## August 30, 2004

### r-flatness = 2-associativity

#### Posted by Urs Schreiber

I am further discussing loop space-, surface- and and 2-group-holonomy with those interested. While trying to convince the 2-group-theorists that 2-associativity admits more solutions than currently considered in the literature, Orlando Alvarez keeps coming up with ever more such solutions to loop space r-flatness(which ensures that loop space connections assign well-defined surface holonomy)! But what is the systematic way to find these solutions? Well, the answer should be given by 2-Lie algebra theory…

It seems to me that hence some effort demonstrating the equivalence of loop space r-flatness with 2-group 2-associativity is in order.

And Ithink that I have now figured out how to do the higher order expansion of (differential) 2-group 2-associativity, demonstrating that it is indeed (essentially) the same as r-flatness in loop space.

One key observation is that for the purpose of defining surface holonomies the 2-associativity law (‘exchange law’)

(1)$\left({f}_{1}\circ {f}_{1}^{\prime }\right)\cdot \left({f}_{2}\circ {f}_{2}^{\prime }\right)=\left({f}_{1}\cdot {f}_{2}\right)\circ \left({f}_{1}^{\prime }\cdot {f}_{2}^{\prime }\right)$

(where a simple dot indicates the horizontal and an open dot the vertical product)

needs only be required to hold in the limit where the surface elements ${f}_{2}$ and ${f}_{2}^{\prime }$ become infinitesimal, since that’s sufficient to consistently integrate their contributions up to a macroscopic surface holonomy.

(Note that this makes good sense also in the light of the fact that the right factor in the horizontal product is parallel transported to the left factor along a single edge, something which manifestly should not in general be done to an extended object.)

For a surface element ${f}_{1}$ and another one ${f}_{2}^{\prime }$ to the right of it the 2-associativity condition says that the commutator

(2)$\left[{g}_{2}^{-1}{f}_{1}{g}_{1},{f}_{2}^{\prime }\right]=0$

must vanish (when all elements take values in the same group and $t$ is trivial).

Because ${f}_{2}^{\prime }$ should be infinitesimal we write (following Girelli & Pfeiffer)

(3)${f}_{2}\approx \mathrm{exp}\left(-{ϵ}^{2}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{iB}}_{\mu \nu }\left(\sigma \right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

and let ${f}_{1}$ be the horizontal product of many small surface elements as in this figure

What looks like TESLA superconducting cavities is a ‘horizontal’ chain of small surface elements (drawn, however, diagonally) making up the total surface strip ${f}_{1}$. ${x}^{\mu }$ and ${x}^{\nu }$ are the coordinates parallel to the sides of the little squares and $\sigma =\frac{1}{2}\left({x}^{\mu }+{x}^{\nu }\right)$ runs ‘horizontally’ along the diagonal.

The whole trick is now to get an expression for ${f}_{1}$ in the limit where it becomes the horizontal product of all these squares

(4)${f}_{1}\to {f}_{1}\cdot {f}_{1}^{\prime }\cdot {f}_{1}^{\prime \prime }\cdots$

and where the left factor ${g}_{2}^{-1}{f}_{1}{g}_{1}$ becomes

(5)${g}_{2}^{-1}{f}_{1}{g}_{1}\to \left({g}_{2}{g}_{2}^{\prime }{g}_{2}^{\prime \prime }\cdots {\right)}^{-1}\left({f}_{1}\cdot {f}_{1}^{\prime }\cdot {f}_{1}^{\prime \prime }\cdots \right)\left({g}_{1}{g}_{1}^{\prime }{g}_{1}^{\prime \prime }\cdots \right)\phantom{\rule{thinmathspace}{0ex}}.$

This problem however becomes very simple once it is realized that the terms obtained by truncating ‘$\cdots$’ are recursively related as

(6)$\left({g}_{2}{g}_{2}^{\prime }{g}_{2}^{\prime \prime }{\right)}^{-1}\left({f}_{1}\cdot {f}_{1}^{\prime }\cdot {f}_{1}^{\prime \prime }\right)\left({g}_{1}{g}_{1}^{\prime }{g}_{1}^{\prime \prime }\right)={g}_{2}^{\prime \prime -1}\left(X\right){g}_{1}^{\prime \prime }\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{g}_{1}^{\prime \prime -1}{f}_{1}^{\prime \prime }{g}_{1}^{\prime \prime }\phantom{\rule{thinmathspace}{0ex}},$

where $X$ is the same expression with at most single primes, and so on.

