the “susy generator” or something instead of deformed exterior derivative?

Ah ha! :) In your paper

On deformations of 2d SCFTs

you have a footnote referring to the equation

(4)

d_{K} \to e^{- W} d_{K} e^{W}, d_{K}^{†} \to e^{W^{†}} d_{K}^{†} e^{- W^{†}} (1.2)

that says

Throughout this paper we use the term “deformation” to mean the operation (1.2) on the superconformal generators, the precise definition of which is given in §3.2 (p.15). These “deformations” are actually isomorphisms of the superconformal algebra, but affect its representations in terms of operators on the exterior bundle over loop space.

Not to mention the fact that you give a precise meaning to the word “deformation” for which $d + i_{X}$ is not one, it looks like you are referring to $d_{K}$ as a “superconformal generator”. Why can’t we just use this terminology for $d_{K}$ ?

If we refer to $d_{K}$ as a deformation of $d$ , then we need to refer to $e^{- W} d_{K} e^{W}$ as a “deformation of a deformation of d.” I understand that it is not incorrect to refer to $d_{K}$ as a deformation of $d$ , but if our goal is to make things clearer, should we try to avoid clumsy phrases like the above? Then again, I am not too sure I am fond of the term “superconformal generator” either because it is so scary :) Is there a less intimidating term we can use? Something like “the square root of the Lie derivative”? :)

Another reason why I would like to try to keep the exterior derivative as a more sacred operator is that I am enamored by the beauty of and profoundness of the generalized Stokes’ theorem. We should leave this temple unspoiled if we can :)

If we are going to deform $d$ , then we had better deform the boundary map $\partial$ as well. Although we could, in principle, deform $\partial$ in a way to have a deformed Stokes’ theorem, I don’t think anyone would think this to be a natural thing to do.

Random thought…

I wonder if there is some natural operator $H_{X} : C_{p} \to C_{p + 1}$ such that

(5)

\int_{S} i_{X} α = \int_{H_{X} S} α .

I can imagine $X$ determining a flow which sweeps the $p$ -chain $S$ along forming a $(p + 1)$ -dimensional chain. If there was such a thing, then maybe it wouldn’t be so unnatural to define

(6)

\partial_{K} = \partial + H_{K}

so that

(7)

\int_{S} d_{K} α = \int_{\partial_{K} S} α .

Hmm…

Anyway…

Stokes’ theorem?

In principle, yes. I think I haven’t come across its analogue in the CFT language yet, but it should be there.

If this is a hole, I think it is a significant hole and maybe we should fill it.

Eric

Posted by: Eric on July 11, 2004 3:54 PM | Permalink | Reply to this

Re: Scandinavian but not Abelian

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Hi again,

I wonder if there is some natural operator $H_{X} : C_{p} \to C_{p + 1}$ such that

(1) $\int_{S} i_{X} α = \int_{H_{X} S} α .$

I can imagine $X$ determining a flow which sweeps the $p$ -chain $S$ along forming a $(p + 1)$ -dimensional chain.

I get the strong feeling I am reinventing old well-known material here, but this is kind of interesting :)

Given a $p$ -chain $S$ and a (smooth) vector field $X$ , then the flow generated by $X$ will drag $S$ along sweeping out a $(p + 1)$ chain. Let $ϕ (t) : ℳ \to ℳ$ be the flow and let $ϕ (t)_{*} S$ denote the $p$ -chain $S$ carried along the flow to time $t$ . Then define

(2)

H_{X} (t) S = ⋃_{0 \leq t' \leq t} ϕ (t')_{*} S

to be the $(p + 1)$ -chain obtained by sweeping $S$ along $X$ for a time $t$ .

I could very well be (and probably am) wrong, but it looks like we have

(3)

\int_{S} i_{X} α = \frac{d}{dt} {[\int_{H_{X} (t) S} α] ∣}_{t = 0} .

This feels right. If it is, that would be pretty neat.

