## October 14, 2004

### Tensionless strings and Polyakov’s loops

#### Posted by Urs Schreiber

I am trying to learn about tensionless strings. Here are some facts, observations, thoughts and questions:

There is a long tradition going back to a paper by Schild

A. Schild: Classical null strings (1976)

of dealing with tensionless strings by writing down an action for null surfaces, quantizing that and extracting constraints. Quantization results for the tensionless ‘super’-string are for instance reported by Gamboa et al. in

J. Gamboa & C. Ramirez & M. Ruiz-Altaba: Null spinning strings (1990)

for the NSR version and in

J. Barcelos-Neto & M. Ruiz-Altaba: Superstrtings with zero tension (1989)

for the GS version. There is a long series of papers by Lindström and collaborators expanding on these ideas. The most recent one with lots of references is

A. Bredthauer & U. Lindström & J. Persson & L. Wulff: Type IIB tensionsless superstrings in a pp-wave background (2004).

In all these papers it is found that the constraints of the tensionless string are those of a 1-dimensional continuum of massless (N=2) particles uncoupled except for the reparameterization constraint:

There is the mass shell constraint at every point of the string (I display the equations for flat space):

(1)${P}^{2}\left(\sigma \right)=0$

two Dirac equations at every point

(2)${\psi }_{\alpha }\cdot P\left(\sigma \right)=0$

and finally the constraint which enforces reparameterization invariance:

(3)$X\prime \cdot P\left(\sigma \right)+\frac{i}{2}{\psi }_{\alpha }{\psi }_{\alpha }^{\prime }\left(\sigma \right)=0\phantom{\rule{thinmathspace}{0ex}}.$

Lindström goes into great detail in discussing the properties of the actions that these constraints derive from, but it seems noteworthy that when the constraints of the ordinary tensionful string are expressed in terms of canonical momenta and coordinates and then the terms multiplied by the tension $T$ are deleted, one obtains precisely the above constraints. So this seems to be very natural and plausible.

Many people have thought about representations of these constraints on some Hilbert space and on the physical states implied by them, but it seems to me that no really coherent picture has emerged yet. Gamboa discusses how different ordering prescriptions lead to different critical dimensions (namely the ordinary one or none at all) while Lindström et al. in

J. Isberg & U. Lindström & B. Sundborg: Space-time symmetries of quantized tensionless strings (1992)

discuss further problems and conclude that

Though we do not have an explicit construction of the Hilbert space we believe that non-trivial solutions should exist.

I am not sure at the moment what the status of this question is. But I’ll further comment on it below.

Before doing so, however, it is worth mentioning that there is a different approach to the tensionless limit of string theory, discussed in

A. Sagnotti & M. Tsulaia: On higher spins and the tensionless limit of string theory (2004)

which does not seem to be equivalent to the Schild-Gamboa-Lindström approach.

The idea here is that once the ordinary tensionfull string is quantized in terms of worldsheet oscillators ${\alpha }_{m}^{\mu }$, taken to be independent of the tension the string tension crucially appears only in the relation between ${\alpha }_{0}$ and the center-of-mass momentum $p$

(4)${\alpha }_{0}=\sqrt{2{\alpha }^{\prime }}p\phantom{\rule{thinmathspace}{0ex}}.$

Sending $T=1/2\pi {\alpha }^{\prime }\to 0$ and assuming that ${\alpha }_{m\ne 0}$ and $p$ are ‘of the same order’ hence scales up ${\alpha }_{0}$ and leads to a contraction of the Virasoro algebra which becomes

(5)$\left[{\ell }_{0},{\ell }_{m}\right]=0$
(6)$\left[{\ell }_{m},{\ell }_{n}\right]=m{\delta }_{m+n,0}{\ell }_{0}$

for the reduced generators

(7)${\ell }_{0}={p}^{2}$
(8)${\ell }_{m}=p\cdot {\alpha }_{m}\phantom{\rule{thinmathspace}{0ex}}.$

Note that these generators are not completely unrelated but still crucially different from the (bososnic part of) the above mentioned constraints used in the Schild-Gamboa-Lindström approach. At least I currently don’t see any isomorphism between these two at least superficially different approaches.

