The String Coffee Table

To rebut fears that having my own blog would let me forget about the coffee table i would like to share some thoughts that I would like to here your comments on. The theme is effective theories and how they fit together with gravity.

As long as we are not entirely sure about the nature of the final theory (I know, string theory is the final theory but let us forget about this for a second) we have to keep an open mind about the fact that all theories that we deal with are only effective theories and that they are obtained by some coarse graining procedure from more fundamental theory. Then the usual renormalization group reasoning tells us that our low energy measurements should be independant of irrelevant deformations of this procedure. Especially, as long as our low energy observational scale is well below the coarse graining scale, low energy physics should be invariant under moving that scale by means of integrating out or in high energy degrees of freedom.

To be a bit more specific about the kinds of models I have in mind let me tell you about a U Chicago exam problem that Gavin Polhemus (the coauthor on my first string theory paper) once told me about:

Find the energies of the lowest energy modes of a particle in a 2d box that has the shape of a highly excentric ellipse!

Say, the particle is confined to $x^{2} + y^{2}/e^{2} \lt 1$ for small $e\ll 1$. Then the idea is to realize that the bounces in the $y$ direction are much faster than those in the $x$ direction and are thus at a higher energy scale and can be integrated out.

By that I mean we use the Born Oppenheimer approximation and treat $x$ as a classical constant variable while solving the Schr"odinger equation for $y$. This is the ordinary particle in the box of size $2e\sqrt{1-x^{2}}$ and this has a zero point energy of

The trick is now to interpret $E_0(x)$ as an effective potential for a one dimensional problem envolving only $x$.

Note one feature of this integrating out procedure: In the original 2d problem, no matter where in the ellipse the particle is, there is no potential energy, all there is is kinetic energy. However, in the effective 1d theory, suddenly what used to be the kinetic energy of the motion in the $y$-direction is now interpreted as potential energy of $x$. This in fact is quite generic of effective theories: Effective potentials often come from kinetic energy in the microscopic theory.

Now you probably wonder why I care, in the end it should not matter what type of energy we have. Well, this is only true as long as we are not dealing with gravity. Gravity appearantly can diffentiate between kinetic and potential energy. One way to see this is to consider the typical cosmology setting of a FRW metric coupled to a scalar field. This problem is again one dimensional as everything (the scale factor and the scalar) only depend on time.

Usually, the Friedmann equations are expressed in terms of the energy density and pressure of a perfect fluid:

and we have to specify $\rho$ and $P$ in terms of the properties of the scalar field. It turns out that $\rho = T + V$ and $P=T-V$ where $T$ is the kinetic and $V$ is the potential energy of the scalar field. Thus $P$ and therefore the evolution of $a$ depend on how the energy is distributed between kinetic and potential energy.

Now imagine that we coupled the above QM system to gravity: We consider two scalar fields $X$ and $Y$ that are free except that they are confined to $X^2 + Y^2/e^2\lt 1$. Then I would expect one evolution corresponding to pure kinetic energy. Alternatively I would integrate out $Y$ and obtain a different cosmology with partly kinetic and partly potential energy. In terms of an equation of state this the first one would be $w=P/\rho = 1$ whereas the second would be roughly $w=0$.

Please find the flaw in this argument, otherwise I would conclude that dealing with effective theories is not save in a gravitational setting!!!

There are two related issues: First of all, strictly speaking, the effective potential is only defined up to an additive constant. But we know we are not supposed to fiddle with that as that would be a cosmological constant and gravity definitely depends on that.

The other issue is less related but still makes me a bit uneasy: There are all kinds of gravitational entropy bounds on the marked that all roughly say that the entropy of an object is bounded by $S=A/4$, where $A$ is the horizon area the black hole you could turn this object into. The reason I am uneasy with these bounds is that in thermodynamics we usually deal only with differences of entropy and only in a quantum treatment you adopt the convention of setting the entropy of the ground state to 0. This corresponds to a normalization of the partition function and of course only holds if the ground state is non-degenerate.