By performing the lattice computations similar to those done by Girelli&Pfeiffer, but taking one more order in $ϵ$ into account one finds that

(7)${g}_{1}^{-1}{f}_{1}{g}_{1}\approx \mathrm{exp}\left(-{ϵ}^{2}\phantom{\rule{thinmathspace}{0ex}}\left(1-ϵ\phantom{\rule{thinmathspace}{0ex}}i{A}_{\sigma }\right)i{B}_{\sigma \tau }\left(1+ϵ\phantom{\rule{thinmathspace}{0ex}}i{A}_{\sigma }\right)\right)$

and

(8)${g}_{2}^{-1}\mathrm{exp}\left(-{ϵ}^{2}\phantom{\rule{thinmathspace}{0ex}}\mathrm{iY}\right){g}_{1}\approx \mathrm{exp}\left(-{ϵ}^{2}\phantom{\rule{thinmathspace}{0ex}}i\left(1-ϵ\phantom{\rule{thinmathspace}{0ex}}i{A}_{\sigma }\right)\left(Y+{F}_{\sigma \tau }\right)\left(1+ϵ\phantom{\rule{thinmathspace}{0ex}}i{A}_{\sigma }\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

Plugging this into the above recursion formula one finds that ${g}_{2}^{-1}{f}_{1}{g}_{1}$ for ${f}_{1}$ a long thin ‘horizontal’ strip becomes

(9)$\cdots \approx \mathrm{exp}\left(-iϵ{\int }_{0}^{\sigma }d{\sigma }^{\prime }\phantom{\rule{thinmathspace}{0ex}}{W}_{A}^{-1}\left({\sigma }^{\prime },\sigma \right)\left({F}_{\sigma \tau }\left({\sigma }^{\prime }\right)+{B}_{\sigma \tau }\left({\sigma }^{\prime }\right)\right){W}_{A}\left({\sigma }^{\prime },\sigma \right)\right)\phantom{\rule{thinmathspace}{0ex}},$

where $W$ is the holonomy of $A$ along the strip.

Inserting this result into the 2-associativity condition finally yields

(10)$\left[{\int }_{0}^{\sigma }d{\sigma }^{\prime }\phantom{\rule{thinmathspace}{0ex}}{W}_{A}^{-1}\left({\sigma }^{\prime },\sigma \right)\left({F}_{\sigma \tau }\left({\sigma }^{\prime }\right)+{B}_{\sigma \tau }\left({\sigma }^{\prime }\right)\right){W}_{A}\left({\sigma }^{\prime },\sigma \right)\phantom{\rule{thinmathspace}{0ex}},{B}_{\sigma ,\tau }\left(\sigma \right)\right]=0\phantom{\rule{thinmathspace}{0ex}}.$

This, I claim, is the ‘exchange law’ in 2-group theory in terms of the target space fields $A$ and $B$ which ensures that 2-group holonomy of continuous surfaces is well-defined.

It is essentially the same as the r-flatness condition found by Orland Alvarez. The latter involves another integral with $\sigma$ and ${\sigma }^{\prime }$ exchanged.

But for all solutions to r-flatness found so far the integrand actually vanishes for all ${\sigma }^{\prime }$ seperately. All these solutions hence also solve the above 2-associativity condition.

The interesting question arises whether the above is actually equivalent to the vanishing of the integrand all by itself.

Posted at 7:02 PM UTC | Permalink | Followups (44)

## August 27, 2004

### News release on emergent string spacetime

#### Posted by Urs Schreiber

My friend Rüdiger Vaas, who is working as a science journalist for the german science magazin Bild der Wissenschaft, where he is responsible for coverage of gr-qc and hep-ph, hep-th related things, informs me that his new article

Rüdiger Vaas: Time before Time (2004)

on notions of the term ‘universe’ and beginning and end of time in science and philosophy pays tribute to some of the hard lobbying work that Robert Helling and I did a while ago here on the SCT in order to promote awareness in the lay public of the fact that spacetime is an emergent phenomenon in string theory.

I know, at least in the US press string theory tends to be rather over- than under-hyped, but since the interested German reader of popular science magazines is currently bound to believe that it is established that ‘the universe is a spin-network’, I find the small reference to Robert Helling’s exposition ‘D-geometry’ on p.14 of Rüdiger’s article already worth mentioning.

Apparently the semi-popular article discussing the amazing fact that string theory identifies spacetime and its generalizations with 2-dimensional superconformal field theories of central charge $c=15$ is still lacking. Its dramatic implication for the general world view is a maybe a little less accesible than the claim that ‘the universe is a spin network’ (which, remarkably, is wrong even in loop quantum gravity).

Posted at 7:08 PM UTC | Permalink | Followups (1)

### On loop space and 2-group holonomy

#### Posted by Urs Schreiber

I had a lot of interesting discussion about what I have written in the previous entry and I promised to write up a comprehensive pdf on what I am talking about, since much of it is currently scattered over various SCT entries and spr posts. It took me a little longer than it should have, but now the result are the notes

There is still some work to be done, but it seems to me that an interesting picture is emerging.

## August 24, 2004

### Why does 2-group theory miss surface holonomies with nonvanishing B+F?

#### Posted by Urs Schreiber

Recall that there are (at least) three roads to well-defined surface holonomies in loop space formalism, but apparently only one road in 2-group theory, namely that called $B+{F}_{A}=0$.

What happens to the other two?

I believe the answer is: One of them is in some sense a special case of the abelian theory and hence not strictly missed. But the other one is missed by 2-group theory because of its inherently finite, non-differential, nature. It could be incorporated if we allowed ‘lattice artifacts’ in 2-group theory that vanish in a ‘continuum limit’ where elementary morphisms become infitesimal, which amounts to considering some sort of ‘weak’ 2-group in some sense.

