## July 30, 2004

### On Holiday

#### Posted by Urs Schreiber

I am very pleased about the current intensive discussion at and around the Coffee Table about lots of interesting things, which was very fruitful for me and, I think, pretty productive. Many thanks to all those who have contacted me by private email.

I am looking forward to continuing these discussions, but I would like you to know that I will take a little break and go on holiday for two weeks.

My girlfriend has decided that we should escape this year’s rainy german summer in a drastic way and head for the Arctic Circle and way beyond to meet maybe a spermwhale or one of his cetacean cousins.

I expect the density of internet cafés to drop considerably north of Narvik, so you won’t hear from me until about 12th of August.

To compensate for that dreadful lack of babble I’ll close this entry with some remarks on recent issues:

In a recent entry I had tried to make up my mind on Hawking’s talk about the black hole information problem. Now I have finally had a look at the paper by Maldacena that this talk is heavily influenced by:

Juan Maldacena: Eternal Black Holes in Anti-de-Sitter (2001)

(Thanks to Luboš Motl and Jens Fjelstad for this reference. The following is a quote from a post of mine on sci.physics.strings)

There is a nice and simple-to-understand insight presented in that paper, which is apparently the key to Hawking’s talk and which is roughly the following (for those who haven’t seen it):

Correlators computed in the boundary CFT cannot decay to zero in the far future, since the CFT is unitary. By AdS/CFT correspondence these correlators are equivalently computed in the bulk theory. Here one has to do the full path integral over gravity and the “matter” field whose operators are inserted at the boundary (as well as other fields, really, which are however ignored in approximation).

One assumes that the gravitational path integral can be approximated well by its saddle points, so that we are left with computing the matter bulk correlators using QFT on these curved backgrounds.

Now, on black hole backgrounds the “matter” correlators are known to decay to 0. Maldacena notes that there is no contradiction with the nonvanishing CFT result because one has to sum up contributions from all gravitational saddle points, which includes the ordinary AdS background, on which the correlators don’t decay and are in fact in accord with the boundary CFT result.

In summary, Maldacena shows/discusses that nontrivial topologies don’t contribute to the correlators of “matter” fields in the far future.

What he does not claim is that the purely gravitational path integral over nontrivial topologies vanishes - something which one might get the impression that Hawking is saying in his talk.

In any case, this seems to clarify it: When computing correlators on the boundary of AdS using the bulk theory nontrivial topologies don’t contribute, because there the correlators vanish. That’s pretty obvious, actually.

I can see now that this is what Hawking is saying, but from his talk alone I found it hard to get this point. In particular, I am now wondering what Hawking claims to have added to Maldacena’s observation.

In another recent entry I had tried to convince you that super Pohlmeyer invariants are more interesting than one might have thought. Now I have prepared some LaTeX notes on this stuff which provide a little more details.

Ok, that’s it. See ya.

## July 27, 2004

### A bird, a plane? No, super-Pohlmeyer!

#### Posted by Urs Schreiber

Recently I had talked about how I think Pohlmeyer invariants are related to boundary states of D-branes with gauge field and fluctuations turned on. I concluded by saying:

It just remains to note that by using the relation to DDF invariants it is easy to generalize the Pohlmeyer invariants to the superstring (section 2.3.3 of hep-th/0403260), which should carry all of the above disucssion over to its supersymmetric extension.

Indeed the super Pohlmeyer invariants, as I will call them, which are obtained this way seem to be useful and interesting. Here I will look at them in more detail.

The idea is as follows: Similar to the considerations presented in hep-th/0312260 we start with the supersymmetric vertex operator for the gauge field and construct a suitable path ordered multi-integral of this guy around the closed string such that the Ishibashi conditions are satisfied, which say that the result must be super-reparameterization invariant along the string (which in loop space differential geometric language means that the result must be ${d}_{K}$-closed).

The authors of hep-th/0312260 discuss one way to do this. Currently I fail to see if and how the boundary state (3.9) which they give is gauge covariant. Maybe it need not be, if only the correct states decouple in the amplitudes calculated from it.

