## June 30, 2004

### Cabinet de Curiosites dot com - STRINGS 04

#### Posted by Urs Schreiber

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 $\Lambda \left(p\right)$ such that $\Lambda \left({p}_{0}\right)=\Lambda$. BRST invariance of $\Lambda$ can then be expressed in terms of some string field $\varphi \left(p\right)$ as

(1)$\left({Q}_{B}+{\overline{Q}}_{B}\right)\mid \Lambda \left(p\right)⟩=\left(p-{p}_{0}\right)\mid \Phi \left(p\right)⟩\phantom{\rule{thinmathspace}{0ex}}.$

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

(2)$⟨ℬ\mid \left[\left({c}_{0}-{\overline{c}}_{0}\right),{Q}_{B}+{\overline{Q}}_{B}\right]\mid \Lambda ⟩=\left(p-{p}_{0}\right)⟨ℬ\mid \left({c}_{0}-{\overline{c}}_{0}\right)\mid \Lambda ⟩\phantom{\rule{thinmathspace}{0ex}}.$

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

(3)$\left(p-{p}_{0}\right)⟨ℬ\mid \left({c}_{0}-{\overline{c}}_{0}\right)\mid \Lambda ⟩=0\phantom{\rule{thinmathspace}{0ex}}.$

Fourier transforming this expression by introducing the object

(4)$F\left(x\right)=\int \mathrm{dp}\phantom{\rule{thinmathspace}{0ex}}{e}^{-\mathrm{ipx}}⟨ℬ\mid \left({c}_{0}-{\overline{c}}_{0}\right)\mid \Phi \left(p\right)⟩$

it is equivalent to

(5)$\nabla \cdot \left({e}^{i{p}_{0}x}F\left(x\right)\right)=0\phantom{\rule{thinmathspace}{0ex}}.$

But this tells us that there is a conserved quantity ${e}^{i{p}_{0}x}F\left(x\right)$ 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.

Posted at 8:29 AM UTC | Permalink | Followups (2)

## June 21, 2004

### Nonabelian 2-form connection from SCFT deformation

#### Posted by Urs Schreiber

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

(1)${d}_{K}:=G+i\overline{G}={ℰ}^{†\left(\mu ,\sigma \right)}{\partial }_{\left(\mu ,\sigma \right)}+iT{ℰ}_{\left(\mu ,\sigma \right)}{X}^{\prime \left(\mu ,\sigma \right)}\phantom{\rule{thinmathspace}{0ex}},$

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

It is also easy to check that sending

(2)${d}_{K}↦{e}^{-W}{d}_{K}{e}^{W}$

with ${W}^{\left(A\right)}=i\oint {A}_{\mu }{X}^{\prime \mu }$ or ${W}^{\left(B\right)}=\oint \frac{1}{2}{B}_{\mu \nu }{ℰ}^{†\mu }{ℰ}^{†\nu }$ 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:

(3)${e}^{-{W}^{\left(A\right)}-{W}^{\left(B\right)}}{d}_{K}\left(\sigma \right){e}^{{W}^{\left(A\right)}+{W}^{\left(B\right)}}{=}_{\left(\mathrm{open}\mathrm{string}\right)}{d}_{K}+i{ℰ}^{†\mu }{A}_{\mu }\left(\sigma \right)\left(\delta \left(\sigma -\pi \right)-\delta \left(\sigma \right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

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 ${M}_{N}\left(H\right)$, namely the algebra of $N×N$ 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 ${M}_{N}\left(H\right)$ 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

(4)${R}^{-1}\circ {d}_{K}\circ R$

where

(5)$R=\mathrm{P}\mathrm{exp}\left({\int }_{0}^{\pi }d\sigma \phantom{\rule{thinmathspace}{0ex}}\left(i{A}_{\mu }{X}^{\prime \mu }+\frac{1}{2}{\left(\frac{1}{T}{d}_{A}A+B\right)}_{\mu \nu }{ℰ}^{†\mu }{ℰ}^{†\nu }\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$