However, what you call the ground state depends on you level of course graining! For example, in the past I worked on protein folding. There the typical question is, given some sequence of amino acids, what is the ground state? We often joked that the ground state of a protein is some $H_2O$ and $CO_2$, but that was wrong: You can still gain some energy by nuclear processes that turn the protein into a lump of iron.

Obviously, what this joke plays with is that bio-physicists use a different coarse graining thant chemists (they assume molecular bounds are inert) that use a different coarse graining than nuclear physicists (as they treat chemical elements as inert). The upshot is that what looks like a ground state in one coarse graining might actually be composed of many micro states in a more microscopic theory. Thus the biophysicist would assign entropy 0 to the protein in the ground state whereas the chemist would assigne a positive entropy to it and the nuclear physicist would assign an even bigger entropy to it. A string theorist might even find more stringy degrees of freedom and would assign an even bigger entroby to it. And hey, maybe also string theory is only an effective theory and there is an even more microscopic theory behind it that would know about even more microstates!

Well, thanks to Beckenstein, we don’t have to worry about this, we can meassure it: We just throw the protein into a black hole and measure how the horizon area increases. This you can translate into an upper bound of the entropy of the protein and this in turn tells you that you cannot infinitely find more fundamental theories with more and more degrees of freedom. Thus you can for example measure if there is room for a more fundamental theory than string theory! Isn’t that great? And all this only envolves low energy physics: You just have to slowly lower a protein into a black hole.

You think I am joking? Well, have a look at papers like this. Here, Andreas derives bounds on the quotient of viscosity and entropy density from holographic arguments. And he even finds that actual fluids (as found in the CRC) are only one order of magnitude away from the bound! But this should make you wonder how the absolute entropy in those tables is defined. And indeed, the tables assume that the chemical composition is kept fixed when the entropy is extrapolated to zero temperature. But we just found that from a nuclear perspective the entropy should be bigger because we have to count nuclear degrees of freedom as well. Hmm.

Please help, I’m worried.

Andrew Neitzke and Luboš Motl recently made me aware of the importance of identifying objects in 2-gauge theory that carry three ‘group indices’.

One expects that a stack of 5-branes is characterized by

- the number $n$ of coincident 5-branes

- a condensate $\{m_{ijk}\}$, $i,j,k \in \{1,2,\dots,n\}$ of BPS membranes with precisely three disconnected boundary components attached to the $i$th, the $j$th and the $k$th 5-brane.

With respect to the 2-gauge theory living on the stack of 5-branes it is clear that the parameter $n$ specifies the gauge group, like in $\mathrm{SU}(n)$.

The big question is: What in the 2-gauge theory is described by $m_{ijk}$?

Intuitively one would expect $m_{ijk}$ to measure the probability for a membrane ending on the $i$the 5-brane to coalesce with another membrane ending on the $j$the 5-brane and turning into a membrane ending on the $k$th 5-brane, roughly. This suggests that $m_{ijk}$ somehow controls the product of Lie algebra elements, maybe.

Aware of that, Andrew Neitzke identified the map $M$ in equation (21) of Christiaan Hofman’s paper hep-th/0207017 as an apparently natural candidate for an incarnation of $m_{ijk}$.

There are three obvious questions:

- Can we check if this makes sense?

- Are there other candidates?

- Are these other candidates possibly just different aspects of the same thing?

In the following I want to briefly sketch some thoughts on these questions:

First of all I would like to make the point that while equation (15) in Christiaan Hofman’s paper is very suggestive and certainly a great idea, it does not really follow (as far as I can see at least) from the considerations that he gives in (1)-(14). For one, from (14) it would follow that there should be terms proportional to $F_A$ in (15). Another difficulty is that the 2-form $B$ is ‘shuffled’ inside the multi-integral, including its $\mathfr{g}$-factor, which is not what happens if $\oint(B)$ is applied from the left on the multi-integral, as one might expect from a connection 1-form.