The issue arises due to the fact that the supersymmetric vertex operator $V$ of the gauge field looks something like

(1)${A}_{\mu }\left(k\right){V}^{\mu }={A}_{\mu }\left(k\right)\left(\partial {X}^{\mu }+ck\cdot \psi {\psi }^{\mu }\right){e}^{ik\cdot X}\phantom{\rule{thinmathspace}{0ex}},$

where the fermionic term on the right is the exterior derivative of $A$ for the mode $k$ - but not the gauge covariant exterior derivative, but the ordinary one. This makes it very non-manifest that exponentiating this object through a Wilson line gives something gauge covariant (and does it?).

On the other hand, my point is that if we use the DDF$↔$Pohlmeyer relation to construct a Wilson-line-like boudnary state for non-abelian $A$ something manifestly gauge covariant is obtained.

This works as follows:

Start with the supersymmetric DDF oscillator in equation (2.58) of hep-th/0403260 and take its Fourier transform as done in equation (2.51) for the bosonic part of that operator. The resulting quasi local invariant observable now contains also a fermionic contribution, but this is a total $\sigma$-derivative, as one can easily see. That’s crucial.

Namely when one now inserts this integrated and Fourier transformed vertex into an ordinary Wilson line, the derivative term leads to boundary terms in the multi-integrals, in just such a way that the covariant field strength appears in the fermionic terms. More precisely, the result is

(2)$W\left[A{\right]}_{\mathrm{super}}=\mathrm{P}\mathrm{exp}\left({\int }_{0}^{2\pi }d\sigma \phantom{\rule{thinmathspace}{0ex}}\left(i{A}_{\mu }+\left({F}_{A}{\right)}_{\mu \nu }\frac{k\cdot \psi {\psi }^{\nu }}{k\cdot \partial X}\right)\partial {X}^{\mu }\right)\phantom{\rule{thinmathspace}{0ex}}.$

This is the explicit form of the super Pohlmeyer invariant. Written this way it is very unobvious that this is really an invariant, so that it commutes with the fermionic Virasoro generators. But, being constructed purely from DDF invariants by the same trick that the bosonic Pohlmeyer invariants are, it is still true, and so in particular this does respect the Ishibashi conditions.

What is manifest, however, is the gauge covariance of this object under $A↦{\mathrm{UAU}}^{†}+U\left({\mathrm{dU}}^{†}\right)$.

Another nice consequence of the fact that this object comes from exponentiated DDF operators is that it is easy to check for divergences when this is applied to a bare boundary state. The calculation turns out to be completely analogous to that in section 3.5. of hep-th/0407122, which means that to lowest order the condition for vanishing of the divergences in the super Pohlmeyer invariant are just the Yang-Mills equations for $A$.

The authors of hep-th/0312260 obtain the same result, but with a rather different method. They regularize their Wilson line so that divergences are canceled by construction. But then invariance is potentially broken and they confirm that it is preserved by the regularizartion precisely if the equations of motion hold.

So maybe the super Pohlmeyer invariant discussed above is just a complementary view on the same result obtained by these authors. It certainly describes some gauge field background, it has the form of a Wilson line, is well defined as an operator and satisfies the correct physical conditions. Plus - it is manifestly gauge covariant.

## July 23, 2004

### More remarks on flat loop space connections

#### Posted by Urs Schreiber

As a result of the recent disucssion about the relation between loop space connections and surface holonomy (I) as well as about the work by Alvarez, Ferreira & Guillén (II) I have decided to add a section which briefly addresses these topics to the preprint hep-th/0407122. A first draft of this section is reproduced in the following. In particular I would like to know if anyone thinks that I am right with claiming that a flat connection on loop space is sufficient to have well defined surface holonomies of topological spheres even when ${\pi }_{2}\left(ℳ\right)$ is nontrivial.

## July 22, 2004

### Three ways to loop flatness

#### Posted by Urs Schreiber

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 $\sigma =0,2\pi$ 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)${d}^{\left(A\right)\left(B\right)}=d+i{\oint }_{A}\left(B\right)\phantom{\rule{thinmathspace}{0ex}}.$

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 $\sigma =0$.