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

(6)${R}^{-1}\circ {d}_{K}\circ R=$
(7)${d}_{K}+$
(8)$+{i{ℰ}^{†\mu }{U}_{A}\left(0,\sigma \right){A}_{\mu }{U}_{A}\left(\sigma ,0\right)\mid }_{\sigma =0}^{\sigma =\pi }+$
(9)$+{\int }_{0}^{\pi }d\sigma \phantom{\rule{thinmathspace}{0ex}}{U}_{A}\left(0,\sigma \right)\left({ℰ}^{†\mu }{B}_{\mu \nu }{X}^{\prime \nu }\right){U}_{A}\left(\sigma ,0\right)+\cdots \phantom{\rule{thinmathspace}{0ex}}$

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

where ${U}_{A}\left(\sigma ,\kappa \right)$ 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.)

## June 13, 2004

### Pohlmeyer invariants and states of the IIB matrix model

#### Posted by Urs Schreiber

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 $\mid {\alpha }_{p}⟩$ 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 ${X}^{i}\left(\sigma \right)$ and ${X}^{\mu }\left(\sigma \right)$ for the string’s worldsheet coordinate fields transverse and longitudinal to the branes, respectively, and ${P}_{i}\left(\sigma \right)$, ${P}_{\mu }\left(\sigma \right)$ for the respective canonical momenta, then the defining condition on $\mid {\alpha }_{p}⟩$ is

(1)${X}^{i}\left(\sigma \right)\mid {\alpha }_{p}⟩=0={P}_{\mu }\left(\sigma \right)\mid {\alpha }_{p}⟩\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\forall \sigma \in \left(0,2\pi \right),\phantom{\rule{thinmathspace}{0ex}}\mu \in \left\{0,1,\cdots ,p\right\},\phantom{\rule{thinmathspace}{0ex}}i\in \left\{p+1,\cdots 9\right\}\phantom{\rule{thinmathspace}{0ex}}.$

This can be derived in full generality by some CFT gymnastics but is also pretty obvious since ${X}^{i}\left(\sigma \right)\mid {\alpha }_{p}⟩=0$ just says that the $\mathrm{Dp}$-branes are sitting at ${X}^{i}=0$ and ${P}_{\mu }\left(\sigma \right)\mid {\alpha }_{p}⟩=0$ is just the T-dual (${X}^{\prime }↔P$) 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 ${\varphi }^{i}\left({X}^{\mu }\right)$ be the scalar field which describes the embedding of the brane in the transverse space, then the above condition obviously generalizes to

(2)$\left({X}^{i}\left(\sigma \right)-{\varphi }^{i}\left({X}^{\mu }\left(\sigma \right)\right)\right)\mid {\alpha }_{p}\left(\varphi \right)⟩=0$

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

(3)$\mid {\alpha }_{p}\left(\varphi \right)⟩=\mathrm{exp}\left(i\oint d\sigma \phantom{\rule{thinmathspace}{0ex}}{\varphi }^{i}\left({X}^{\mu }\left(\sigma \right)\right){P}_{i}\left(\sigma \right)\right)\mid {\alpha }_{p}⟩\phantom{\rule{thinmathspace}{0ex}}.$

By applying T-duality $P↔{X}^{\prime }$ again this tells us that the boundary state $\mid {\alpha }_{p}\left(A\right)⟩$ which describes the flat branes with an abelian gauge field ${A}_{\mu }$ turned on is

(4)$\mid {\alpha }_{p}\left(A\right)⟩=\mathrm{exp}\left(i\oint d\sigma \phantom{\rule{thinmathspace}{0ex}}{A}_{\mu }\left({X}^{\mu }\left(\sigma \right)\right){X}^{\prime \mu }\left(\sigma \right)\right)\mid {\alpha }_{p}⟩\phantom{\rule{thinmathspace}{0ex}}.$