So if we are to physically interpret the map $M$ we first need a strict derivation of equation (15) from some loop space reasoning. Christiaan’s Hochschild complex considerations certainly show that (15) is a good idea, but how does it really arise?

The key observation is probably the shuffling property of the $B$-term. It implies that this term does not come from any multiplication from the left, but arises from a derivation inside the multi-integral, the same way that the $A$ term arises. This could then also explain why the $F_A$ terms don’t appear: They could cancel against one part of the $B$ term.

Thinking about this for a moment seems to admit only a single solution:

Recall that in the context of superstrings the natural differential operator on loop space is not the loop space exterior derivative $\mathbf{d}$, but the polar combination of the worldsheet supercharges, which reads

where $\iota_K$ is the operator of inner multiplication with the loop space vector $K^{(\mu,\sigma)} = X^{\prime\mu}(\sigma)$ ($X$ is the loop and $X^\prime$ its $\sigma$-derivative). $T$ is the string’s tension.

This operator was first considered in the second half of Witten’s ‘SUSY and Morse theory’ paper and is nowadays familiar from boundary state formalism. A boundary state for some gauge field background is simply an inhomogenous differential form on loop space of the form

Usually the $B$ term is absent here, but as I have tried to argue in hep-th/0407122 including it (which is very natural) immedietaly gives us local nonabelian connections on loop space induced by the nonabelian 2-form $B$ that have well-defined surface holonomy and are equivalent to local 2-connection in the theory of 2-groups.

So this is in a sense the generalization of the ordinary Wilson line for $A$ along the loop/string. The ordinary Wilson line is what Christiaan Hofman convincingly argued to have its place in between the factors of the multi-integrals. Now lets take his approach and the above one together and generalize in the obvious way: This leads us to consider multi-integrals of target space $(p_i+1)$-forms $\omega_i$ of the form

This is very close to what Christiaan Hofman does, it essentially just replaces the ordinary Wilson line by its supersymmetric version. The interesting point is that acting with the modified exterior derivative $\mathbf{d}_K$ which comes from the string’s supercharges on such a multi-form produces precisely the terms that Christiaan Hofman postulates in his equation (15):

where $R$ denotes some additional terms that don’t look like multi-integrals of the above form. These terms drop out however if we scale $T \to \infty$ with keeping $T B$ constant.

So this construction has a nice side and a surprising side: The nice thing is that we can very naturally derive the terms in Hofman’s equation (15), the surprising thing is that we can do so precisely only by taking the unexpected limit of large tension. Maybe this is a sign that the above needs to be improved, maybe it is a sign of some effect. (E.g. it could be that the above applies to membranes that stretch between a stack of 5-branes and some other attachment point, thus inducing large tension on their boundary strings.)

In any case, this demonstrates that it is possible in principle to derive the multi-derivations that the map $M$ which we are interested in is part of from a consistent scheme of loop space differential geometry, even one which has the right physical objects like $\mathbf{d}_K$ and $W(\sigma,\sigma^\prime)$ appearing.

Incidentally, this suggest that the $A$ and $B$ appearing here are not nessecarily the same that would also appear in an honest connection on loop space, which would be represented by a loop space 1-form of the form $\oint_{A^\prime} (B^\prime)$ and give rise to a covariant derivative

the way I have discussed before. So it seems that we really end up with two sets of a (1+2) form. This is actually nice, because the definition of a nonabelian gerbe also involves two such pairs!

But there is still a problem with idenitfying the map $M$ with a version of the $m_{ijk}$: In all of the above formulas the product in $\mathfr{g}$ is really implicit already in the multi-integrals. After all, these integals are valued in the enveloping algebra of $\mathfr{g}$ and not in some tensor product. What $M$ really does is just implementing the wedge product on the scalar coefficients. Otherwise it seems hard to give the above objects a sensible interpretation.