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)${\left({d}^{\left(A\right)\left(B\right)}\right)}^{2}={\left(d+i{\oint }_{A}\left(B\right)\right)}^{2}={\oint }_{A}\left({d}_{A}B\right)+{\oint }_{A}\left({F}_{A},B\right)-{\oint }_{A}\left(B,{F}_{A}\right)-{\oint }_{A}\left(B\right){\oint }_{A}\left(B\right)\phantom{\rule{thinmathspace}{0ex}}.$

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 ${F}_{A}\ne 0$ 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)${d}_{A}B=0\phantom{\rule{thinmathspace}{0ex}}.$

With this requirement, all that remains is the exterior product of the 1-form ${\oint }_{A}\left(B\right)$ 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, $\left({\oint }_{A}\left(B\right){\right)}^{2}=0$ 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 5:46 PM UTC | Permalink | Followups (10)

### To loose or not to loose information

#### Posted by Urs Schreiber

Over here and here a link to a transcript of Stephen Hawking’s talk on the apparent black hole information loss problem can be found.

The key argument in, well, a nutshell, is that the Euclidean path integral for gravity over topologically trivial manifolds gives an invertible mapping from initial to final configuration, while that over topologically non-trivial manifolds does not.

Hawking concludes from that that the total path integral will be unitary.

I am looking forward for seeing this detailed in a paper, because I am not sure what to make of it. Something unitary plus something non-unitary certainly does not give us something unitary, so this cannot be what is meant. I would understand the final claim if we had restricted ourselfs to the path integral over trivial topologies, but is this what is meant?

[Almost immediate update: Ah, now I get it, the idea is that the contribution from the nontrivial topologies completely factors to a constant and can be devided out. ]

In the talk, the AdS/CFT correspondence is mentioned frequently. Right at the beginning it seems like Hawking is crediting AdS/CFT and hence Maldacena for giving the solution to the information paradox and that his talk is merely supposed to elucidate how this happens in detail on the gravity side of the duality.

What I find puzzling is that AdS/CFT makes the ‘gravitational’ path integral well defined by giving it a UV-completion, namely string theory. Hawking on the other hand argues purely from the Euclidea path integral for Einstein-Hilbert gravity as well as its canonical quantization. But as far as I know the Euclidean path integral is only gradually better behaved than the Lorentzian one, Wick rotation in a scenario where no background metric and much less timelike isometries are present is a mystery, really, and finally nobody knows how and even if that canonical Hamiltonian operator of pure gravity can be defined, which Stephen Hawking argues to generate the unitary time evolution.

Some work by Maldacena on 3-dimensional AdS gravity is mentioned which seems to support the main claim that information loss and non-unitarity is related to nontrivial topologies, but I don’t know about the details here.

The last but one part of the talk is concerned with a rough (looks hand-waving, indeed, but it is not clear to me which omissions are due to the nature of the talk or actually due to unsolved problems) argument how one could go about actually calculating a solution which shows the unitary formation and evaporation of something that would be a black hole for practical purposes.

The very last part of the talk is about merchandising in theoretical physics. :-)

Posted at 11:21 AM UTC | Permalink | Followups (15)

## July 19, 2004

### Talking to ’t Hooft about tossing TVs

#### Posted by Urs Schreiber

Today Prof. Gerard ’t Hooft gave a talk at University Duisburg-Essen on Black Holes in Elementary Particle Physics. Maybe due to the media hype about Hawking’s announcement of his new idea about the black hole information ‘paradox’, ’t Hooft decided to throw his TV set away, and not only his but lots of them, in fact enough that they would form a spherical shell collapsing to a black hole.

Using this picture to emphasize the process in which ‘known physics’, represented by well understood TV sets, passes the horizon and hence a border beyond which all kinds of apparent paradoxes lurk, he talked about some standard facts of high energy physics and then briefly mentioned some of his intriguing observations and speculations concerning physics of the stretched horizon, the collision of infalling particles with outgoing Hawking radiation as well as the possibility of a deterministic hidden variable model of quantum theory, which, as he says, he develops as a hobby.

After the talk we went to a nearby Biergarten and I had the chance to ask some more detailed questions.

