The String Coffee Table

The annual string theory conference is taking place at College de France in Paris this year. Fortunately this is a little closer to where I live than Tokyo, so that I don’t have to miss it. Refreshed by a marvelous weekend with my girlfriend at the banks of Seine I am now switching from right to left brain hemisphere and listen to up to eleven plenary talks per day about that theory which once fell from here into the 20th century and which is now being pulled back to the ortochronous frame in order to catch up with the accelerating cosmic expansion.

While this is still work in progress my humble task shall be to accelerate the expansion of my personal horizon.

In the age of internet communication one interesting aspect of conferences is always the identification and meeting of e-pen pals. I was glad to meet Jacques Distler and Robert Helling in person for the first time, after quite a while of virtual acquaintance.

In case you haven’t seen it, first check Jacques’ musings (I, II) on ${\mathrm{Strings}}_{04}$ which were produced close to real-time and where of course much more erudite comments on the plenary talks can be found than I am able to produce.

In fact, I’ll only mention the first of today’s talks, which was by Ashoke Sen on 2 D-string theories. As opposed to many other talks wich were concerned with model building, this one stood out as one that nicely addressed the ‘big picture’ of string theory, albeit just in a toy model. The main point was to show how the continuum worldsheet description maps in detail to the Matrix Model point of view. One crucial technique used by Sen was the correspondence between rigid target space gauge symmetries to conserved D-brane charges. This works as follows (assuming that I recall the details correctly):

A rigid gauge transformation in closed string field theory is generated by a ghost number 1 gauge parameter field $\Lambda$, which, since it corresponds to a rigid transformation, has vanishing momentum. Assume more generally that such a field has some fixed momentum $p_0$. Construct a 1-parameter family of fields $\Lambda (p)$ such that $\Lambda (p_0)=\Lambda$. BRST invariance of $\Lambda$ can then be expressed in terms of some string field $\varphi (p)$ as

Next consider some boundary state $\langle mathcaB\mid$ describing some brane. Using the BRST invariance of $\langle mathcaB\mid$ one sees that

But this expression vanishes identically, because the ghost 0-modes are not saturated:

Fourier transforming this expression by introducing the object

it is equivalent to

But this tells us that there is a conserved quantity $e^{i{p}_{0}x}F(x)$ for every gauge parameter field $\Lambda$ at fixed momentum.

Sen compares these conserved quantities with those arising in the Matrix Model of 2D string theory and finds lots of interesting equivalences. But the details are beyond the scope of my notes and my remaining time this evening.

Over at sci.physics.strings Charlie Stromeyer made me think about $p$-gerbes a little bit. A literature search revealed that apparently the SCFT deformation formalism that I am playing with could be useful for understanding nonabelian Kalb-Ramond fields.

In particular, in

Ch. Hofman: Nonabelian 2-forms (2002)

a connection on loop space induced by a nonabelian 2-form $B$ field on target space is proposed.

I would like to show how something very similar can be derived from the string theory perspective by simply deforming the SCFT algebra of the open string appropriately. (A detailed version of the following can be found LaTeXified in this draft.)

The key observation is the following: As discussed before we can combine the left- and rightmoving supercharge $G$ and $\overline{G}$ on the worldsheet to obtain the deformed exterior derivative on loop space

where $K^{(\mu ,\sigma )}=X^{\prime \mu }(\sigma )$ is the reparameterization Killing vector on loop space.

It is also easy to check that sending

with $W^{(A)}=i\oint A_{\mu }{X}^{\prime \mu }$ or $W^{(B)}=\oint \frac{1}{2}{B}_{\mu \nu }{ℰ}^{†\mu }{ℰ}^{†\nu }$ turns on a gauge field and a $B$ field, respectively, for the closed string. This can straightforwardly be generalized to the open string by using an appropriate boundary state and by setting $B=\mathrm{dA}/T$ all the worldsheet bulk terms cancel and the above deformation gives just the covariant exterior derivative for the Chan-Paton endpoints of the string:

This has a totally obvious generalization to the non-abelian case. Simply replace the exponentiated integral with the corresponding path ordered expression and set $B\to d_{A}A/T+B$. (Don’t take a trace, since the matrices must act on the CP factors.)