The nice thing about the tensionless limit used by Sagnotti&Tsulaia is that there is a well defined Hilbert space, BRST operator and space of physical states and that this contains triplets of massless higher spin fields of exactly the form as expected from general considerations first given in

A. Bengtsson, Phys. Lett. B 182 (1986) 321 .

I would hence like to know how the two appraoches to tensionless strings are related, and how we can decide which one is the correct one to describe a given physical situation. For instance, which one would be expected to describe the tensionless strings propagating on 5-branes?

In this context, I would like to share the following observation concerning a relation between the Schild-Gamboa-Lindström constraints to the loop space formulation of Yang-Mills theory as developed by Polyakov, Migdal and Makeenko:

Let $\Omega \left(ℳ{\right)}_{{x}_{0}}$ be space space of loops based at ${x}_{0}$ in the manifold $ℳ$, which I assume to be flat target space for the moment. There is at least formally an exterior derivative $d$ on this space of the form

(9)$d={\int }_{\gamma }{\mathrm{dX}}^{\mu }\left(\sigma \right)\wedge \phantom{\rule{thinmathspace}{0ex}}\frac{\delta }{\delta {X}^{\mu }\left(\sigma \right)}\phantom{\rule{thinmathspace}{0ex}}d\sigma \phantom{\rule{thinmathspace}{0ex}}.$

There is a nice way to make this object regular and rigorously defined, which follows the approach by Chen as described in

S. Rajeev: Yang-Mills theory on loop space (2004)

but is a little different: Given a family of ${p}_{i}+1$ forms ${\omega }_{i}$, $i=1,\dots ,n$ on $ℳ$ one can define the multi pull-back form

(10)$\Omega =\oint \left({\omega }_{1},\dots ,{\omega }_{n}\right)$

on $\Omega \left(ℳ\right)$ defined as the loop-ordered integral of the pull-back of these forms to the given loop $\gamma$. It turns out that acting with the somewhat formal $d$ on such multi pull-back forms produces something which is still a multi pull-back form and is in particular given by the nice formula

(11)$d\oint \left({\omega }_{1},\dots ,{\omega }_{n}\right)=\sum _{k}\left(-1{\right)}^{\sum _{i

As far as I know this equation was first given in

Getzler & Jones & Petrack: Differential forms on loop spaces and the cyclic bar complex, Topology 30,3 (1991) 339 .

The similar equation used by Chen is obtained by restricting the summation to the lowest value of $k$. Nevertheless, Chen’s concept of making loop space differential geometry rigorous by taking such an action of $d$ on formal power series of multi pull-back forms also applies here.

In a similar fashion, one can give an action of the exterior coderivative on loop space, formally of the form

(12)${d}^{†}=-{\int }_{\gamma }\iota \left({\mathrm{dX}}^{\mu }\left(\sigma \right)\right)\phantom{\rule{thinmathspace}{0ex}}\frac{\partial }{\partial {X}^{\mu }}\left(\sigma \right)$

on multi pull-back forms. One needs to apply some sort of regularization, though. I don’t know yet the best way to do this, but one first interesting step should be to follow Polyakov

A. Polyakov: Gauge Fields and strings (1987)

recently recalled in

A. Polyakov: Confinement and Liberation (2004)

and just pick out the regularized divergent part coming from two coincient functional derivatives. Doing so one can obtain a well-defined action of ${d}^{†}$ on multi pull-back forms on loop space. The details of this can be found discussed in my notes on Nonabelian surface holonomy from path space and 2-groups.

Of course the regularization involved here is precisely that needed to make sense of the Gamboa-Lindström constraint ${P}^{2}\left(\sigma \right)=0$. Even better, when taking polar combinations of the Dirac constraints of Gamboa’s tensionless strings one finds that these are formally just $d$ and ${d}^{†}$ on ${\Omega }_{{x}_{0}}\left(ℳ\right)$. (The fact that we are dealing with based loop space should be completely inessential for the tensionless string, since the tensionless string may develop ‘spikes’ similar to what the membrane does.)

If we write ${ℒ}_{\xi }$ for the modes of the generator of reparameterizations of $p$-forms on ${\Omega }_{{x}_{0}}\left(ℳ\right)$ we can hence rewrite the Schild-Gamboa-Lindström constraints on states $\mid \psi ⟩$ of the tensionless string regarded as differential forms on ${\Omega }_{{x}_{0}}\left(ℳ\right)$ as

(13)$d\mid \psi ⟩=0$
(14)${d}^{†}\mid \psi ⟩=0$
(15)${ℒ}_{\xi }\mid \psi ⟩=0\phantom{\rule{thinmathspace}{0ex}}.$

This are the two Dirac and the reparameterization constraint. The mass shell constraint is implied automatically

(16)$⇒\left\{d,{d}^{†}\right\}\mid \psi ⟩=0\phantom{\rule{thinmathspace}{0ex}}.$

Let me again emphasize that this ‘quantization’ of the tensionless string assumes a regularization of the constraints that may require further analysis and may well turn out to involve corrections to the above definition of ${d}^{†}$. For the time being I suggest to regard these equations as tensionless-string-inspired and see what results.