It seems to be right for the case where $α = df$ is a 1-form, i.e. for the left-hand side we have

(4)

\int_{p} i_{X} df = {(i_{X} df) ∣}_{p} = {df (X) ∣}_{p},

which is the directional derivative of $f$ along $X$ evaluated at point $p$ . For the right-hand side we have

(5)

\frac{d}{dt} {[\int_{H_{X} (t) p} df] ∣}_{t = 0} = \frac{d}{dt} {[\int_{\partial H_{X} (t) p} f] ∣}_{t = 0} = \frac{d}{dt} [f (ϕ (t)_{*} p) - f (p)],

which is also the directional derivative of $f$ along $X$ evaluated at $p$ . So it works for this case, but this might be too special because $i_{X} df = ℒ_{X} f$ .

I think the case when $α$ is exact is covered in Frankel (I’ll need to check), but I don’t recall seeing this relation

(6)

\int_{S} i_{X} α = \frac{d}{dt} {[\int_{H_{X} (t) S} α] ∣}_{t = 0} .

for $α$ not exact.

Eric

Posted by: Eric on July 11, 2004 4:41 PM | Permalink | Reply to this

Re: Scandinavian but not Abelian

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I think the case when $α$ is exact is covered in Frankel (I’ll need to check), but I don’t recall seeing this relation

(1) $\int_{S} i_{X} α = \frac{d}{dt} {[\int_{H_{X} (t) S} α] ∣}_{t = 0}$

for α not exact.

Ok. I just checked Frankel and he has the expression

(2)

\frac{d}{dt'} {[\int_{S (t')} α] ∣}_{t' = t} = \int_{S (t)} ℒ_{X} α,

where $S (t) = ϕ (t)_{*} S$ . This is pretty much different from what I am proposing in the quote above. However, I did check to make sure that the two expressions are compatible and they are (or seem to be).

From the expression in Frankel, we have

(3)

\frac{d}{dt'} {[\int_{S (t')} α] ∣}_{t' = t} = \int_{S (t)} ℒ_{X} α = \int_{S (t)} (i_{X} d α + d i_{X} α) = \int_{S (t)} i_{X} d α + \int_{\partial S (t)} i_{X} α .

Now considering the first of the two terms on the right-hand side above we have (using my expression)

(4)

\int_{S (t)} i_{X} d α = \frac{d}{dt'} {[\int_{H_{X} (t') S (t)} d α] ∣}_{t' = t} = \frac{d}{dt'} {[\int_{\partial [H_{X} (t') S (t)]} α] ∣}_{t' = t}

and considering the second term we have

(5)

\int_{\partial S (t)} i_{X} α = \frac{d}{dt'} {[\int_{H_{X} (t') \partial S (t)} α] ∣}_{t' = t} .

It seems kind of miraculous the way it works out, but unless I made a mistake, it appears that (while keeping track of orientation) we have

(6)

\partial [H_{X} (t') S (t)] + H_{X} (t') \partial S (t) = S (t') - S (t)

so that

(7)

\frac{d}{dt'} {[\int_{\partial [H_{X} (t') S (t)] + H_{X} (t') \partial S (t)} α] ∣}_{t' = t} = \frac{d}{dt'} {[\int_{S (t') - S (t)} α] ∣}_{t' = t} = \frac{d}{dt'} {[\int_{S (t')} α] ∣}_{t' = t}

as it should :)

This gives me a boost of confidence that perhaps my expression is not too far off (and might actually be correct).

Recall that the reason I am interested in this in the first place is that I am trying to see if there is some natural way to extend Stokes’ theorem into something that might be called a “super Stokes’ theorem” :) I really like the sound of that :) The super Stokes’ theorem should look something like

(8)

\int_{S} d_{K} α = \int_{\partial_{K} S} α .

From the above, it appears that this is not possible in the continuum. However, there certainly is a natural extension of this to the discrete theory. Yet another reason why the discrete theory is superior to the continuum ;)

I claim that in the discrete theory that you can certainly write down a very natural “super Stokes’ theorem”, where the word “super” is not meant to mean “great”, it relates to supersymmetry :)

Eric

Posted by: Eric on July 11, 2004 6:46 PM | Permalink | Reply to this

Re: Scandinavian but not Abelian

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By the way, if any of this can be made to make sense and their is a “super Stokes’ theorem”, then this justified a proposal to call $d_{K}$ the “super exterior derivative” and $\partial_{K}$ the “super boundary” :)

Just a thought :)

This way we can refer to

(1)

d + A + i_{K}

as the “covariant super exterior derivative.” Phew! What a mouthful :)

Eric

Posted by: Eric on July 11, 2004 6:56 PM | Permalink | Reply to this

Re: Scandinavian but not Abelian

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Would it be so controversial to think of some name besides ‘deformed exterior derivative’ for $d + ι_{X}$ ?