So it seems that if we want to identify the $m_{ijk}$ with anything determining the group product, we must to so for all such products, not just inside the map $M$. That is, we have to somehow generalize the product in $\mathfr{g}$ in general.

There seems to be no room for such a step in the theory of nonabelian gerbes. That’s why Christiaan Hofman fixes the freedom contained in the definition of the product before discussing gerbes in his paper.

But we know that 2-bundles for strict structure 2-groups are equivalent to nonabelian gerbes. On the other hand, 2-bundles are more general than that. In particular, the structure 2-group of a 2-bundle is in general weak and/or coherent (which is essentially the same) instead of strict.

But a coherent 2-group is a 2-group in which, lo and behold, the group product operation is more flexible than ordinarily! In particular, the group product here need not be associative. Instead, there is a natural transformation called the associator which tells you how the group product fails to be associative.

This is precisely the kind of degree of freedom that we are looking for. Now there is a very interesting result for coherent 2-groups:

In

J. Baez & A. Lauda: Higher Dimensional Algebra V: 2-Groups (2004)

it is proved in section 8.3 that every coherent 2-group is specified up to equivalence by the following data:

- a group $G$

- an abelian group $H$ and an action of $G$ on $H$ by automorphisms

- an element $[a]$ of the cohomology group $H^3(G,H)$ .

Incidentally, this $[a]$ specifies the associator.

This looks rather similar to the data mentioned above wich specifies the stack of 5-branes. The group $G$ would by $SU(n)$ or something. Since $H$ is abelian it does not contain a whole lot of information. But then there is the object $[a]$, which, indeed, carries three ‘group indices’.

So it seems that coherent 2-groups might provide just the right kind of degrees of freedom to account for those on a stack of 5-branes, including the $n^3$ scaling.

To check this conjecture in more detail one would have to define the notion of 2-connection in a 2-bundle which has a coherent 2-group as structure group. This has not been done yet. But we are getting closer I think.

My apologies if the above was too speculative for anyone’s taste. I don’t have any hard results here, but I think the above observations are intersting enough to warrant thinking about them if one is interested in the $n^3$-puzzle.

[This is a followup to Luboš’s blog entry Self-dual strings and M5-brane anomalies. Let’ see if it is possible to have an inter-blog discussion.]

When I read Berman’s & Harvey’s hep-th/0408198 a while ago I learned a bit more about how difficult it really is to understand the situation with $Q_5 \gt 1$ coincident 5-branes.

Important for me at that point was the reference to

X. Bekaert & M. Henneaux & A. Sevrin: Chiral forms and their deformation (2000)

which demonstrates that it is impossible to have a local deformation of an abelian theory of self-dual 2-forms to a non-abelian one.

This result is perhaps reminiscent of the insight of

F. Girelli & H. Pfeiffer: Higher gauge theory - differential versus integral formulation (2004)

that the ‘naive’ (depending on what you still consider naive) local extension of the YM Lagrangian to nonabelian 2-forms constituting a local connection for a strict structure 2-group is really equivalent to ordinary gauge theory again and does not yield the expected extension.

The same problem from another point of view is that one can write down the dynamical fields that are expected to describe the coincident 5-branes by deriving data describing a nonabelian gerbe from M5-brane anomaly cancellation as in

P. Aschieri & B. Jurčo: Gerbes, M5-Brane Anomalies and $E_8$ Gauge Theory (2004) ,

but then it is not known (yet) how to construct gauge invariant quantities, holonomies and ‘Lagrangian-like’ objects from that data.

Though there are some first hints, I believe. I think I can show that the nonabelian gerbes (without connection) considered by Aschieri & Jurčo are equivalent to the data describing 2-transitions in 2-bundles (without connection), which were very recently introduced in

T. Bartels: Categorified gauge theory: 2-Bundles (2004) .

That’s good, because the 2-categoric context of 2-bundles should allows us to simply categorify the objects that we are looking for in the ordinary incarnation to get their nonabelian 2-form version.