Namely what results is this: Of course we can solve all the constraints at once. The reparameterization constraint is trivially solved by restricting attention to rep invariant functionals of loops. We should be able to generate all of these in terms of Wilson loops along the string. So pick any 1-form $A$ on $ℳ$ and consider the loop space 0-form

(17)${W}_{A}\left[\gamma \right]=\sum _{n}^{\infty }\oint \left({\mathrm{iA}}^{{a}_{1}},\dots ,{\mathrm{iA}}^{{a}_{n}}\right){T}_{{a}_{1}}\cdots {T}_{{a}_{n}}\phantom{\rule{thinmathspace}{0ex}},$

which are just the Wilson loops of a given gauge connection along the string. These should span at least most of the space of rep invariant functions on ${\Omega }_{{x}_{0}}\left(ℳ\right)$, I assume. (I can almost prove it :-).

There is a simple way to now solve the $d\mid \psi ⟩$ constraint: It can be checked that with the above definition $d$ is nilpotent, as it should be. In fact, something interesting is going on: $d$ naturally splits into two mutually anticommuting nilpotent parts

(18)$d=\stackrel{˜}{d}+\stackrel{˜}{A}$

defined by

(19)$\stackrel{˜}{d}\oint \left({\omega }_{1},\dots ,{\omega }_{n}\right)=\sum _{k}\left(-1{\right)}^{\sum _{i

and

(20)$\stackrel{˜}{A}\oint \left({\omega }_{1},\dots ,{\omega }_{n}\right)=\sum _{k}\left(-1{\right)}^{\sum _{i

One can check (see my above mentioned notes for the proof) that

(21)${\stackrel{˜}{d}}^{2}=0={\stackrel{˜}{A}}^{2}$
(22)$\left\{\stackrel{˜}{d},\stackrel{˜}{A}\right\}=0\phantom{\rule{thinmathspace}{0ex}}.$

This looks interesting, because it indicates some ‘holomorphic’-like structure of $\Omega \left(ℳ\right)$ since a similar proliferation of independent supercharges usually happens for Kähler configuration spaces where

(23)$d=\partial +\overline{\partial }$

and the Dolbeault operators $\partial$ and $\overline{\partial }$ are nilpotent by themselves and mutually anticommute. Note that both $\stackrel{˜}{d}$ as well as $\stackrel{˜}{A}$ annihilate the constant 0-form on loop space, so that they share all the formal properties of $d$.

Anyway, obviously the $d$ constraint can be solved by closing the above Wilson line to obtain the loop space 1-form

(24)$d{W}_{A}=i\sum _{n,m=0}^{\infty }\oint \left({\mathrm{iA}}^{{a}_{1}},\dots ,{\mathrm{iA}}^{{a}_{n}},{F}_{A}^{a},{\mathrm{iA}}^{{a}_{n+1}}\dots ,{\mathrm{iA}}^{{a}_{n+m}}\right)\phantom{\rule{thinmathspace}{0ex}}.$

But this breaks the non-zero modes of the $ℒ\mathrm{constraints}$. So this cannot be the whole story for the tensionless ‘super’-string. There will need to be fermionic correction terms. However, for the bosonic tensionless string according to Schild, Gamboa and Lindström we have to solve (according to my above regularization presciption at least)

(25)$\left\{d,{d}^{†}\right\}{W}_{A}\left[\gamma \right]={d}^{†}d{W}_{A}\left[\gamma \right]=0\phantom{\rule{thinmathspace}{0ex}}.$

So we need to act with the above defined ${d}^{†}$ on $d{W}_{A}$. The result reproduces Polyakov’s old insight that the vanishing of the Laplacian of the Wilson line involves the Yang-Mills equations:

(26)${d}^{†}d{W}_{A}=i\sum _{n,m=0}^{\infty }\oint \left({\underset{⏟}{\mathrm{iA},\dots ,\mathrm{iA}}}_{n},\nabla \cdot {F}_{A},{\underset{⏟}{\mathrm{iA},\dots ,\mathrm{iA}}}_{m}\right)\phantom{\rule{thinmathspace}{0ex}}.$

It follows that according to the above regularization scheme physical states of the bosonic tensionless strings are given by Wilson lines along the string with respect to gauge connections that satisfy the classical YM equations of motion.