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:

(5)$\mid {\alpha }_{p}\left(A\right)⟩=\mathrm{Tr}\phantom{\rule{thinmathspace}{0ex}}\mathrm{P}\phantom{\rule{thinmathspace}{0ex}}\mathrm{exp}\left(i\oint d\sigma \phantom{\rule{thinmathspace}{0ex}}{A}_{\mu }\left({X}^{\mu }\left(\sigma \right)\right){X}^{\prime \mu }\left(\sigma \right)\right)\mid {\alpha }_{p}⟩\phantom{\rule{thinmathspace}{0ex}}.$

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 ${\stackrel{^}{\varphi }}^{i}$ amounts to setting

(6)$\mid {\alpha }_{p}\left(\stackrel{^}{\varphi }\right)⟩=\mathrm{Tr}\phantom{\rule{thinmathspace}{0ex}}\mathrm{P}\phantom{\rule{thinmathspace}{0ex}}\mathrm{exp}\left(i\oint d\sigma \phantom{\rule{thinmathspace}{0ex}}{\stackrel{^}{\varphi }}^{i}\left({X}^{\mu }\left(\sigma \right)\right){P}_{i}\left(\sigma \right)\right)\mid {\alpha }_{p}⟩\phantom{\rule{thinmathspace}{0ex}}.$

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

(7)$Z\left[A\right]:=\mathrm{Tr}\phantom{\rule{thinmathspace}{0ex}}\mathrm{P}\phantom{\rule{thinmathspace}{0ex}}\mathrm{exp}\left(\frac{i}{\sqrt{2}}\oint d\sigma {A}_{\mu }\left({P}^{\mu }\left(\sigma \right)±{\mathrm{iX}}^{\prime \mu }\left(\sigma \right)\right)\right)$

for constant matrices $A$. In principle, one could also use $A=A\left(X\left(\sigma \right)\right)$ 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

(8)${\int }_{\circ }{d}^{N}\sigma \phantom{\rule{thinmathspace}{0ex}}\left(P+i{X}^{\prime }{\right)}^{{\mu }_{1}}\left({\sigma }^{1}\right)\cdots \left(P+i{X}^{\prime }{\right)}^{{\mu }_{1}}\left({\sigma }^{N}\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{Tr}\left[{A}_{{\mu }_{1}}\cdots {A}_{{\mu }_{N}}\right]$

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

(9)${\int }_{\circ }{d}^{N}\sigma ={\int }_{0}^{2\pi }d{\sigma }^{1}\phantom{\rule{thinmathspace}{0ex}}{\int }_{{\sigma }^{1}}^{{\sigma }^{1}+2\pi }d{\sigma }^{2}\phantom{\rule{thinmathspace}{0ex}}{\int }_{{\sigma }^{2}}^{{\sigma }^{1}+2\pi }d{\sigma }^{3}\phantom{\rule{thinmathspace}{0ex}}\cdots {\int }_{{\sigma }^{N-1}}^{{\sigma }^{1}+2\pi }d{\sigma }^{N}\phantom{\rule{thinmathspace}{0ex}}$

(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 $\left(P+{\mathrm{iX}}^{\prime }\right)\left({\sigma }^{i}\right)$ 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 $\mid {\alpha }_{9}⟩$ be the boundary state of a space-filling (stack of) brane(s), annihilated by all the ${P}_{0,1,\cdots ,9}$, then the application of the Pohlmeyer operator $Z\left[A\right]$ to the respective boundary state

(10)$Z\left[A\right]\mid {\alpha }_{p}⟩=\mathrm{Tr}\phantom{\rule{thinmathspace}{0ex}}\mathrm{P}\mathrm{exp}\left(\frac{i}{\sqrt{2}}{A}_{\mu }{X}^{\prime \mu }\left(\sigma \right)\right)\mid {\alpha }_{9}⟩=\mid {\alpha }_{9}\left(A\right)⟩$

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 $P\left(\sigma \right)$ 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

(11)$\left[{A}^{\mu },\left[{A}_{\mu },{A}_{\nu }\right]\right]=0\phantom{\rule{thinmathspace}{0ex}},$

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 $D\left(-1\right)$-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.