I have to admit that I haven’t read any of ’t Hooft’s papers concerning the above mentioned issues, so I learned for the first time about his calculation which indicates that, somehow, the scattering of Hawking radiation at infalling matter (one form - even though not the only one relevant I’d think - of back reaction which is not usually taken into account in related discussions, but which certainly should be) has some surprising resemblance to string scattering amplitudes - well, except for the curious fact that the analogy requires a imaginary string tension.

Very interesting are also his ideas about the foundations of quantum mechanics, holography and string theory.

He says that he expects that there is a deterministic and local (yes, local) hidden variable theory behind it all, which would be apparent if only we knew the correct degrees of freedom of nature. Since we don’t, we only see a statistical average of this deterministic process, and this translates in a non-local way to the quantum mechanical wavefunction, roughly.

To me this philosophy sounded a lot like approaches by Lee Smolin to get quantum mechanical dynamics from the classical statistics of ensembles of large matrices that encode the deterministic interrelation of all particles (well, probably, if at all, of all D0 branes) in the universe. But when I asked Prof. ’t Hooft about this he said he wasn’t fully familiar with Smolin’s approach.

Anyway, ’t Hooft’s idea now is that the full deterministic theory has no information loss, but that on the ‘coarse grained’ level of familiar quantum theory information is lost all the time in virtual black holes that are abundant in vacuum fluctuations. The point is that, he says, this way information about degrees of freedom in the bulk diasappears. The only information left is that at some holographic boundary! This way, I think, he tries to give a ‘dynamical’ explanation of holography.

I asked if and how he sees string theory fit into this picture, and he said that he thinks that since in string theory essentially only the S-matrix is a well defined observable, and since this means that only on-shell information at the ‘boundary’ is available while local physics in the bulk is fundamentally out of reach of present day string theory, this fits in perfectly with the above picture, where ordinary quantum mechanics is kind of an ‘effective theory’ on the boundary while the true bulk theory is a deterministic hidden-variable thingy.

I have to say that when first confronted with speculations like this some alarm bells go off - but then I realize that when ’t Hooft discovered holography a while back this idea must have sounded - before Maldacena came along and gave an explicit realization - just as weird, and now it is widely accepted and even standard lore.

So maybe in this little chat over a glass of beer I was actually shown a glimpse of the big physics picture of the future, without my poor mind being able to fully grasp it.

On the other hand, when asked what he thinks about how his ideas about string/gauge duality and holography have come to life in string theory, he answered, humbly and jokingly, that he almost fails to recognize his original ideas.

There was much more discussion, but that’s all I am going to report here. It was a big pleasure to talk to such an outstanding person as ’t Hooft is, and I have some things to think about now. First of all, I’ll toss away my TV set…

Posted at 9:31 PM UTC | Permalink | Followups (10)

## July 15, 2004

### Mathematical loop space literature

#### Posted by Urs Schreiber

Readers of my writings will have noticed that I mention the tem loop space from time to time.

Now, I have to admit that my understanding of loop space is very much that of a physicists, not that of a mathematician. My main tool to deal with the subtleties of the very concept of loop space is that I know how the exterior geometry on loop space is related to 2d superconformal field theory (by way of switching from Heisenberg to Schrödinger QFT picture), which is well understood.

Even though I have the suspicion that this, from the mathematical point of view rather unsophisticated, attitude is actually quite useful, my recent occupation with nonabelian connections on loop space indicates that I should try to get a more high-brow understanding of this space.

In particular, thinking about flat connections on loop space - and my claim is that a nonabelian 2-form on target space does correspond to a flat connection on loop space (which also follows from work by Girelli and Pfeiffer) - made me wonder about contractibility of paths in loop space.

A torus in target space is a closed path in loop space. This closed path can certainly be contracted to a point in loop space (this point corresponding to a constant function from the circle into target space) by shrinking the torus to a point.

But there are several closed paths in loop space that correspond to the same torus in target space, one corresponding to every foliation of that torus by circles.

All of these paths can be deformed into each other, but not all can be deformed into each other without using the constant path (in loop space). Namely the ‘horizontal’ and the ‘vertical’ slicings of the torus, corresponding to its 2 1-homotopy classes, can apparently never be continuously deformed into each other without using the ‘vanishing torus’.