I don’t know if this is of any relevance, but it seems to look interesting. In particular this very nicely fits into the formulation of tensionless strings in non-abelian 2-form backgrounds in terms of nonabelian principal bundles on loop space, as discussed in my notes mentioned above.

Instead of states of the form $\mid \psi ⟩={W}_{A}$ one could also consider taking superpositions of these weighted by the usual Yang-Mills action, i.e. use $\mid \psi ⟩=\int \mathrm{DA}\mathrm{exp}\left(-\int {F}_{A}^{2}\right){W}_{A}$. Then the expression $\left\{d,{d}^{†}\right\}\mid \psi ⟩$ involves the Migdal-Makeenko equation, recently discussed with methods similar to those used here in

A. Agarwal & S. Rajeev: A cohomological interpretation of the Migdal-Makeenko equations (2002) .

I wanted to say a couple more words about some details, but I have to hurry to get my lunch!

Posted at October 14, 2004 12:54 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/453

### Re: Tensionless strings and Polyakov’s loops

>> I would hence like to know how the two appraoches to tensionless strings are related, and how we can decide which one is the correct one to describe a given physical situation. For instance, which one would be expected to describe the tensionless strings propagating on 5-branes?

As already pointed out by Jacques, those are strongly coupled and so presumably any attempt to describe them in terms of a world-sheet CFT would fail.

Posted by: WL on October 14, 2004 4:05 PM | Permalink | Reply to this

### Re: Tensionless strings and Polyakov’s loops

Ganor discussed tensionless strings in 6D which have CFTs on their worldsheets, hep-th/9602120, hep-th/9605201. These are supposed not to be the ordinary self-dual strings coming from membranes stretching between two 5-branes, but to come from ‘zero size instantons’ which translate into membranes stretched between the 5-brane and a ‘9-brane’ ${ℤ}_{2}$ fixedpoint. These seem to be the type of strings that I would like to think about in the context of non-abelian 2-forms, since, as Ganor discusses, for $k$ such instantons, these strings transform in the fundamental of $\mathrm{SO}$, which should be what the non-abelian 2-form is acting on, if I understand correctly.

Concerning the self-dual case I do understand that strong coupling prevents a sensible summing up of would-be worldsheet diagrams. I see why you say that this suggests that no worldsheet theory can even be written down.

On the other hand, somehow this seems weird to me. Maybe you can help me clarify my confusion:

The tensionless string on the 5-brane is the boundary of a membrane stretching between two 5-branes. This membrane does have a worldvolume action. Shouldn’t the boundary string inherit a ‘boundary action’ from that?

Hm, actually since the $\left(2,0\right)$ theories arise as the low energy limit of LSTs which again can be defined in terms of M(atrix) theory on ${T}^{5}$ it would seem that there might be a ‘matrix string’ description of the self-dual string? By the usual relation of Matrix Theory to the membrane Lagrangian this might point to a way to connect the self-dual string on the 5-brane to a Lagrangian description. (?)

Posted by: Urs Schreiber on October 14, 2004 7:30 PM | Permalink | PGP Sig | Reply to this

### Re: Tensionless strings and Polyakov’s loops

These are supposed not to be the ordinary self-dual strings coming from membranes stretching between two 5-branes, but to come from ‘zero size instantons’ which translate into membranes stretched between the 5-brane and a ‘9-brane’ ${ℤ}_{2}$ fixedpoint.

Those are not fundamentally different from the 5-5 strings. In both cases, they couple to a 2-form with self-dual field strength. In the latter case, this is part of a (2,0) tensor multiplet; in the former, it is part of a (1,0) tensor multiplet.

In the (2,0) case, there are 5 real scalars in the tensor multiplet and, away from the origin in the moduli space, we have strings whose tension is proportional to the distance from the origin. In the (1,0) case, there is a single real scalar in the tensor multiplet whose VEV, again, controls the tension of the string.

At the origin of the moduli space, we have a 6-D CFT, not some exotic string theory.