Posted at 8:53 PM UTC | Permalink | Followups (1)

## June 11, 2004

### Open string backgrounds from boundary state deformations

#### Posted by Urs Schreiber

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 ${d}_{K}^{\left(†\right)}\sim {\mathrm{iT}}_{F}±{\overline{T}}_{F}$ 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 $W=i\oint d\sigma \phantom{\rule{thinmathspace}{0ex}}{X}^{\prime \mu }\left(\sigma \right){A}_{\mu }\left(X\left(\sigma \right)\right)$ corresponds to turning on a gauge field background, which couples trivially to closed strings, manifesting itself as a gauge-trivial shift $B↦B+dA$ 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

(1)$⟨\psi ,\varphi {⟩}_{\mathrm{openstring}}:=⟨\psi ,\varphi \mid \alpha {⟩}_{\mathrm{closedstring}}\phantom{\rule{thinmathspace}{0ex}},$

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

One curious observation is that the boundary state $\mid {\alpha }_{0}⟩$ 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 $U\left(A\right)=\mathrm{Tr}𝒫\mathrm{exp}\left(\oint d\sigma \phantom{\rule{thinmathspace}{0ex}}{X}^{\prime \mu }{A}_{\mu }\right)$ 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 $\mid \alpha ⟩=U\left(A\right)\mid {\alpha }_{0}⟩$ if we want to describe open superstrings on $N$ flat D9 branes with a $U\left(N\right)$ 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

(2)${U}^{f}\left(A\right)=\mathrm{exp}\left(\oint d\sigma \phantom{\rule{thinmathspace}{0ex}}{ℰ}^{†}\cdot F\cdot {ℰ}^{†}\right)\mathrm{exp}\left(-\oint d\sigma \phantom{\rule{thinmathspace}{0ex}}ℰ\cdot F\cdot ℰ\right)$

where ${ℰ}^{\left(†\right)}$ 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 $\mid {\alpha }_{D0}⟩$ of a single D0-brane simply by applying the unitary ‘translation’ operator with respect to the non-commutative coordinates ${M}^{i}$:

(3)$\mathrm{Tr}𝒫\mathrm{exp}\left(\oint {M}^{i}{\partial }_{i}\left(\sigma \right)\right)\mid {\alpha }_{D0}⟩\phantom{\rule{thinmathspace}{0ex}}.$

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

(4)$\mathrm{Tr}𝒫\mathrm{exp}\left(\oint {M}^{i}\left({\mathrm{iX}}_{i}\left(\sigma \right)+{\partial }_{i}\left(\sigma \right)\right)\mid {\alpha }_{D0}⟩$

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…

Posted at 9:12 PM UTC | Permalink | Followups (1)

## June 10, 2004

### String entropy and black hole correspondence

#### Posted by Urs Schreiber

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.

Gary T. Horowitz & Joseph Polchinski: A Correspondence Principle for Black Holes and Strings (1996)

and

Thibault Damour & Gabriele Veneziano: Self-gravitating fundamental strings and black-holes (1999)

I have once written a little more detailed description of this correspondence principle on sci.physics.research:

In its more refined form this ‘principle’ amounts to noting that there is a critical excitation energy where a massive string collapses under its own gravity to the size of the order of the string scale (becoming a ‘string ball’) and that precisely at this point its rms radius coincides with its Schwarzschild radius and furthermore all its thermodynamical properties (temperature, entropy, radiation, decay rate) coincide, up to some unknown factors of order unity, with that of a BH of the same mass (e.g. hep-th/9907030).