[Update 06/23/04: So I am thinking here of the ‘bare’ boundary state that all the operator $R$ below acts on as a module of the algebra $M_N(H)$, namely the algebra of $N\times N$ matrices with values in the Heisenberg algebra $H$ of worldsheet oscillators. The matrix entries of course correspond to the strings stretching between two of the $N$ branes in the stack.

The relation between boundary states and modules of (non-commutative) algebras and how this relates to various brane configurations is discussed nicely in

Yonatan Zunger: Constructing exotic D-branes with infinite matrices in type IIA string theory (2002) .

Even though it is not explicitly stated there, the fact that the above matrices $M_N(H)$ take non-commutative values is related, of course, to the noncommutativity introduced by the spatial extension of the string.]

Hence I am saying that the connection on loop space induced by a nonabelian 2-form that follows from string theory considerations is the 1-form component of

where

Performing some loop-space gymnastics this can be evaluated and yields

(up to terms of higher form degree, which don’t have an interpretation as connection terms)

where $U_A(\sigma ,\kappa )$ is the holonomy of $A$ along the string from $\kappa$ to $\sigma$.

The first term is the unperturbed loop space $K$-deformed exterior derivative. The second is the gauge connection of $A$ on the Chan-Paton factors. The third is obviously the connection associated with the nonablian $B$-field.

It has a nice and plausible heuristic interpretation: The CP factor is parallel-transported, using the gauge field $A$, along the string from the endpoint to the point $\sigma$ in the worldsheet bulk. There it is multiplied with the B-field density at that point and then it is parallel-transported by $A$ back to the string’s endpoint.

This is pretty much as expected from general considerations on 2-form gauge theory. For instance see the text related to figure 1) of

Amitabha Lahiri: Parallel transport on non-Abelian flux tubes (2003) .

It is almost precisely the same action as that of the $B$ field connection in the above paper by Hofman, the only difference being that here the restoring parallel transport is also present, which looks very plausible.

The correct gauge invariance of the above construction is manifest, it reduces to known constructions in the appropriate special cases and is the only obvious natural generalization of these.

(As before, the above considerations don’t take quantum divergencies into account. But Hashimoto has shown that demanding Wilson lines of the above form to have a well defined action is equivalent to demanding the background field’s equations of motion.)

I have been jabbering about a possible relation between Pohlmeyer invariants and the IIB matrix model for a while now, without being able to give any systematic evidence for this intuition. Now that I have learned a bit about boundary state techniques I think the relation is pretty obvious:

[Update 16/06/04: Details can be found in section 3.6.1 ‘Boundary DDF/Pohlmeyer invariants’ of these notes.]

Consider a stack of $N$ flat $\mathrm{Dp}$ branes without any gauge fields or other stuff turned on. As is very well known (and as is sort of reviewed in the last entry) such can conveniently be characterized by the closed string source term $\mid {\alpha }_p\rangle$ that comes with it, which, equivalently, is the ‘mirror’ state of a closed string that yields the desired open string boundary condition at the brane. More precisely, when we write $X^i(\sigma )$ and $X^{\mu }(\sigma )$ for the string’s worldsheet coordinate fields transverse and longitudinal to the branes, respectively, and $P_i(\sigma )$, $P_{\mu }(\sigma )$ for the respective canonical momenta, then the defining condition on $\mid {\alpha }_p\rangle$ is

This can be derived in full generality by some CFT gymnastics but is also pretty obvious since $X^i(\sigma )\mid {\alpha }_p\rangle =0$ just says that the $\mathrm{Dp}$-branes are sitting at $X^i=0$ and $P_{\mu }(\sigma )\mid {\alpha }_p\rangle =0$ is just the T-dual ($X^{\prime }\leftrightarrow P$) assertion (up to the 0-mode, which I just ignore here).