Is that right?

See, that’s the level of understanding of global properties of loop space that I am currently sitting at…

For my own good I’ll list a couple of math books on loop space that certainly contain the answer to this and many other questions:

Jean-Luc Brylinski: Loop spaces, characteristic classes, and geometric quantization (1993)

John Frank Adams Infinite Loop Spaces (1978)

Hans Baues: Geometry of Loop Spaces and the Cobar Construction (1980)

L. Feher, A. Stipsicz: Topological Quantum Field Theories & Geometry of Loop Spaces
(1992)

- Sergeev, A. G. [Ed.] Loop spaces and groups of diffeomorphisms (1997)

Posted at 3:00 PM UTC | Permalink | Followups (16)

## July 13, 2004

### Nonabelian B-field equations of motion

#### Posted by Urs Schreiber

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

In the previous entry I discussed the boundary state which should describe superstrings in the background of a nonabelian 2-form field. Consistency requires that ${F}_{A}+B=0$ and hence the boundary state reads

(1)$\mid \text{D9}\left(A,B\right)⟩=P\mathrm{exp}\left(i{\int }_{0}^{2\pi }d\sigma \phantom{\rule{thinmathspace}{0ex}}{A}_{\mu }{X}^{\prime \mu }\left(\sigma \right)\right)\mid \text{D9}⟩\phantom{\rule{thinmathspace}{0ex}},$

i.e. it is obtained from the boundary state $\mid \text{D9}⟩$ of the bare D9 brane by acting on it with the untraced Wilson loop of the nonabelian 1-form gauge field $A$.

This is essentially (up to the $B$-contribution which follows from hep-th/0401175) a straightforward generalization of the technique described in

Koji Hashimoto: Generalized supersymmetric boundary state (2000)

for nonabelian $A$, nonvanishing $B$ field and the special case ${F}_{A}+B=0$. What I wrote previously serves to demonstrate that this boundary state has indeed the intended physical interpretation by showing that pulling the worldsheet supercurrent through the above Wilson line does indeed deform the exterior derivative in loop space in a way expected from 2-form gauge theory.

The next logical step is to check for the equations of motion of the background fields $A$ and $B$. In the above paper Koji Hashimoto demonstrated - for the case of abelian $A$ and $B$ = 0 - that these can be read off from the divergeces of the deformed boundary state itself.

Here I want to check for these divergences for the case of the nonabelian deformation given above. It turns out that in the nonabelian case this is actually much simpler than in the abelian case. For nonabelian $A$ we can, without loosing information, restrict attention to constant gauge fields, ${\partial }_{\mu }A=0$, for convenience. This removes a lot of worldsheet fields whose contractions would mess up the calculation.

Expanding the above $\mid \text{D9}\left(A,B\right)⟩$ in $A$ yields

(2)$\mid \text{D9}\left(A,B\right)⟩=\mid \text{D9}⟩+1+i{A}_{\mu }{\int }_{0<\sigma <2\pi }d\sigma \phantom{\rule{thinmathspace}{0ex}}{X}^{\prime \mu }\left(\sigma \right)\mid \text{D9}⟩-$
(3)$-{A}_{\mu }{A}_{\nu }{\int }_{0<{\sigma }_{1}<{\sigma }_{2}<2\pi }\phantom{\rule{-0.1667 em}{0ex}}\phantom{\rule{-0.1667 em}{0ex}}\phantom{\rule{-0.1667 em}{0ex}}\phantom{\rule{-0.1667 em}{0ex}}{d}^{2}\sigma \phantom{\rule{thinmathspace}{0ex}}{X}^{\prime \mu }\left({\sigma }_{1}\right){X}^{\prime \mu }\left({\sigma }_{2}\right)\mid \text{D9}⟩-$
(4)$-i{A}_{\mu }{A}_{\nu }{A}_{\lambda }{\int }_{0<{\sigma }_{1}<{\sigma }_{2}<{\sigma }_{3}<2\pi }\phantom{\rule{-0.1667 em}{0ex}}\phantom{\rule{-0.1667 em}{0ex}}\phantom{\rule{-0.1667 em}{0ex}}\phantom{\rule{-0.1667 em}{0ex}}{d}^{3}\sigma \phantom{\rule{thinmathspace}{0ex}}{X}^{\prime \mu }\left({\sigma }_{1}\right){X}^{\prime \mu }\left({\sigma }_{2}\right){X}^{\prime \mu }\left({\sigma }_{3}\right)\mid \text{D9}⟩+\cdots \phantom{\rule{thinmathspace}{0ex}}.$