You certainly have a point, since in any case the deformation here is different from these other deformations - at least on the surface of it. (But there is a subtle relation: Actually it is possible to get $d + i ι_{K}$ from similarity transformations and taking some linear combinations from $d$ alone. This is the content of equations (701) and (702) of this. But I so far haven’t managed to make any good use of this.)

I wonder if there is some natural operator $H_{X} : C_{p} \to C_{p + 1}$ such that

(1) $\int_{S} ι_{X} α = \int_{H_{X} S} α .$

I don’t know if this works for general vector fields, or for general Killing vector fields, $X$ , but I think something like that should work for $X = K$ the reparametrization Killing vector on loop space.

The resaon is essentially that $ι_{K}$ is nothing but the T-dual of $d$ !

I give the explanation of that weird sounding statement in that deformation paper. The point is that under the exchange of form creator with annihilators and of partial derivatives with multiplication by $X^{'}$ , the canonical supercommutation relation remain intact. So algebraically there is little difference between $d$ and $ι_{K}$ and as $d$ generates a differential calculus based on the 0-form, $ι_{K}$ should generate a differential calculus based on the top form.

You write:

(2) $H_{X} (t) S = ⋃_{0 \leq t^{'} \leq t} ϕ (t^{'})_{*} S$

to be a ( $p + 1$ )-chain

Wait, now I am confused: Are you claiming that a linear combination of $p$ -chains can be a $p + 1$ -chain? I don’t see how this should work, even in the discrete case, but maybe I am misunderstanding your notation.

As I said above, I would expect that the best way to get something like Stokes for $ι_{K}$ would be to regard this operator as another exterior derivative, a T-dual one, and devise a T-dual Stokes law for it.

Posted by: Urs Schreiber on July 12, 2004 10:14 AM | Permalink | PGP Sig | Reply to this

Re: Scandinavian but not Abelian

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Too bad! :)

I was so looking forward to hearing what you had to say that I almost couldn’t sleep last night. I rush in (half dressed) to check SCT this morning and this is all you have to say?!?!? :)

You write:

(1) $H_{X} (t) S = ⋃_{0 \leq t' \leq t} ϕ (t')_{*} S$

to be a $(p + 1)$ -chain

Wait, now I am confused: Are you claiming that a linear combination of $p$ -chains can be a $p + 1$ -chain? I don’t see how this should work, even in the discrete case, but maybe I am misunderstanding your notation.

The notation could probably use some work, but think of what I am saying. We have a $p$ -chain $S$ and a vector field $X$ that determines a 1-parameter flow $ϕ (t) : ℳ \to ℳ$ . Following Frankel, we can define

(2)

S (t) = ϕ (t)_{*} S

to be the $p$ -chain $S$ carried along $X$ for a time $t$ . The $p$ -chain $S$ is going to “sweep out” a $(p + 1)$ -dimensional region as it gets carried along $X$ . The region is going to be time dependent and I denote the corresponding $(p + 1)$ -chain by

(3)

H_{X} (t) S .

There is a little more to the story because a chain is more than just a point set, but involves orientation as well. This is not too difficult to work out. In the continuum, I found that I could not construct some $h_{X} : C_{p} \to C_{p + 1}$ that satisfies

(4)

\int_{S} i_{X} α = \int_{h_{X} S} α .

The best I could do in the continuum was (the closely related version)

(5)

\int_{S} i_{X} α = \frac{d}{dt} {[\int_{H_{X} (t) S} α] ∣}_{t = 0} .