I think so for the following reason:

Consider a ‘tesselation’ of a surface where ‘plaquettes’ are labeled by group elements $h$ and edges are labeled by group elements $g$ as in hep-th/0206130 and hep-th/0309173.

For notational simplicity assume that both labels take values in the same group.

We want to multiply all the $h$ labels together to obtain the full surface holonomy. But the ordering of these will matter. For $h$ labels sitting next to each other vertically they can be just multiplied like beads on a string using the geometrically induced order.

Labels horizontally next to each other have to be inserted into this string of beads. In order to move them around we need some parallel transport, namely that provided by the group elements of the edge labels. Parallel transporting some surface label $h$ from target to source of some edge $g$ gives $gh{g}^{-1}$.

But this zipper must have the special properties that it does not matter where beads are inserted, otherwise the surface holonomy would be ill-defined.

Therefore consider a plaquette ${f}_{1}$ with an edge ${g}_{1}$ going along its upper and an edge ${g}_{2}$ along its lower boundary and with a surface element ${f}_{2}^{\prime }$ to its right (where I use labels as on p. 5 of hep-th/0206130).

Either we move ${f}_{2}^{\prime }$ to the upper boundary of ${f}_{1}$ by parralel transporting it with ${g}_{1}$, or to its lower boundary by using ${g}_{2}$. The resulting string of beads must be the same in both cases, which means that the crucial condition

(1)$\left(\star \right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{f}_{1}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{g}_{1}{f}_{2}^{\prime }{g}_{1}^{-1}={g}_{2}{f}_{2}^{\prime }{g}_{2}^{-1}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{f}_{1}$

must be satisfied.

Here I have used non-standard wording, because I think the zipper-process going on here is instructive, but it can be checked that this equation is equivalent to that found in 2-group literature saying that the order of horizontal and vertical composition must not matter.

So given the group labels of our graph and the above rule to multiply them all up, what one needs to do in order to obtain the most general well-defined surface holonomy is find all solutions of $\left(\star \right)$

One way to solve this equation is that used in 2-group theory, namely that obtained by choosing arbitrary edge labels and setting the surface labels equal to the edge-holonomy around the surface. This yields $B+{F}_{A}=0$ in the continuum.

Of course there is a much more trivial way to solve $\left(\star \right)$: If we choose arbitrary edge labels and let the surface labels take values generated by an abelian ideal of the group algebra, then surface labels may be commuted with everything (but not edge labels among themselves) and $\left(\star \right)$ is easily seen to be satisfied. This is one of the cases used in hep-th/9710147.

In fact, the authors of that paper assume in addition the curvature of the edge labels to vanish, which is necessary to have a flat connection on loop space but not necessary in order to have a well-defined surface holonomy in this case.

So this case kind of belongs to the abelian theory, but it is maybe noteworthy that the ege labels need not be gauge equivalent to the trivial set of labels at all for this to give a well-defined surface holonomy. In this sense this does have some genuine non-abelian character and deserves to be mentioned as one interesting solution of $\left(\star \right)$.

What about the second condition considered in hep-th/9710147, that where the edge labels have vanishing loop holonomy and surface labels are ‘covariantly constant’ with respect to the edge labels?

This does solve $\left(\star \right)$, too, but only if we let the size of the surface elements tend to 0!

Namely in the covariantly constant case we can pick one reference plaquette on the surface and the label for any other surface element is defined to be the parallel transport of that reference label to the given position. The path of the parallel transport does not matter (in the limit of small lattice spacing) due to the assumption that loop holonomies vanish (that the 1-form connection is flat).

But then one can see (I don’t bother to spell the obvious but tiresome details out), that the right hand side of $\left(\star \right)$ is essentially the same group-valued function as the left hand side, but translated by one lattice unit. In other words, left and right hand side differ by something proportional to our lattice spacing.

This means that in the continuum limit this does give a well defined surface holonomy (as we already know from the loop space point of view), but it also means that strict 2-group theory does not see this possibility to get such a unique holonomy for $B+{F}_{A}\ne 0$.

I think that alone is (unless I screwed up somewhere) an interesting fact. But it also makes me speculate:

The small translation between the left and the right hand side of $\left(\star \right)$ just discussed can, due to the covariant constancy of the surface labels, also be interpreted as a short parallel transport obtained by adjoining the group element of the edge translating between the two nearby positions of the left and the right hand side.

This means that both sides of this equation are equal up to isomorphism in this sense, doesn’t it?

Possibly I am wrong, being a category-theory layman, but this sounds like some ‘weak’ notion of category or something like that.

Any ideas?

Posted at 5:14 PM UTC | Permalink | Followups (11)

## August 23, 2004

### 1+1D LQG is really about boundary states

#### Posted by Urs Schreiber

Over on Peter Woit’s blog we might have the chance to continue some of the discussion (I, II, III) about alternative and non-perturbative quantizations of gravity with Lee Smolin. I am looking forward to seeing his replies.