Divergences in these expressions appear as follows:

The bare boundary state $\mid \text{D9}⟩$ is a Fock state with lots of worldsheet oscillators acting on the worldsheet vacuum and designed in such a way that

(5)$\left({\alpha }_{n}^{\mu }+{\overline{\alpha }}_{-n}^{\mu }\right)\mid \text{D9}⟩=0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\forall \phantom{\rule{thinmathspace}{0ex}}n,\mu \phantom{\rule{thinmathspace}{0ex}}.$

On this we are acting with ${X}^{\prime }$ which has the oscillator expansion

(6)${X}^{\prime \mu }\left(\sigma \right)=-\sqrt{{\alpha }^{\prime }/2}\sum _{n\ne 0}\left({\alpha }_{n}^{\mu }-{\overline{\alpha }}_{-n}^{\mu }\right){e}^{\mathrm{in}\sigma }\phantom{\rule{thinmathspace}{0ex}}.$

By using the above boundary condition on the oscillators annihilators can be turned into creators when applied to the boundary state, so that

(7)${X}^{\prime \mu }\left(\sigma \right)\mid \text{D9}⟩=-\sqrt{2{\alpha }^{\prime }}\sum _{n>0}\left({\alpha }_{-n}^{\mu }{e}^{-\mathrm{in}\sigma }-{\overline{\alpha }}_{-n}^{\mu }{e}^{\mathrm{in}\sigma }\right)\mid \text{D9}⟩\phantom{\rule{thinmathspace}{0ex}}.$

So when acting on this again with ${X}^{\prime \nu }\left(\kappa \right)$ there is a divergence from pulling the annihilators at $\kappa$ through the creators at $\sigma$:

(8)${X}^{\prime \mu }\left(\kappa \right){X}^{\prime \nu }\left(\sigma \right)\mid \text{D9}⟩={\alpha }^{\prime }{\eta }^{\mu \nu }\sum _{n>0}n\mathrm{cos}\left(n\left(\sigma -\kappa \right)\right)\mid \text{D9}⟩+:{X}^{\prime \mu }\left(\kappa \right){X}^{\prime \nu }\left(\sigma \right):\mid \text{D9}⟩\phantom{\rule{thinmathspace}{0ex}}.$

The sum over cosines is what has to be inserted into the above path-ordered integral in prder to compute the divergences. Obviously the first two terms have trivially no divergence and in the third they also obviously vanish. So the first nontrivial contribution comes at order $𝒪\left({A}^{3}\right)$.

I have taken the liberty to simply feed the relevant mixed ordered integrals over these cosines into Mathematica. This yields for the term at $𝒪\left({A}^{3}\right)$ a contraction proportional to

(9)$\sim {\alpha }^{\prime }{A}_{\mu }{A}_{\nu }{A}_{\lambda }\left({\eta }^{\mu \nu }-2{\eta }^{\mu \lambda }+{\eta }^{\nu \lambda }\right)\sum _{n>0}\frac{1}{n}{\int }_{0}^{2\pi }d\kappa \phantom{\rule{thinmathspace}{0ex}}{e}^{-\mathrm{in}\kappa }\left({e}^{\mathrm{in}\kappa }-1{\right)}^{2}{X}^{\prime }\left(\kappa \right)\mid \text{D9}⟩\phantom{\rule{thinmathspace}{0ex}}.$

So the divergences vanish iff

(10)${\eta }^{\mu \nu }\left[{A}_{\mu }\left[{A}_{\nu },{A}_{\lambda }\right]\right]=0+𝒪\left({\alpha }^{\prime 2}\right)\phantom{\rule{thinmathspace}{0ex}}.$

Assuming that this result is independent of our choice to work with constant $A$ this means that

(11)${\mathrm{div}}_{A}{F}_{A}=0+𝒪\left({\alpha }^{\prime 2}\right)\phantom{\rule{thinmathspace}{0ex}},$

i.e. that $A$ satisfies the equations of motion of nonabelian YM theory, up to higher order corrections.