By similar reasoning one gets the boundary states which describe fluctuations of the D-branes as well as gauge field excitations:

Let ${\varphi }^i(X^{\mu })$ be the scalar field which describes the embedding of the brane in the transverse space, then the above condition obviously generalizes to

which is simply solved by acting with the ordinary translation operator on the boundary state:

By applying T-duality $P\leftrightarrow X^{\prime }$ again this tells us that the boundary state $\mid {\alpha }_p(A)\rangle$ which describes the flat branes with an abelian gauge field $A_{\mu }$ turned on is

Since this is just the Wilson line of $A$ along the string it is clear that for general $A$ we should take the trace over the $\sigma$-ordered exponential:

By T-dualizing back once again this tells us that displacing the $N$ $\mathrm{Dp}$ branes by different amounts, i.e. by matrix valued coordinates ${\hat{\varphi }}^i$ amounts to setting

Now that I have bored everyone who knows about boundary states to death let me come to the Pohlmeyer invariants:

As one may recall, these are nothing but the objects

for constant matrices $A$. In principle, one could also use $A=A(X(\sigma ))$ in the above definition of the Pohlmeyer invariants - and they would still be invariants (classically or, up to some extra conditions, also quantumly), but there is a certain beauty to the fact that by just choosing (arbitrarily) large constant matrices $A$ in the above expression one still obtains a ‘complete’ set of invariants - and this is how they are defined.

It is clear that these Pohlmeyer invariants are very similar to the unitary operators that were used above to take ordinary boundary states to those describing gauge fields and brane fluctuations. All one has to note to see this in full detail is the following simlple fact:

When the exponential in the Pohlmeyer invariant is Taylor expanded one gets terms of the form

where the ordered periodic integral ${\int }_{\circ }{d}^N\sigma$ may be rewritten as

(cf. equation (2.17) of hep-th/0403260). It must be noted that ${\sigma }^1$ here seems to play a preferred role, but in fact it does not and we may choose any of the $\sigma$s as ${\sigma }^1$ as long as we preserve the correct periodic order of the $\sigma$s. The point is that this implies that even at the quantum level we may re-order the $(P+{\mathrm{iX}}^{\prime })({\sigma }^i)$ terms in the above expression. That’s because their commutator is a total ${\sigma }^i$-derivative which vanishes under the total ${\int }_0^{2\pi }d{\sigma }^i$-integral.

This somewhat technical consideration has the sole purpose of showing that when applied to a state which is annihilated either by $P$ or by $X^{\prime }$, we may simply cancel the respective operator from the exponential of the Pohlmeyer invariant.

More precisely, let $\mid {\alpha }_9\rangle$ be the boundary state of a space-filling (stack of) brane(s), annihilated by all the $P_{\mathrm{0,1},\cdots ,9}$, then the application of the Pohlmeyer operator $Z[A]$ to the respective boundary state

is nothing but that stack of branes with the constant gauge field $A$ turned on, because, by the above considerations, we can move all the $P(\sigma )$ that enter the Pohlmeyer operator to the right, where they annihilate the boundary state.

The nice thing is the following: It is known from

T. Maeda & T. Nakutsu & T. Oonishi: Non-linear Field Equation from Boundary State formalism (2004)

(taken together with hep-th/9909027) that applying the above exponential is well defined (has no divergencies from nearby quantum fields that appear in the expression) precisely (to lowest non-trivial order) only when the classical equations of motion hold, i.e. when the matrices $A$ satisfy

which is nothing but an operator version of the familiar vanishing of the $\beta$-functionals of the string’s $\sigma$-model.

Taking this result and T-dualizing all directions (including the time-like one) one finally sees that applying the Pohlmeyer invariant to $N$ coincident $D(-1)$-branes (instantons) yields the configuration of $N$ such instantons distributed according to the non-commutative coordinates ${\varphi }^{\mu }=A_{\mu }$ iff (to lowest non-trivial order), the classical equations of motion of the IIB matrix model hold.