In particular, it should be possible to have a 2-morphisms from the 2-groupoid of 2-paths in the base space of a 2-bundle to the structure 2-group and thus get a holonomy of non-abelian 2-forms. It remains to be seen if the non-abelian cocycle data introduced that way is still equivalent to Aschieri&Jurč’s nonabelian gerbes with connection, but the success in the case without connection suggests that this must be true.

The good thing is that if we pick the base 2-space of our 2-bundle to be $B$ with point space

being ‘spacetime’ (i.e. the 5-brane worldvolume) and the arrow space

the space of based loops over $\mathcal{M}$ (i.e. the configuration space of closed strings in the 5-brane) then such a 2-connection gives rise to a connection on the loop space $B^2$ (as the entire 2-bundle gives rise to an ordinary bundle over $\Omega$\mathcal{M}) and we know some things about how to get holonomies from such nonabelian path-space connections and how they can in principle be more general than the local strict 2-group holonomies considered by Girelli&Pfeiffer (though it remains to be better understood exactly how they are more general).

In fact, this leads me to the cubic scaling of degrees of freedom that Lubš talked about in his post.

So if I understand correctly the usual asymptotic scaling with $n^2$, where $n$ is the dimension of the Cartan sub-algebra of the gauge group) is simply an indication of the fact that fields in the adjoint rep and in particular the gauge bosons themselves have to make up $n \times n$ matrices and thus appear in bunches of $n^2$.

So if objects in theories on $n$ $Q_5$ branes scale faster than $n^2$ this might indicate that the gauge connection in these theories requires for its specification more than a $\mathrm{Lie}(\mathrm{SU}(n))$-valued differential form. (What Luboš addresses as fields carrying ‘three indices’).

But that’s exactly what is the case in general for the 2-connections and the connections on loop space! Bekaert,Henneaux&Sevrin in their paper mention the famous old result by Teitelboim (Phys. Lett. 167B (1986) 63) that no ‘straightforward’ (as they say) non-abelian extension of the 2-form field exists, which is based on the assumption that the holonomy of a 2-form gauge field $B$ along a string worldsheet $\Sigma$ is

even in the nonabelian case.

But the boundary state deformation considerations that I gave in hep-th/0407122 as well as the relation to the 2-group connections shows that (what Alvarez et. al had, in a special case, considered before and what was also expressed by Hofmann) that this fails because parallel transport of the non-abelian $B$ to a reference fiber has to be included, generalizing the above to

This form of a path space connection evades Teitelboim’s no-go theorem if the 2-form $B$ and the 1-form $A$ satisfy a certain condition, dubbed r-flatness by Alvarez, which can be shown to be equivalent to the exchange law arising in a sesqui-connection (a 2-connection into a sesqui-group).

There is some discussion necessary concerning the uniqueness of this form, but in any case it shows that the gauge fields expected to describe this non-abelian parallel transport of strings involves more than one $\mathrm{Lie}(\mathrm{SU}(n))$-valued differential form and hence more than $n^2$ ‘fields of data, namely (at least) two such forms. This holds locally and indeed the general results on nonabelian gerbes say that there must be even more forms involved.

Now this does not yet prove the $n^3$ scaling, but this is kind of suggestive to be part of the explanation why the scaling is $\gt n^2$.

I was recently trying to learn about nonabelian gerbes. That turns out to be quite hard. The canonical reference is

Breen & Messing: Differential geometry of gerbes (2001)

which I have still not entirely deciphered. A much more digestible presentation is given in

P. Aschieri & L. Cantini & B. Jurco: Nonabelian bundle gerbes, their differential geometry and gauge theory (2003)

which I am able to follow, though it took me two days to work through it.

Anyone wondering what this has to do with strings should have a look at

P. Aschieri & B. Jurco: Gerbes, M5-Brane Anomalies and $E_8$ Gauge Theory (2004) .

John Baez kept telling me that there is an alternative to gerbe language that might have some conceptual advantages: That’s 2-bundles, the categorification of ordinary fiber bundles. Unfortunately, 2-bundles were top-secret - until a couple of days ago. Now they have been revealed to the world in the paper

Toby Bartels: Categorified gauge theory: 2-bundles (2004) .