This is the best you can do using standard continuum differential geometry because there, you think of $X_{p}$ as being a tangent vector at a point $p$ , whereas a tangent vector should really be associated with a little infinitessimal line segment. If instand of standard differential geometry, you were dealing with synthetic differential geometry that does treat little infinitessimal line segments, then you can get rid of the time derivative out front. Since little line segments are fundament in the discrete theory, things work out perfectly natural and you obviously do not need a time derivative out front (meaning you can have a “T-dual” Stokes theorem (I need to read you notes because I have no idea what “T-dual” means, but what you say sounds very interesting, i.e. $i_{X}$ is “T-dual” to $d$ .)).

Without reading your notes yet, it seems that if you can get a T-dual Stokes theorem, then it might mean that T-dual differential geometry is more closely related to synthetic differential geometry than to standard differential geometry. That would be interesting if true :)

Eric

PS: This

(6)

\int_{S} i_{X} α = \frac{d}{dt} {[\int_{H_{X} (t) S} α] ∣}_{t = 0} .

is NEAT!!! I am pretty sure it is correct too.

Posted by: Eric on July 12, 2004 1:11 PM | Permalink | Reply to this

Re: Scandinavian but not Abelian

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Too bad! :)

Oops, you are right. I wasn’t paying attention closely enough.

To my defence I could cite that my brain was really occupied with another calculation concerning nonablian loop space connections, as well as with some responses that I got on my latest spr post (unfortunately only by private email). But I shouldn’t.

You are right, the formula you give seems to be correct. It becomes pretty obvious when one picks some Cartesian set of coordinates and everything else rectangular, too.

For instance if $α = f (x) {dx}^{1} \land {dx}^{2}$ and $X = \partial_{x^{2}}$ and $S = {x^{2} = 0}$ , then $H_{X} (t) = {0 \leq x^{2} \leq t}$ and $ι_{X} α = - f (x) {dx}^{1}$ and we get

(1)

\int_{S} ι_{X} α = \int_{- ∞}^{∞} f (x) {dx}^{1}

and

(2)

\int_{H_{X} (t) S} α = - \int_{0}^{t} {dx}^{2} \int_{- ∞}^{∞} {dx}^{1} f (x) .

This very manifestly satisfies your formula. And if it works for little cubes then, because we are physicists, it works for everything.

Yeah, cool. This formula looks like it should have been known for ages, but I, too, cannot remember having seen it stated explicitly anywhere.

I think if we allow ourselfs to be a little cavalier with notation then we can even put it in a form that is very suggestive of the application that you have in mind, namely I suggest writing

(3)

\int_{S} ι_{X} α = \int_{H_{X}^{'} (0) S} α .

Then it would indeed be possible to write down the ‘super Stokes’ theorem’ as

(4)

\int_{S} (d + ι_{X}) α = \int_{H_{X}^{'} (0) S + \partial S} α .

Yes, good, I like that. Is there anything we can say about the chain $H_{X}^{'} (0) S + \partial S$ ?

Seems to be something like the infinitely tight ‘wrapping’ of $S$ .

BTW, I think I made some progress with nonabelian connections on loop space. I found that one should maybe first concentrate on connections which are flat on loop space, i.e. which assign the identity group element to every (contractible) closed curve in loop space, i.e. to every torus.

This is not quite as trivial as it may sound. Indeed, for a loop space connection to be flat both the 1-form $A$ and the 2-form $B$ are generically non-flat by themselves.

Moreover, for a true boundary effect in string theory, i.e. a scenario where all the nontrivial background is really living on the brane and coupled only to ends of open strings, the flat loop space connection is precisely what we need and want.

As far as I can see everybody including me is pretty much in the dark concerning the physical interpretation of nonablian 2-form backgrounds in string theory, but what I just wrote makes a lot of sense to me, now that I think about it. In particular it removes the confusion how anything nonablian could couple to a closed string. (Since, as you may have heard, in string theory non-abelianness comes from open strings that attach to several D-branes. The nonablian $N \times N$ matrices are essentially coincidence matrices describing which end of which string ends on which of the $N$ branes.)

So maybe flat nonablian connections on loop space is precisely what we should really be looking for. And I think this case is non-trivial and maybe only here all the desired properties hold.