Meanwhile, I would like to maybe help focus the discussion by, following the important suggestion by Hermann Nicolai, again concentrating on a simple toy example of full 3+1d gravity, namely 1+1 dimensional gravity coupled to scalar matter.

As readers of this blog will remember, it was Thomas Thiemann’s paper hep-th/0401172 which made a couple of people aware of a crucial, in principle very well-known but rarely noticed technicality in the LQG approach:

This is the postulate that ‘to quantize’ a theory does not require to have an operator representation of canonical coordinates and canonical momenta.

This plays a big role in LQG because there the reparameterization constraints are implemented not by quantizing the generators of reparameterizations (as is done, for instance, in the standard quantization of 1+1 dimensional gravity which leads to the string worldsheet theory and where the crucial and characteristic quantum effects come precisely from operator ordering effects in the operator representation of the generators) but by constructing operators on a non-seperable Hilbert space which represent the classical symmetry.

While some theorists feel that this step should be uncontroversial, for instance Josh Willis argued that ‘relaxed canonical quantization’ is nothing to be worried about, it should be noted that when it is applied to systems whose standard (e.g. path integral or BRST) quantization we do understand, like the free particle or 1+1 dimensional gravity, the results are blatantly different from the standard results. Recall that these standard results for well understood systems like the free particle is what we experimentally know to be correct.

There is one paper which tries to address how this difference could disappear in some sort of limit. This is gr-qc/0207106, where the concept of ‘Shadow States’ was introduced as a means to get back from a ‘shadow state’ in the non-seperable LQG-like Hilbert space to that of the corresponding ordinary Hilbert space in standard quantization. But a closer inspection of that paper (and in particular the lower half of p. 14) showed that this is only made to work by including information obtained in the standard quantization. But this is not available for systems where the standard canonical quantization is not available, like for 3+1D gravity.

In the discussion Thomas Thiemann conceded that hence the use of ‘relaxed canonical quantization’ is a step based on the hope that experiment will (or would) find that at the Planck scale we have to modify standard quantum theory. Now, quantum gravity in general is a highly speculative business, but evidence that non-standard quantum theory will be a key ingredient are rather rare.

Since this sort of discussion seems to have been delayed by the general complexity of quantum gravity in more than 2+1 dimensions, I find it very helpful to discuss all these conceptual questions in lower dimensions, preferably in 1+1 of them, where standard quantum gravity is under full control, while still being non-trivial.

In particular, the reason why I decided to warm up the former discussion again is that I would like to point out one particular aspect of LQG quantization in 1+1 dimensions, which previously did not receive any attention: That’s its formal relation to boundary state formalism in string theory.

Recall that in LQG in 3+1 dimensions people construct a space of states that are invariant under spatial reparameterizations, and that they then try to construct a Hamiltonian constraint operator (but as far as I am aware this object is still elusive) to act with it on the space of reparameterization invariant states.

Furthermore recall that when 1+1 dimensional gravity is quantized as in string theory, physical states are not spatially reparameterization invariant. In the standard mode notation the reparameterization invariance assumed in LQG would amount to requiring that physical states be annihilated by ${L}_{n}-{\overline{L}}_{-n}$ for all integer $n$. But, as is very well known, physical states can be annihilated at most by half of these generators.

So if we still insist to follow the LQG prescription of restricting to rep-invariant states we find non-physical states, in the sense of OCQ, namely the boundary states. They become BRST closed when multiplied by an appropriate ghost sector and are in this sense again physical, but the important point is that they are not annihilated by the worldsheet Hamiltonian constraints.

Remarkably, this is precisely the situation in current 3+1 dimensinal LQG: The space of spatially rep-invariant states can be written down, but the operator version of the Hamiltonian constraint on this space is not available.

Still, boundary states in string theory play an important role and are interesting all by themselves.

Since Thomas Thiemann has originally suggested that Pohlmeyer invariants serve as observables for the LQG-like quantization of the string it is maybe interesting to observe that Pohlmeyer invariants, do play a role in the standard quantization of (super-)string as boundary state deformation operators, which turn on non-abelian gauge fields on D-branes.

## August 20, 2004

### Super-Pohlmeyer invariants and boundary states for non-abelian gauge fields

#### Posted by Urs Schreiber

I have just submitted the following to the preprint server:

Urs Schreiber: Super-Pohlmeyer invariants and boundary states for non-abelian gauge fields (2004)

Abstract:

Aspects of the supersymmetric extension of the Pohlmeyer invariants are studied, and their relation to superstring boundary states for non-abelian gauge fields is discussed. We show that acting with a super-Pohlmeyer invariant with respect to some non-abelian gauge field $A$ on the boundary state of a bare D9 brane produces the boundary state describing that non-abelian background gauge field on the brane. Known consistency conditions on that boundary state equivalent to the background equations of motion for $A$ hence also apply to the quantized Pohlmeyer invariants.

Introduction:

This paper demonstrates a relation between two apparently unrelated aspects of superstrings: boundary states for nonabelian gauge fields and (super-)Pohlmeyer invariants.