I very much like this approach for several of reasons. Here is the central idea in a single sentence:

While an ordinary bundle is a map $p: E \to B$ from the total space $E$ to the base space $B$, a 2-bundle is a 2-map $p: E \to B$ from the 2-space $E$ to the 2-space $B$.

That’s it. Now you turn the 2-category-crank and everything else just drops out. By going from groups to 2-groups and from open covers to 2-covers, etc. one gets the definition of locally trivial principal $G$-2-bundles by copying word for word the respective definitions for ordinary principal bundles. And indeed that’s what Toby Bartels does, parts of his section 2 must have been deliberately copy-and-pasted from section 1, with the prefix ‘2-’ inserted here and there.

So could you just write a paper on 3-bundles by copy-and-pasting the above paper and replacing 2s by 3s? Not quite. Categorification adds additional ‘logic’ at every step, expressed in terms of coherence laws which tell you that while all expressions which are the same are equal, some are less equal than others (so to say). Coherence laws tend to manifest themselves in terms of intimidating diagrams of ($n$-)morphisms that may have a striking resemblance to diagrams of compounds in organic chemistry, but actually when you stare at them long enough most of them become quite intelligible.

What is nice is that the precise nature of the coherence laws for 2-bundles is not needed to understand the concept and to write down examples, for instance. Anyone interested in the care and feeding of such laws should have alook for instance at

J. Baez & A. Lauda: Higher Dimensional Algebra V: 2-groups (2003)

and

J. Baez & A. Crans: Higher Dimensional Algebra VI: Lie 2-algebras (2003) .

So does a $G$-2-bundle for non-abelian $G$ give us everything a gerbe would give us?

Almost everything. Apart from aspects which I cannot overlook yet, one crucial difference between 2-bundles and gerbes is that a gerbe more or less automatically comes with some analog of a connection, while the above 2-bundles don’t.

The reason why gerbes and connections go hand in hand is that gerbes are characterized by elements in Deligne hypercohomology, which are objects that include a higher order generalization of transition functions as well as 1-forms and 2-forms that generalize the ordinary 1-form of a connection on a fiber bundle. But this automatism has a drawback, it seems: Nobody knows (at the moment) how, in the non-abelian case (which can be understood using twisted abelian 2-gerbes), you get a holonomy from such 1+2 forms. In the abelian case this is well understood and pretty nifty, but nonabelian gerbes are, truly, harder to understand.

The only known way to nonabelian surface holonomy at the moment seems to be the one that I have talked about a lot here, which involves ordinary locally trivial bundles with connection on the space of paths/loops. It would be nice if this approach could be related to gerbes or 2-bundles somehow.

I have only a very vague idea how the connection between path space bundles and nonabelian gerbes could be obtained. But for 2-bundles it is very easy to see and quite beautiful:

In order to see that one needs to know that a ‘2-space’ $X$ is essentially an ordinary space $X^1$ (consisting of points) together with a space $X^2$ consisting of ‘paths’ or ‘morphisms’ between points (see section 2.1.1 of Toby Bartle’s papers for more details). So for instance spacetime $\mathcal{M}$ together with the space of paths (strings!) $P\mathcal{M}$ in $\mathcal{M}$ gives us a 2-space $B$ with $B^1 = \mathcal{M}$ and $B^2 = P\mathcal{M}$.

Next a 2-map between 2-spaces is essentially two maps, one going between the point-spaces the other going between the morphism-spaces and such that source and target of any path/morphism is respected by the two maps.

A 2-group $G$ is a 2-space which is also a group, roughly, which in particular means that the ‘paths’ in $G^2$ are group elements that have a source and a target group element.