If I find the time I’ll write that up in more detail.

Posted by: Urs Schreiber on July 12, 2004 3:26 PM | Permalink | PGP Sig | Reply to this

Re: Scandinavian but not Abelian

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Too bad! :)

Oops, you are right. I wasn’t paying attention closely enough.

Don’t worry about it :) I know you’ve got a million things on your mind. I almost feel guilty for bugging you with this stuff (note: I only said “almost” :)).

I like the new (?) way to view $i_{X}$ very much and the neat geometrical interpretation should carry over to loop space very naturally, which I think may help significantly in making everything more understandeable. Especially considering the somewhat prominent role of $i_{X}$ and $ℒ_{X}$ on loop space.

Yes, good, I like that. Is there anything we can say about the chain $H_{X}' (0) S + \partial S$ ?

Well, since it is hard to picture $H_{X}' (0)$ , I would suggest to instead consider the visualizable chain

(1)

H_{X} (t) S + \partial S

for some finite $t$ . We can even try to understand the operator

(2)

\partial_{X} (t) = H_{X} (t) + \partial .

The first thing to note is the important property

(3)

H_{X} (t)^{2} = H_{X} (t) H_{X} (t) = 0 .

The operator $H_{X} (t)$ has the nice geometrical picture of sweeping a $p$ -chain $S$ along forming a $(p + 1)$ -chain. If you sweep a $p$ -chain once and then sweep it again, you get a degenerate $(p + 2)$ -chain, the integral over which will always vanish.

The neat thing about $\partial_{X} (t)$ is that it squares to

(4)

\partial_{X} (t)^{2} = H_{X} (t) \partial + \partial H_{X} (t) = ϕ_{*} (t) - 1,

i.e.

(5)

\partial_{X} (t)^{2} S = S (t) - S .

This gives a beautiful interpretation of the Lie derivative and in fact is the transpose of the Lie derivative once you put a $d / dt$ out in front of the integral.

On loop space, reparameterization invariance means that

(6)

ϕ (t)_{*} S = S

where $ϕ (t)$ is the flow generated by sweeping points around the closed string. Therefore, it seems you could similarly state parameterization invariance via

(7)

\partial_{K} (t)^{2} = 0 .

Neat, huh? :)

BTW, I think I made some progress with nonabelian connections on loop space.

Nice to hear. I bet you wish you could clone yourself now more than ever :)

Remember, these are the best years of your life so no complaining ;)

Eric

Posted by: Eric on July 12, 2004 4:02 PM | Permalink | Reply to this

Re: Scandinavian but not Abelian

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Since things are a little easier to understanding using some finite $t$ , I just got the idea to define some operator

(1)

i_{X} (t) : Ω^{p} \to Ω^{p - 1}

via

(2)

\int_{S} i_{X} (t) α : = \int_{H_{X} (t) S} α .

Then we could define operators

(3)

d_{X} (t) = d + i_{X} (t)

and

(4)

ℒ_{X} (t) = d_{X} (t)^{2} = d i_{X} (t) + i_{X} (t) d .

Using your “cavalier” notation, then the regular Lie derivative is actually

(5)

ℒ_{X} = ℒ'_{X} (0) .

Fun fun :)

Eric

Posted by: Eric on July 12, 2004 4:13 PM | Permalink | Reply to this

Re: Scandinavian but not Abelian

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In an email to Urs, I said

You mentioned that i_X was somehow “T-dual” to d. Do you know where I might be able to find a discussion about this?

to which you replied

You will find the discussion as soon as you ask about it in the SCT! :-)

Ok. Here you go! :)

Eric

Posted by: Eric on July 15, 2004 3:57 PM | Permalink | Reply to this

Re: Scandinavian but not Abelian

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The answer to this question is given in section 4.2.1 of hep-th/0401175, and in particular in equation (4.12).

The canonical supercommutation relations on loop space are

(1)

[\partial_{(μ, σ)} X^{' (ν, κ)}] = - δ_{μ}^{ν} δ^{'} (σ - κ)

and

(2)

[ℰ^{† (μ, σ)}, ℰ_{(ν, κ)}] = δ_{ν}^{μ} δ (σ - κ) .