On the one hand side superstring boundary states describing excitations of non-abelian gauge fields on D-branes are still the subject of investigations [1,2,3] and are of general interest for superstring theory, as they directly mediate between string theory and gauge theory.

On the other hand, studies of string quantization focusing on non-standard worldsheet invariants, the so-called Pohlmeyer invariants, done in [4,5,6,7] and recalled in [8], were shown in [9,10] to be related to the standard quantization of the string by way of the well-known DDF invariants. This raised the question whether the Pohlmeyer invariants are of any genuine interest in (super-)string theory as commonly understood.

Here it shall be shown that the (super-)Pohlmeyer invariants do indeed play an interesting role as boundary state deformation operators for non-abelian gauge fields, thus connecting the above two topics and illuminating aspects of both them.

A boundary state is a state in the closed string’s Hilbert space constructed in such a way that inserting the vertex operator of that state in the path integral over the sphere reproduces the disk amplitudes for certain boundary conditions (D-branes) of the open string. In accord with the general fact that the worldsheet path integral insertions which describe background field excitations are exponentiations of the corresponding vertex operators, it turns out that the boundary states which describe gauge field excitations on the D-brane have the form of (generalized) Wilson lines of the gauge field along the closed string [11,12,1,2,3].

Long before these investigations, it was noted by Pohlmeyer [7], in the context of the classical string, that generalized Wilson lines along the closed string with respect to an auxiliary gauge connection on spacetime provide a ‘complete’ set of invariants of the theory, i.e. a complete set of observables which (Poisson-)commute with all the Virasoro constraints.

Given these two developments it is natural to suspect that there is a relation between Pohlmeyer invariants and boundary states. Just like the DDF invariants (introduced in [13] and recently reviewed in [9]), which are the more commonly considered complete set of invariants of the string, commute with all the constraints and hence generate physical states when acting on the worldsheet vacuum, a consistently quantized version of the Pohlmeyer invariants should send boundary states of bare D-branes to those involving the excitation of a gauge field.

Indeed, up to a certain condition on the gauge field, this turns out to be true and works as follows:

Posted at 7:29 PM UTC | Permalink | Followups (1)

## August 19, 2004

### Everybody agrees on the non-abelian boundary state!

#### Posted by Urs Schreiber

[Update next day: What I am saying here is the content of pp. 236-237 of

As I have probably mentioned before, I am currently writing up some notes on how super-Pohlmeyer invariants give boundary states for non-abelian gauge fields when applied to the boundary state of a bare D9 brane.

However, up until about two minutes ago I thought there was one big unsolved problem. I was just about to write in the conclusion section a paragraph about how this apparent problem remains unsolved, when suddenly the gods of reserch showed mercy and enlighted me. It’s abasingly simple. There is no problem at all, in fact everything is in much better shape than I thought.

This is what I was puzzled about:

As I menioned before and as far as I am aware, there are two versions of what the boundary state for a non-abelian gauge field should look like. One is that given in equation (3.7) of hep-th/0312260, the other the non-abelian generalization of the state considered in JHEP2000/04/023 which I generalized in hep-th/0407122 to include a nonabelian $B$ field.

Now in the introduction of hep-th/0312260 the ideas in JHEP2000/04/023 are addressed as ‘another approach’ and somehow this seriously confused me. I was under the impression that there are two different proposals for how the boundary state for a nonabelian gauge field looks like and worried about how that could be. I still don’t know what precisely the authors of that paper meant by ‘another approach’ - but the simple truth is that the boundary states considered in all these papers are all one and the same, identical!

Even though it’s very simple, let me spell it out:

The boundary state given in hep-th/0312260 is the Wilson line

(1)$\mathrm{P}\mathrm{exp}\left(\int d\sigma \phantom{\rule{thinmathspace}{0ex}}d\theta \phantom{\rule{thinmathspace}{0ex}}V\left(X\left(\sigma ,\theta \right)\right)\right)$

over the string, with the integrals being super-integrals over the bosonic parameter $\sigma$ and the Grassmann parameter $\theta$ over the gluon super-vertex

(2)$V\left(X\left(\sigma ,\theta \right)\right)={A}_{\mu }\left({D}_{\mu }{X}^{\mu }\right){e}^{ik\cdot {X}^{\mu }}\left(\sigma ,\theta \right)=\left(-a{A}_{\mu }{ℰ}^{†\mu }-i\theta \left({X}^{\prime \mu }+i{ℰ}^{†\mu }\phantom{\rule{thinmathspace}{0ex}}k\cdot {ℰ}^{†}\right)\right){e}^{ik\cdot X}\left(\sigma \right)\phantom{\rule{thinmathspace}{0ex}},$

where in my wacky notation ${ℰ}^{†}\propto \psi +i\overline{\psi }$ is the polar combination of the worldsheet fermions which constitutes a 1-form on loop space.