In both cases, these theories can be embedded in little string theories. Those have another mass-scale, the tension of the “fundamental” little string. At energy scales well below that scale, the little string theory reduces to a field theory (which may, indeed, have “heavy” string-like excitations). At a generic point in the moduli space of the field theory, the physics is infrared-free. But at the origin, one has a nontrivial interacting CFT.

Posted by: Jacques Distler on October 15, 2004 3:05 AM | Permalink | PGP Sig | Reply to this

### Re: Tensionless strings and Polyakov’s loops

I’ve had a look at your and Hanany’s hep-th/9611104. From this and from what you say above I think I can understand why every tensionless string in an $N=\left(2,0\right)$ or $N=\left(1,0\right)$ theory in 6 dimensions necessarily has to be self-dual and hence strongly coupled. Let’s see:

You discuss how for $N=\left(2,0\right)$ there are 21 anti-self-dual 2-forms and 5 self-dual ones coming from the NS-NS and the R-R 2-forms and the periods of the self-dual R-R 4-form on 2-cycles of the K3. The charges of any string under these 2-forms have to lie on a Lorentzian lattice where the 21 anti-self-dual 2-forms give the negative-signature dimensions and the 5 self-dual 2-forms the positive signature dimensions of the lattice.

So any charge vector in this lattice looks like

(1)$q=\left({q}_{-},{q}_{+}\right)$

with

(2)${q}^{2}=\mid {q}_{+}{\mid }^{2}-\mid {q}_{-}{\mid }^{2}\phantom{\rule{thinmathspace}{0ex}}.$

The string is BPS-saturated if

(3)${q}^{2}\ge -2$

and its tension $T$ is

(4)${T}^{2}=\frac{1}{8{\pi }^{2}}\mid {q}_{+}{\mid }^{2}\left({M}_{\mathrm{pl}}^{\left(6\right)}{\right)}^{4}\phantom{\rule{thinmathspace}{0ex}}.$

- From this it follows that the string is tensionless iff ${q}_{+}=0$, which then implies that it couples exclusively to one chirality of 2-forms, namely the anti-self-dual ones.

If something similar is true for the $N=\left(1,0\right)$ case it explains why all these tensionless strings are strongly coupled.

How comes Ganor talks about CFTs on these $\left(1,0\right)$ strings?

Posted by: Urs Schreiber on October 15, 2004 3:25 PM | Permalink | PGP Sig | Reply to this

### Re: Tensionless strings and Polyakov’s loops

If something similar is true for the $N=\left(1,0\right)$ case it explains why all these tensionless strings are strongly coupled.

There’s no “if” involved. The 5-9 strings couple to a (1,0) tensor multiplet. The single real scalar in the tensor multiplet is the position of the 5-brane along the M-theory interval (in the Horava-Witten realization of the ${E}_{8}×{E}_{8}$ heterotic string).

The spectrum of 2-forms in (1,0) supergravity has one anti-self-dual 2-form in the gravity multiplet and $\left(n+1\right)$ self-dual 2-forms if there are $n$ 5-branes in the bulk of the interval (there’s always at least one tensor multiplet, whose scalar component is the length of the interval between the 9-branes).

The fundamental heterotic (9-9) string couples to an unconstrained 2-form (the sum of a self-dual and the anti-self-dual 2-form). The rest of the strings (5-9 and 5-5 strings) couple to self-dual 2-forms.

Posted by: Jacques Distler on October 15, 2004 6:37 PM | Permalink | PGP Sig | Reply to this

### Re: Tensionless strings and Polyakov’s loops

One more comment:

I have just found (what you are perhaps well aware of) that in

P. Argyres & K. Dienes: On the worldsheet formulation of the six-dimensional self-dual string (1996)

a (admittedly speculative and untested) proposal for a CFT description of the self-dual string is made. In the introduction some discussion is given why this could be meaningful. In particular an argument by Dijkgraaf, Verlinde & Verlinde is recalled (hep-th/9603126, hep-th/9604055) according to which a weakly-coupled description of this string can be justified. Also the existence of classical GS strings in six dimensions is mentioned as evidence for such a worldsheet description.

I don’t know, but I thought I might mention it.

Posted by: Urs Schreiber on October 14, 2004 8:27 PM | Permalink | PGP Sig | Reply to this

### Re: Tensionless strings and Polyakov’s loops

Welcome back to physics. I was beginning to get worried about you ;)

Posted by: Eric on October 17, 2004 9:46 PM | Permalink | Reply to this

Post a New Comment