And this, after all pretty trivial, observation is what I wanted to get at. It shows that the Pohlmeyer invariants are related to coherent states of closed strings which constitute a distribution of space-time ‘events’ (D(-1)-branes) that is a solution of the classical equations of motion of the IIB matrix model and can be argued to describe a discrete/noncommutative approximation to 1+9D spacetime.

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

Today Eric Forgy has asked me to summarize some aspects of results by Koji Hashimoto concerning the description of open string backgrounds by deformations of boundary states.

With the time and energy that I have I will certainly not succedd in giving a complete description starting from first principles, but I’ll try to convey the basic ideas and add some comments concerning my perspective on these matters. (The details are left to the comment section ;-)

So here’s the story: While thinking about how the deformations of closed string worldsheet theories of the form which we discussed a while ago in this entry generalize to open strings I began of course to think about boundary state formalism and some literature search then turned up the papers

Koji Hashimoto: Generalized supersymmetric boundary state (2000)

T. Maeda & T. Nakatsu & T. Oonishi: Non-linear Field Equations from Boundary State Formalism (2004)

Koji Hashimoto: The shape of nonabelian D-branes (2004)

Here is my perspective on what’s going on:

We had seen that all (massless NS, at least) backgrounds of the closed string come from deformations of the super-Virasoro constraints of the form $d_K\to e^{-W}{d}_K{e}^W$, $d_K^{†}\to e^{W^{†}}{d}_K^{†}{e}^{-W^{†}}$, where $d_K^{(†)}\sim {\mathrm{iT}}_F\pm {\overline{T}}_F$ are polar combinations of the left- and rightmoving supercurrents.

Some of these transformations are pure gauge. Namely if $W^{†}=-W$ then (and only then!) is the above transformation a global unitary tranformation of the entire super-Virasoro algebra. It has been shown how such unitary transformations encode gauge shifts and dualities of the background fields, precisely as expected (in retrospect, in my case) from string field theory considerations.

In particular, it can be shown that the gauge transformation $W=i\oint d\sigma X^{\prime \mu }(\sigma )A_{\mu }(X(\sigma ))$ corresponds to turning on a gauge field background, which couples trivially to closed strings, manifesting itself as a gauge-trivial shift $B\mapsto B+dA$ of the 2-form NS-NS field. This can be understood by regarding a closed string as consisting of two glued open strings. The gauge field couples to the endpoints of these open strings and since they stick together (in this picture) one coupling cancels the other.

But this alrready suggests how the deformation generalizes to the open string: We have to rewrite the opens string theory as that of a closed string split in half.

This is precisely the moral content of the boundary state formalism. I have a brief description of this technique in what is currently appendix C.2 ‘Boundary states’ of my OSFT notes:

Imagine an open string propagating with both ends attached to some D-brane. The worldsheet is topologically the disk (with appropriate operator insertions at the boundary). This disk can equivalently be regarded as the half sphere glued to the brane. But from this point of view it represents the worldsheet of a closed string with a certain source at the brane. Therefore the open string disk correlator on the brane is physically the same as a closed string emission from the brane with a certain source term corresponding to the open string boundary condition. The source term at the boundary of the half sphere can be represented by an operator insertion in the full sphere. The state corresponding to this vertex insertion is the boundary state.

This is very similar in sprit to the method of mirror charges in elementary electrostatics. Simple but profound.

So what we have to do in fact consider just closed strings, and the deformations known of these, and then consider the modified inner product

where $\mid \alpha \rangle$ is the closed string state which encodes the desired boundary condition for the open string.

One curious observation is that the boundary state $\mid {\alpha }_0\rangle$ which describes a flat space-filling D9 brane is nothing but the constant 0-form on loop space! For anyone knowing the respective formulas this is trivial, but to me this seems to be important, conceptually. For details of what I am talking about see what is currently section 3.6 ‘Boundary states and loop space formalism’ in my OSFT notes.

So let $U(A)=\mathrm{Tr}𝒫\exp (\oint d\sigma X^{\prime \mu }{A}_{\mu })$ be the deformation inducing a gauge field background for the closed string, as in equation (3.51) of my hep-th/0401175.