P in the integral above denotes path ordering, but because the integral is in superspace this is super-path ordering which is enforced by the super-step function $\Theta \left({\sigma }_{n+1}-{\sigma }_{n}+i{\theta }_{n+1}{\theta }_{n}\right)$ which expands to

(3)$\Theta \left({\sigma }_{n+1}-{\sigma }_{n}+i{\theta }_{n+1}{\theta }_{n}\right)=\Theta \left({\sigma }_{n+1}-{\sigma }_{n}\right)+i{\theta }_{n+1}{\theta }_{n}\delta \left({\sigma }_{n+1}-{\sigma }_{n}\right)\phantom{\rule{thinmathspace}{0ex}}.$

And there you go. I didn’t think about this carefully when reading their paper, but it is very easy to see that this second term quadratic in the Grassmann variables glues gluon-super vertices together in such a fashion that the gauge-covariant derivative appears in the Wilson line, so that after performing the Grassmann integrals one is left with

(4)$\mathrm{P}\mathrm{exp}\left(\int d\sigma \phantom{\rule{thinmathspace}{0ex}}d\theta \phantom{\rule{thinmathspace}{0ex}}V\left(X\left(\sigma ,\theta \right)\right)\right)=\mathrm{P}\mathrm{exp}\left(\int d\sigma \phantom{\rule{thinmathspace}{0ex}}\left({\mathrm{iA}}_{\mu }{X}^{\prime \mu }+c\left({F}_{A}{\right)}_{\mu \nu }{ℰ}^{†\mu }{ℰ}^{†\nu }\right)\right)\phantom{\rule{thinmathspace}{0ex}},$

which is precisely the boundary state that I consider in hep-th/0407122 and hep-th/upcoming in various contexts.

Phew.

## August 18, 2004

### Microscopic relation between loop space connections and 2-group holonomies

#### Posted by Urs Schreiber

Currently I am discussing aspects of surface holonomy in terms of loop space connections with Jens Fjelstad. One aspect that keeps puzzling us is the following:

Introduction:

Recall that a curve in loop space corresponds to a, possibly degenerate, surface in target space and that taking the holonomy of a connection on loop space over this curve hence associates a group element to that surface which is addressed as surface holonomy. But in general the group element obtained this way is not unique.

Some of this non-uniqueness can be understood and removed easily: As I noted in hep-th/0407122 from various points of view one is lead to restrict attention to connections on loop space which are flat. This condition suffices to ensure that homotopy equivalent curves in loop space with the same endpoints assign the same surface holonomy.

Moreover, it turns out that no generality is lost by restriction to flat connections: A large and ‘natural’ set of connections $𝒜$ on loop space is of the form

(1)$𝒜={\int }_{0}^{2\pi }d\sigma \phantom{\rule{thinmathspace}{0ex}}{W}_{A}\left(\sigma \right){B}_{\mu \nu }\left(X\left(\sigma \right)\right){W}_{A}^{-1}\left(\sigma \right){X}^{\prime \mu }\left(\sigma \right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{dX}}^{\nu }\left(\sigma \right)$

with $A$ locally a 1-form and $B$ locally a 2-form on target space $ℳ$, $X:\left(0,2\pi \right)\to ℳ$ the given loop and ${W}_{A}$ the holonomy of $A$ along that loop.

In the above paper I showed that for gauge transformations on loop space to preserve the general form of this connection we need to have

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

where ${F}_{A}$ is the field strength of $A$ and that this does imply that $𝒜$ is flat. But in hep-th/0309173 it was discussed (using 2-group technology) that $B=-{F}_{A}$ is also the necessary condition for a well defined surface holonomy. Hence nothing is lost by restricting to flat connections on loop space of this form.

But in the loop space framework a connection $𝒜$ of the above form with $B=-{F}_{A}$ alone does not seem to ensure a well-defined surface holonomy. Apparently there must be further consistency conditions.

We have tried to discuss here these problems for the case of tori recently. But let’s concentrate on topological spheres for a moment, where things are much simpler:

For the sphere one can convince oneself that all closed curves in (unbased, oriented, parameterized) loop space which map (without overlap) to a given sphere in $ℳ$ are continuously deformable into each other and hence do associate unique surface holonomy. All these closed curves correspond to a foliation of the sphere into circles which share a common base point.

But there are also open curves in loop space which map to that given sphere. These must all start and end at a constant loop, too, but not at the same one.

It is easy to see that these open curves in loop space cannot possibly associate a unique surface holonomy: By making a gauge transformation on loop space the holonomy of the open curve in loop space can be given any value.

So the question that Jens Fjelstad and I are thinking about is:

Question:

Is there a fix to the above problem which makes the surface holonomies associated with open curves in loop space well defined, or, if not, do we have a good reason not to compute surface holonomy using open curves in loop space?

I currently tend to think that the latter is true. For the following reason:

Since the 2-group scheme does associate a unique surface holonomy in every case, it must be that the computation of surface holonomy using closed curves in loop space which correspond to spheres can be mapped to a computation along the lines of section 2.5 in hep-th/0309173, while apparently for open curves in loop space this is not possible.

If this were true it would explain from the 2-group perspective why we should not want to compute surface holonomy using open curves in loop space - it would just not correspond to an honest 2-group calculation.