But this means that a, trivial say, 2-bundle

is essentially an ordinary $G^1$-bundle over $\mathcal{M}$

together with a $G^2$-bundle over $P\mathcal{M}$, the space of paths in $\mathcal{M}$:

plus some consistency relations between these maps. Here $p^1$ and $p^2$ are the two parts of the 2-map $p$ which act on the point-parts and the path-parts of the 2-spaces respectively, and the above is just a verbose version of the simple statement that a 2-bundle is a 2-map $p: E \to B$.

This means, unless I am mixed up, that the notion of 2-bundle automatically gives us an ordinary 1-bundle on the space of paths. That’s great, because it is here that nonabelian holonomy is best understood.

Hence the only part of Toby Bartel’s paper that I am not completely happy with is his statement on p.18 that physically interesting 2-bundles have base spaces $B$ which are essentially souped-up ordinary spaces, with only the trivial constant paths included. I believe that instead the physically interesting case is the one I mentioned above, where $X^2$ is $P\mathcal{M}$. That’s where nonabelian strings live.

If 2-bundles really capture the same information as nonabelian gerbes there should be some translation mechanism between the two concepts. This I would like to better understand.

For starters, where in the 2-bundle approach are the transition $\lambda_{ijk}$ functions on quadruple overlaps $U_{ijkl}$? Toby Bartels does mention the space $U^4$ of all quadruple overlaps of a given cover $U$. It appears in the coherence laws in ‘equation’ (84). Is any part of that diagram to be identified with the action of the $\lambda_{ijk}$ or something?

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 $\alpha^\mu_m$, 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 $\alpha_{m\neq 0}$ 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 $\Omega(\mathcal{M})_{x_0}$ be space space of loops based at $x_0$ in the manifold $\mathcal{M}$, which I assume to be flat target space for the moment. There is at least formally an exterior derivative $\mathbf{d}$ 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 $\mathcal{M}$ one can define the multi pull-back form

on $\Omega(\mathcal{M})$ 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 $\mathbf{d}$ 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 $\mathbf{d}$ 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 $\mathbf{d}^\dagger$ 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 $\mathbf{d}$ and $\mathbf{d}^\dagger$ on $\Omega_{x_0}(\mathcal{M})$. (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 $\mathcal{L}_\xi$ for the modes of the generator of reparameterizations of $p$-forms on $\Omega_{x_0}(\mathcal{M})$ we can hence rewrite the Schild-Gamboa-Lindström constraints on states $|\psi\rangle$ of the tensionless string regarded as differential forms on $\Omega_{x_0}(\mathcal{M})$ 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 $\mathbf{d}^\dagger$. 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 $\mathcal{M}$ 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 $\Omega_{x_0}(\mathcal{M})$, I assume. (I can almost prove it :-).

There is a simple way to now solve the $\mathbf{d}|\psi\rangle$ constraint: It can be checked that with the above definition $\mathbf{d}$ is nilpotent, as it should be. In fact, something interesting is going on: $\mathbf{d}$ 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 $\Omega(\mathcal{M})$ since a similar proliferation of independent supercharges usually happens for Kähler configuration spaces where

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

Anyway, obviously the $\mathbf{d}$ 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 $\mathcal{L} constraints$. 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 $\mathbf{d}^\dagger$ on $\mathbf{d}W_A$. 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 $|\psi\rangle = W_A$ one could also consider taking superpositions of these weighted by the usual Yang-Mills action, i.e. use $|\psi\rangle = \int DA \exp(-\int F_A^2) W_A$. Then the expression $\{\mathbf{d},\mathbf{d}^\dagger\}|\psi\rangle$ 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!

In my last post I didn’t give very many details about the talks at the Quantum Theory of Black Holes workshop held at Ohio State. Last week was pretty busy, so I haven’t gotten around (yet) to writing a more informative post. I plan on doing that soon, but right now I’d like to talk about something else. I want to tell a story about Bryce DeWitt, who passed away recently.