This does certainly not look surprising, even though I think the derivation is pretty nice (assuming that I didn’t screw up somewhere). But I think, since this is in the context of nonabelian 2-form gauge theory, this result is not uninteresting.

To recall, what it really says (all assuming that I am not confused) is that the background field equations for strings in a nonablian 1-form background $A$ and a nonabelian 2-form background $B$ are

(12)${F}_{A}=-B$

and

(13)${\mathrm{div}}_{A}{F}_{A}=0$
(14)$⇒{\mathrm{div}}_{A}B=0\phantom{\rule{thinmathspace}{0ex}}.$

This is curious from the point of view of 2-group gauge theory. The reason is that these equations are not invariant under the 2nd order gauge transformation

(15)$A↦A+\Lambda$
(16)$B↦B-{d}_{A}\Lambda \phantom{\rule{thinmathspace}{0ex}}.$

But I think the discussion from the previous entry clarifies why this second transformation cannot be expected to be true in the boundary state formalism.

Namely there I had argued that the above gauge transformation is really one of the closed string, and it is still in the nonabelian case. But the closed string does not really couple to the nonabelian $A$ and $B$-fields, so that’s not of much physical importance. What does couple to these backgrounds is the open string. And here it matters which value $A$ has and so the above shift $A↦A+\Lambda$ should not be a symmetry.

Or so I think. Comments are welcome.

Posted at 12:23 PM UTC | Permalink | Followups (1)

## July 12, 2004

### Not Abelian - but Flat

#### Posted by Urs Schreiber

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

Over the weekend I had some time to think about what I had written about nonabelian 2-form connections here at the String Coffee Table as well as over on sci.physics.research.

The result are some refinements on my draft Nonabelian 2-form connections from 2d BSCFT deformations, which is beginning to approach a final form.

Using boundary state deformation theory I had read off the loop space connection induced by a nonabelian 2-form background from the deformed worldsheet supercharges. This is a 1-form connection on loop space, which is essentially what Christiaan Hofman writes as

(1)$d+\mathrm{iT}{\oint }_{A}\left(B\right)\phantom{\rule{thinmathspace}{0ex}},$

with the only difference that I derive that (at least in the context that I am working in) one has to define this a little differently, as explained before.

Then one can study gauge transformations of this 1-form connection in the usual way and try to read off what this implies for the target space fields $A$ and $B$.

It can be checked that global gauge transformations on loop space give rise to the ordinary target space gauge transformations obtained by adjoining unitary group elements.

The problem was that for loop-space-local gauge transformations one obtains the expected terms - plus plenty of other terms that don’t have a target space analogue.

As Christiaan Hofman and Hendryk Pfeiffer kindly told me, in one way or another this is a well known problem. What I think is new is the particular string/loop space perspective that I am using. This can be used for the following simple physical argument, which, as I now realize, is the heuristic version of the key for making further progress:

A non-abelian 2-form background must obviously couple to open strings only, since there are no Chan-Paton factors for closed strings (even though the abelian 2-form of course does couple to the closed string). This means that whatever connection we find, it should not couple to closed strings. But noting that a torus worldsheet in target space is nothing but a closed path in loop space, this tells us that we should be looking for loop space connections which are flat. That’s because this implies that the surface holonomy associated with every closed target space surface, which is the same as the line holonomy of the respective closed path in loop space, is always the trivial element $g=1$. The nontrivial physics is then induced only on open strings via the deformed boundary state.

So let’s look for configurations of $A$ and $B$ fields that make $\left(d+i{\oint }_{A}\left(B\right){\right)}^{2}=0$. There are probably several ways to derive these, but I have found that the boundary state deformation theory allows a very simple criteria: The boundary state deformation operator should not depend on worldsheet fermions.