Therefore one should try to translate the computation of the holonomy of the loop space connection $𝒜$ given above to the computations used in 2-group theory and see under which conditions the two can agree. This is what I am going to do here. I am calling this the microscopic relation between the two approaches because the idea is to discretize both computations and see how the adding up of the discrete contributions in both cases compare. It turns out that apparently indeed only for closed curves in loop space which run over loops that all share a common point does the computation of the holonomy of $𝒜$ amount locally (and hence globally) to that of the corresponding 2-group computation.

This works as follows:

The details:

First recall the laws of 2-group computations:

I) review of 2-group computations

Recall that the computation of surface holonomy using 2-group technology works as follows:

The surface whose holonomy is to be computed is covered with a graph (whose average mesh size vanishes as we take the continuum limit) whose edges are labeled by group elements $g\in G$ and whose faces are labeled by group elements $h\in H$ and by a sort of orientation defined by giving two vertices touching that surface.

The labeling has to satisfy some obvious consistency relations together with the not-so-obvious but extremely crucial one which says that

(3)$h={g}_{2}\cdot {g}_{1}^{-1}\phantom{\rule{thinmathspace}{0ex}},$

where ${g}_{1}$ (called the source) is the label of the edge running from one of these two vertices to the other around the surface labeled by $h$ one way and ${g}_{2}$ (called the target) the label of the edge running the other way.

(For simplicity and without lack of generality I will asumme that $G=H$ and that $g$ acts on $H$ by the adjoint action.)

Recalling the lattice definition of ${F}_{A}$ it is clear that this relation is nothing but the lattice version of $B=-{F}_{A}$.

For computing the surface holonomy one picks any two vertices of the graph, picks any edge running between them and then ‘walks over the surfaces’ starting at this edge and again ending at this edge, while using the two elementary rules of 2-group multiplication to multiply up the contributions from the various faces:

1)

Horizontal multiplication of two surfaces $\left({g}_{1},h,{g}_{2}\right)$ and $\left({g}_{1}^{\prime },{h}^{\prime },{g}_{2}^{\prime }\right)$ which share a common vertex produces the total surface with label

(4)$\left({g}_{1}{g}_{1}^{\prime },h\left({g}_{1}{h}^{\prime }{g}_{1}^{-1}\right),{g}_{2}{g}_{2}^{\prime }\right)\phantom{\rule{thinmathspace}{0ex}}.$

Essentially everything is multiplid in the naively obvious way, with the only difference that ${h}^{\prime }$ is adjoined by ${g}_{1}$ before multiplication with $h$. This can be understood simply as a parallel transport of ${h}^{\prime }$ back to the source vertex of $h$, where the two may be compared. One already sees that this precisely the same mechanism that is at work in the definition of the loop space connection $𝒜$ above.

2)

Vertical multiplication of two surfaces with labeling as above which share a common edge gives

(5)$\left({g}_{1},{h}^{\prime }h,{g}_{2}^{\prime }\right)\phantom{\rule{thinmathspace}{0ex}}.$

So the intermediate edge contribution cancels and the surface holonomy is simply multiplied.

Now let us see how this can be recovered from loop space holonomies:

II) relation to loop space computations

The connection $𝒜$ above involves the integral of the 2-form $B$ around the full loop.

This suggests to consider a long chain of horizontal composition of small surfaces $\left({g}_{1}\left(\sigma \right),h\left(\sigma \right),{g}_{2}\left(\sigma \right)\right)$ with $\sigma =nϵ$ an integer multiple of some small parameter which we’ll send to 0 in the continuum limit.

The horizontal product of all these surfaces gives, by rule 1) above

(6)$h\left(0\right)\cdot \left({g}_{1}\left(0\right)h\left(ϵ\right){g}_{1}^{-1}\left(0\right)\right)\cdot \left({g}_{1}\left(0\right){g}_{1}\left(ϵ\right)h\left(2ϵ\right){g}_{1}^{-1}\left(ϵ\right){g}_{1}^{-1}\left(0\right)\right)\cdots$

Due to the smallness of these surfaces we expand as usual

(7)${g}_{1}\left(\sigma \right)\approx 1+ϵ{A}_{\sigma }\left(\sigma \right)$
(8)$h\left(\sigma \right)=1\approx {ϵ}^{2}{B}_{\sigma \tau }\phantom{\rule{thinmathspace}{0ex}}.$

Here $\sigma$ is supposed to be an index roughly parallel to that chain of surfaces, while $\tau$ is orthogonal to it (but still lying in the surface, of course).

When this is inserted into the above expression for the long horizontal product one obtains

(9)$\cdots \propto 1+ϵ\int d\sigma \phantom{\rule{thinmathspace}{0ex}}{W}_{A}\left(\sigma \right){B}_{\sigma \tau }{W}_{A}^{-1}\left(\sigma \right)+𝒪\left({ϵ}^{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$

This is precisely the expression appearing in the loop space connection $𝒜$ above.

Not too surprising, but maybe interesting, the integral over $\sigma$ which enters the above definition of the loop space connection is hence nothing but the computation of the first order term in a continuum horizontal 2-group product.