These canonical commutation relations are preserved under the exchange

(3)

\partial_{(μ, σ)} \leftrightarrow X^{' (μ, σ)}

(4)

ℰ^{† (μ, σ)} \leftrightarrow ℰ_{(μ, σ)} .

Physically this corresponds to exchanging the canonical momentum at each point of the string with its ‘winding’ excitation $X^{'} = \frac{d}{d σ} X .$ Since the canonical brackets are preserved (as long as the 0-mode of $X$ is not involved), this operation preserves the constraint algebra of the string and hence maps consistent string backgrounds to consistent string backgrounds. This duality is known as $T$ -duality. In the above mentioned paper I demonstrate how the usual facts about T-fuality - plus a little more - can be deduced from this algebra isomorphism.

Ok, so this answers your question: The exterior derivative $ℰ^{† (μ, σ)} \partial_{(μ, σ)}$ on loop space and the operator $ℰ_{(μ, σ)} X^{' (μ, σ)}$ of interior multiplication with the reparameterization Killing vector are interchanged under T-duality

(5)

ℰ^{† (μ, σ)} \partial_{(μ, σ)} \leftrightarrow ℰ_{(μ, σ)} X^{' (μ, σ)} .

A subtlety is that the above isomorphism does not work as soon as the undifferentiated coordinate field plays a role. This is related to the fact that you can T-dualize only (as far as I know) along Killing directions in target space. Namely in these cases you can find coordinates so that the target space metric is independent of the dualized directions, so that no $X$ along these directiosn appears in the string’s constraints.

Posted by: Urs Schreiber on July 15, 2004 4:35 PM | Permalink | Reply to this

Re: Scandinavian but not Abelian

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Hmm…

Does this have any meaning for target space? I mean, what if $d$ and $i_{X}$ are defined on target space? Is there some duality

(1)

d \leftrightarrow i_{X}

even in this case or do we have to go to loop space to see it?

Sorry if it is obvious :)

Eric

Posted by: Eric on July 15, 2004 5:57 PM | Permalink | Reply to this

Re: Scandinavian but not Abelian

$MathML-enabled post (click for more details).$

Well, if you forget about loop space you can note that all I really used is that $X^{' (μ, σ)}$ is a Killing vector on loop space, which is not trivially a constant vector field.

So assume in more generality that there is a covariant derivative $\nabla$ on some manifold and a Killing vector $v$ , so that (if application of the derivative is written as commutation with the respective operator)

(1)

[{\hat{\nabla}}_{μ}, v_{ν}] + [{\hat{\nabla}}_{ν}, v_{μ}] = 0 .

This implies that the bracket

$[{\hat{\nabla}}_{μ}, v_{ν}]$

is invariant under the exchange $\nabla_{μ} \leftrightarrow v_{μ}$ , because

(2)

[{\hat{\nabla}}_{μ}, v_{ν}] \to [v_{μ}, {\hat{\nabla}}_{ν}] = - [{\hat{\nabla}}_{ν}, v_{μ}] = + [{\hat{\nabla}}_{μ}, v_{ν}],

by the condition that $v$ is Killing.

So that’s formally what is going on, and it has as such nothing to do with loop space.

However, I would not know what this exchange means when the space it is used on is not loop space.

But maybe you can figure it out… :-)

Posted by: Urs Schreiber on July 15, 2004 6:14 PM | Permalink | Reply to this

Re: Scandinavian but not Abelian

$MathML-enabled post (click for more details).$

Is this anything like saying

(1)

ℒ_{X} = [d, i_{X}]

is invariant under the change $d \leftrightarrow i_{X}$ ? I doubt it because this is true regardless of whether $X$ is Killing.

I know I’m being dense (I feel dense at the moment), but I don’t see how invariance under

(2)

\nabla_{μ} \leftrightarrow X_{μ}

implies (or is related to)

(3)

d \leftrightarrow i_{X} .

In what sense is the interchange a duality.

Sorry :)

Eric

Posted by: Eric on July 15, 2004 7:21 PM | Permalink | Reply to this

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July 9, 2004