This is precisely the case when the sum of $A$-field strength with the 2-form field vanishes:

(2)${F}_{A}+B=0\phantom{\rule{thinmathspace}{0ex}}.$

One can explicitly check that in this case the obnoxious corrections terms cancel each other and one finds that in this case that infinitesimal local gauge transformations in loop space of the form

(3)$U\left(X\right)=1-i{\oint }_{A}\left(\Lambda \right)+\cdots$

do indeed induce the expected

(4)$A↦A+\Lambda$
(5)$B↦B-{d}_{A}\Lambda$

(to first order in $\Lambda$), and one can also explicitly check that in this case the loop space connection is flat.

Now, to my delight, after I had convinced myself of the above relations I opened up

Florian Girelli & Hendryk Pfeiffer: Higher gauge theory - differential versus integral formulation (2004)

and found that the authors of that paper, by completely different reasoning (based on 2-groups introduced by John Baez), find precisely the same condition (their (3.25))!

Cool. So the result is not new (but not very old, either), but I still think that it is kind of interesting to see how the abstract category-theory reasoning meets string theory and in particular the loop space deformation theory that I have been working on.

To summarize: Nonabelian 2-form connections can consistently be defined only when ${F}_{A}+B=0$. This makes physical sense, because it is precisely the condition which says that the connection on loop space is flat, so that closed strings do not couple to the nonablian background, and they shouldn’t. The corresponding deformed boundary state $\mathrm{P}epx\left(\int A\cdot {X}^{\prime }\right)\mid \alpha ⟩$ should describe open strings in the nonabelian 1- and 2-form background.

What do you all think?

## July 9, 2004

### Scandinavian but not Abelian

#### Posted by Urs Schreiber

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)$\nabla =d+{\oint }_{A}\left(B\right)\phantom{\rule{thinmathspace}{0ex}},$

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)$A↦UA{U}^{†}+U\left({\mathrm{dU}}^{†}\right)$
(3)$B↦UB{U}^{†}$

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

(4)$A↦A$
(5)$B↦B+{d}_{A}\lambda$

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 $\left(d+{\oint }_{A}\left(B\right){\right)}^{2}$ 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)$d+{\oint }_{A}\left(B\right)={ℰ}^{†\left(\mu ,\sigma \right)}\left({\partial }_{\left(\mu ,\sigma \right)}+{U}_{A}\left(0,\sigma \right){B}_{\mu \nu }{X}^{\prime \nu }{U}_{A}\left(\sigma ,0\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$
Posted at 8:28 AM UTC | Permalink | Followups (116)

## July 4, 2004

### From Paris to Karlstad (via Bonn)

#### Posted by Urs Schreiber

These days you can see me with dark rings under my eyes, murmering to myself while struggling to keep myself upright - well, almost at least.

I had to hurry home from ${\mathrm{Strings}}_{04}$ at Paris on Thursday in order to tutor a class Friday morning here in Essen and to prepare for the things to come. In a couple of minutes I take a train to Bonn, where tomorrow I give a talk at the theory seminar on ‘Loop space methods in string theory’. After the talk I will have to hurry to Düsseldorf airport in order to get my plane to Karlstad/Sweden, where I’ll attend the conference on Non-commutative geometry and representation theory in mathematical physics, where I’ll also give the above talk. And guess where I have to be next Friday morning again.

I wanted to prepare a handout script for that talk. Unfortunately, while still in Paris I had to learn that my grandmother has deceased. Yesterday was her funeral and obviously my private and academic life are having a hard time sharing the same wetware at the moment.

Nevertheless I managed to get some things done.

First of all, talking to Christiaan Hofman in Paris inspired me to write up a paper on nonabelian 2-form connections on loop space, a draft of which is available here.

Then I made some last additions to the stuff I have been working on with Eric Forgy. Finally our notes are now available on the preprint server as math-ph/0407005 (available tomorrow morning).

This is all very timely, because these references make up important mosaic pieces in the big picture that I want to draw in my talk and which I have tried to describe in my (as yet not completely finished) talk notes Loop space methods in string theory.

If anyone feels sufficiently pitied by beginning of this entry he or she is welcome to consider doing me a favor by reading these notes and giving me some critical feedback.