More precisely, the result is the surface label of the full $ϵ$-thin but macroscopically long strip formed by all the small surfaces that were horizontally multiplied.

Now comes, finally, the crucial point: When we compute the holonomy of the loop space connection we actually compute

(10)$\left(1+ϵ\int d\sigma \phantom{\rule{thinmathspace}{0ex}}{W}_{A}\left(\sigma \right){B}_{\sigma \tau }{W}_{A}^{-1}\left(\sigma \right)\right)\left(\tau =0\right)\left(1+ϵ\int d\sigma \phantom{\rule{thinmathspace}{0ex}}{W}_{A}\left(\sigma \right){B}_{\sigma \tau }{W}_{A}^{-1}\left(\sigma \right)\right)\left(\tau =ϵ\right)\cdots \phantom{\rule{thinmathspace}{0ex}}.$

But this is now seen to be nothing but the vertical 2-group product of all these strips at different $\tau$. Not too surprising either - but the point is that according to the above mentioned rules of 2-groups this is only defined when these strips completely share one total edge, including the vertex where the strip begins and ends.

Anyone wondering what I really mean should please have a look at equation (2.13) in hep-th/0309173, which makes it quite clear.

The conclusion is that the computation of the holonomy of a loop space connection of the form given at the very beginning of this text corresponds to a 2-group surface holonomy computation if and only if the loops that are integrated over all share a common point, namely that common vertex of the lattice approximation!

Phrased differently, when $𝒜$ is integrated over a curve in loop space which contains loops that do not share a common point, then the result cannot correspond to anything computed using 2-group theory. But since the rules of 2-groups are precisely those which guarantee a consistent surface holonomy, it is no wonder that open curves in loop space, which cannot satisfy this condition, do not yield a consistent surface holonomy.

Discussion:

It seems to me that this is telling us that when computing surface holonomy using loop space technology we must for consistency reasons restrict to loops in loop space whose elements are loops that all share a common point. Otherwise the loop space holonomy is not computing anything that can reasonably be interpreted as surface holonomy in target space.

Of course we knew this before, using the very simple argument involving gauge transformations on loop space which I mentioned before, but I find it illuminating to see in ‘microscopic’ detail how this hangs together with local 2-group computations.

On the other hand, in the string theory context and the boundary state formalism which I originally derived the ideas about nonabelian loop space connections in, this raises further quetsions:

Everything seems fine as long as one is considering tree-level amplitudes for open strings. Here the boundary state formalism is telling us to compute sphere diagrams, and everything mentioned above worked nicely for spheres.

But it does not work nicely for higher genus surfaces. There we do not even have foliations into loops which all share a common basepoint.

Probably to resolve this one has to decompose the surface into patches which each seperately allow foliation with common basepoint. But the details of this are not clear to me at all at the moment.

## August 16, 2004

### Escape from the Fog

#### Posted by Urs Schreiber

We went by train to Andenes/Norway, the most norther spot on the Vesteralen island group where the European continental plate drops off sharply enough that sperm whales, which are the incarnation of the idea of deep sea diving, can be spotted within a ship-hour distance from the shore.

With due stops in Berlin and Stockholm such a trip takes about 55 hours but can be spent leisurely in the armchair of a lounge waggon with a glass of wine and the endless swedish landscape passing by.

Without really noticing how it happend I thus found myself thinking about super-Pohlmeyer invariants and boundary states for non-abelian gauge fields. There was something I still didn’t quite understand. But some sort of fog prevented me from seeing clearer.

But after crossing the norwegian border I was called back to reality as the woods gave way to a dramatic Fjord landscape, dropping off almost vertically right next to the rails.

As then the bus took us through Slartibartfast’s award-winning design from Narvik over the Lofoten to Andenes I began finally to see the fog, creeping in like a living being.

It covered Andenes like a blanket and when I fell to sleep in a midsummer-lit grey soup in our cabin it truly felt like the end of the world.

After an eventful next day the bright midsummer night inspired us to rent bikes and try to get out of that blanket. What an experience. Behind a tiny tunnel through the rocks 5 km south of Andenes neir Bleik, the fog ended as if cut through and a meditarranean beach lay in front of us, shone on by a low-hanging sun.

And - believe it or not - the fog in my mind lifted, too, and suddenly I was seeing clearly:

When the transversal components of the gauge field $A$ mutually commute, the super-Pohlmeyer invariant constructed from that $A$ becomes equal to the boundary deformation operator considered in hep-th/0407122. Furthermore, the condition for the ‘analytic’ extension of the super-Pohlmeyer invariant restricted to the sub-phase-space where $R$ is invertible is invariant if the longitudinal excitations are in light cone gauge.

(If anyone wants to know what this gibberish means, have a look at these notes.)

When we awoke next day the fog over Andenes was gone and we had a great couple of days watching whales, reading Moby Dick, learning about whales in the local whale museum, having apply pie and coffee at midnight in the everlasting afternoon, travelling on the Hurtigrute, almost swimming in the sea - and playing the wind harp.