This suggests we study something like $\mid \alpha \rangle =U(A)\mid {\alpha }_0\rangle$ if we want to describe open superstrings on $N$ flat D9 branes with a $U(N)$ gauge field turned on.

And, indeed, this is essentially what Hashimoto showed in JHEP 04 (2000) 023 to be the correct choice, at least with regard to the bosonic degrees of freedom. In addition one want the boundary state to be BRST-closed, i.e. annihilated by the BRST operator, because that sort of makes is an honest physical state of the closed string (a coherent state, in fact, describing the macroscopic excitation of a brane with that gauge field). So one has to add a fermionic component. This can be guessed and checked as in the above paper or derived with superfield formalism as in hep-th/0312260, but in any case the result is that we also need to multipliy by the unitary operator

where ${ℰ}^{(†)}$ are the form creators/annihilators on loop space and $F=d_{A}A$ is the field strength. This is natural, since according to the closed string formalism of section 3.3.2 these operators are asscociated with deformations turning on 2-form backgrounds of the form $\mathrm{dA}$.

So this is already the basic idea: A unitary and hence gauge-trivial deformation of the closed string becomes a non-trivial deformation of the open string formalism by ‘cutting it in half’, i.e. by inserting it only on one side of the inner product, instead of on two sides, so to say.

It should also not be a miracle how such deformations arise in the context of

A. Recknagel & V. Schomerus: Boundary deformation theory and Moduli spaces of D-branes,

because that was already clarified by J. Klusoň as reviewed (and referenced) in section 2.4 ‘CFTs from string field backgrounds’ in my notes (I’ll have to expand on that summary, though…).

The above method generalizes strightforwardly to tachyon backgrounds, finite number of gluon excitations (instead of coherent states) and what not. This and the derivation of the correct background equations of motion from the BRST-closedness condition is shown in great detail by Koji Hashimoto in hep-th/0312260 (thereby solving for the case of open striongs the little excercise that a kind referee suggested to me in the context of closed strings ;-)

It is nice to see how much physics this method captures. For instance on p. 5 of hep-th/0401043 Koji Hashimoto mentions that a set of D0 branes whose distribution is encoded in the matrices $M^i$ can be obtained from the simple boundary state $\mid {\alpha }_{D0}\rangle$ of a single D0-brane simply by applying the unitary ‘translation’ operator with respect to the non-commutative coordinates $M^i$:

Note that if we tranlated these branes in this way and at the same time added a gauge field in the sense of the boundary state

this is nothing but applying a Pohlmeyer operator to the original boundary state. Maybe I was not that far off with my outlook in hep-th/0403260 after all…

As with the previous entry, this one here is a reply to a question on sci.physics.strings which seems to have problems to propagate through USENET.

‘Mike2’ wrote in news:Mike2.17k9lf-100000@physicsforums.com

It seems clear that strings represent structure, and various quantities are calculated along the worldsheet. So can one calculate the entropy associated with the information contained in the quantities along the string?

Yes, there are many states of the string which have the same energy and taking the logarithm of this number gives you the entropy of the string at that energy.

There are lots of very deep questions associated to this entropy.

One is the so-called string-black hole correspondence. It is generally said that a black hole carries the highest amount of entropy per volume. But a simple calculation shows that this is true only up to a very small size of the black hole. As the black hole shrinks (due to evaporation by means of Hawking radiation) it will become very tiny and at some point the entropy of a highly excited single string of a given mass will be equal to that of the black hole of that mass. For even lower masses the string’s entropy will even be greater than that of the corresponding black hole. The point at which that happens is called the string/black hole correspondence point.

The interesting thing is that, despite the crudeness of the calculations used in this sort of correspondence, it gives an easy way to calculate the correct order-of-magnitude entropy of all kinds of black holes, Schwarzschild, rotating, various charged ones, etc. It also provides a nice heuristic picture of black hole entropy at the correspondence point. One can sort of imagine the different “bits of string” sitting on the horizon and the entropy comes from the different ways in which these bits are connected inside the whole by the string.

If you want to find out more about this effect see

Gary T. Horowitz & Joseph Polchinski: