The String Coffee Table

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):

two Dirac equations at every point

and finally the constraint which enforces reparameterization invariance:

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 $p$

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

for the reduced generators

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 $\Omega (ℳ{)}_{x_0}$ 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

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

on $\Omega (ℳ)$ 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

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

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 $P^2(\sigma )=0$. 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 ${\Omega }_{x_0}(ℳ)$. (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 ${\Omega }_{x_0}(ℳ)$ we can hence rewrite the Schild-Gamboa-Lindström constraints on states $\mid \psi ⟩$ of the tensionless string regarded as differential forms on ${\Omega }_{x_0}(ℳ)$ as

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

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

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 ${\Omega }_{x_0}(ℳ)$, 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

defined by

and

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

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

and the Dolbeault operators $\partial$ and $\overline{\partial }$ are nilpotent by themselves and mutually anticommute. Note that both $\tilde{d}$ as well as $\tilde{A}$ 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

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)

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:

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 $\mid \psi ⟩=\int \mathrm{DA}\exp (-\int F_A^2)W_A$. Then the expression $\{d,d^{†}\}\mid \psi ⟩$ 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