Given the audience here I think it is safe to assume that most readers know who Bryce is. It’s probably not a good idea to start an anecdote by saying that I didn’t know Bryce very well, but it’s true. I never took a class from him while I was a grad student at the University of Texas, and I really only spoke to him a few times. The reason I would like to talk about him now is that, on one of those occasions, he gave me a great piece of advice.

The first physics paper paper I wrote was about the AdS/CFT correspondence. Along with Rich Corrado and Bogdan Florea, two other grad students, I calculated the two and three point functions of chiral primary operators in the large N (0,2) theory in six dimensions. Not very much was known about this exotic superconformal field theory, and the AdS/CFT correspondence provided a novel means of obtaining some new information about it. The correlation functions we were studying could be extracted from the on-shell action for fluctuations around the $AdS_{7} \times S^{4}$ solution of 11 dimensional supergravity. What made this calculation difficult (besides a great deal of algebra) was the fact that the on-shell action must be expressed as a functional of the boundary data for bulk fields.

Of course, there is nothing new about writing an on-shell action as a functional of data on the boundary of some region; it is the same thing we do in the Hamilton-Jacobi formalism. But our calculation took place in a gravitational theory, where there are ambiguities as far as what sorts of boundary terms might appear in the action. In other applications people usually (justifiably) discard boundary terms and total derivatives. This is okay because, in a theory with a sensible variational principle, such terms aren’t relevant to the bulk physics. In AdS/CFT, however, those are precisely the terms we want to keep track of.

That brings us to Bryce. We had quite a few questions about boundary terms. We weren’t sure which ones should be included in the action, and we weren’t clear as to how they behaved under the variations that allow us to study fluctuations around the $AdS_{7} \times S^{4}$ background. Bryce seemed like the natural person to ask about these things. At that point, most of what Rich, Bogdan, and I knew about fluctuations around a background geometry came directly from Bryce’s “Dynamical Theory of Groups and Fields” (from the 1963 Les Houches Lectures). Rich and I collected the questions we wanted to ask him, thought about the least foolish sounding ways of asking said questions, and proceeded to his office. We stood in front of Bryce’s blackboard and began to ask him our questions about boundary terms.

Bryce cut us off almost immediately, asking why we would be interested in boundary terms when they couldn’t possibly affect the bulk physics. I think we anticipated that he might ask that, so we began to explain what AdS/CFT was and why we needed to know about boundary terms. I don’t know if he was aware of AdS/CFT or not, but he sounded very skeptical about our questions. For every explanation we gave he seemed to ask two more questions, and it took us about a half hour to convince him that it was legitimate for us to even be interested in boundary terms.

Eventually we got to the point where we could ask Bryce our original question. I don’t actually remember what it was. I think it had to do with the Gibbons-Hawking term that you add to the Einstein-Hilbert action, and how it behaves under variations of the bulk metric. For the purposes of this story, however, it doesn’t actually matter. We finally asked our question, and Bryce sat back in his chair and began to think. Rich and I stood at the blackboard, looking expectant. After a moment or two Bryce leaned forward, looked at the two of us, and said:

“Your problem is too much book learning.”

That wasn’t the answer we were expecting. We might have looked a little stunned, because Bryce was quick to explain what he meant. He didn’t know the answer of the top of his head. We could probably find it in a paper somewhere if we looked hard enough, but we would learn a lot more if we just figured it out ourselves.

I usually tell this story for a laugh, because the thought of Bryce telling Rich and me that our problem is too much “book learning” is pretty funny. But it was a great piece of advice. We ended up figuring it out for ourselves, and we learned a lot more doing it that way than we would have tracking the answer down in some old GR paper. And it couldn’t have come at a better time for me. As a second year grad student I was still stuck pretty firmly in the “homework mentality”, where the answer to every question exists in the back of some book, somewhere. That is not the right mindset for doing research, and Bryce’s “answer” cleared that up for me earlier rather than later.

So thanks, Bryce, for the excellent advice.