(I am not sure how this relates to the D5/D1 brane models, which I don’t know well, but I seem to recall that these brane configurations are describable, and are described, by ‘effective long strings’, too.)

Anyway, the string/BH correspondence principle gives rise to a neat mental picture of the BH degrees of freedom which is actually rather similar to the LQG picture of a ‘pierced horizon’. It is roughly the following:

A highly excited string with the large mass $M\gg 1/{l}_{s}$ (${l}_{s}=\sqrt{{\alpha }^{\prime }}$ is the string scale) is in strikingly good approximation a random walk of $n=\sqrt{N}$ steps of step size ${l}_{s}$, where $N$ is the level number of the string, i.e. $N={M}^{2}{l}_{s}^{2}$. It follows that its entropy is to leading order

(1)${S}_{s}\sim n\sim M{l}_{s}\phantom{\rule{thinmathspace}{0ex}}.$

This is quite unlike the entropy dependence of a black hole, which goes as

(2)${S}_{\mathrm{BH}}\sim {M}^{q}$

with $q>1$. But it so happens that at the above mentioned correspondence point, which is reached when the mass of the string becomes the critical value

(3)${M}_{c}=\frac{1}{{g}^{2}}\frac{1}{{l}_{s}}$

($g$ is the string coupling) and where the string collapses under its self-gravity to a ball of diameter $\sim {l}_{s}$, the entropy of the string and that of a BH of the same size coincide. Still, the entropy of a random-walk-like string, even in the collapsed form, has a simple interpretation, it counts the number of decisions one can make while stepping along the random walk.

Now imagine how that collapsed ‘random walk’ looks like: A chain of n segments, each of length ${l}_{s}$ is restricted to lie within a ball whose diameter is also about ${l}_{s}$. A typical such state looks somewhat star-shaped with all the vertices of the random walk on the outside, forming a sphere. This sphere about coincides with the event horizon of a BH which has the same mass as our string. The edges of the random walk cross the interior of this sphere, pierce the horizon, deposit their vertex there, then return to a point near the corresponding antipode and so on, thereby covering the sphere with all n vertices, all about equally spaced (for a typical state). What is the mean area ${A}_{v}$ of the sphere occupied by one such vertex? It is the number of vertices divided by the area of the horizon, i.e.

(4)${A}_{v}\sim n/{l}_{s}^{\left(d-1\right)}\sim n{g}^{\left(d-1\right)}/{l}_{p}^{\left(}d-1\right)\sim 1/{l}_{p}^{\left(d-1\right)}\phantom{\rule{thinmathspace}{0ex}}.$

Here $d$ is the number of spatial dimensions and ${l}_{p}$ is the Planck length, given in string theory by

(5)${l}_{p}^{\left(d-1\right)}={g}^{2}{l}_{s}^{\left(d-1\right)}\phantom{\rule{thinmathspace}{0ex}}.$

It follows that (for instance) in 1+3 dimensions each of the above vertices occupies an area of about a square Planck length of the event horizon.The entropy of this system is, due to the nature of a random walk and by the above formula, proportional to the number of vertices and hence to the area of the event horizon in Planck units.

This is the picture of black hole microstates at the string/BH correspondence point, i.e. for BH that are about to decay into a string state or for strings that are about to become black holes. (In fact the string ball configuration has been used to predict the signature of decaying black holes that may be detected in accelerators one day).

What happens to this crude picture when the mass is increased further? I am not sure how solid the knowledge obout the answer to this question is, but there is a lot of literature about “strings on the stretched horizon”. The basic idea is that once the BH description takes over the above mentioned vertices are somehow frozen on the event horizon. Since the temperature of the string and the Hawking temperature agree at the correspondence point and hence the rates of change of horizon area with mass do, one can show that further quanta of mass that one throws into a BH at correspondence point correctly translates into further string ‘vertices’ appearing on the horizon. But this only holds in the vicinity of the correspondence point. Farther away one has to take into account the fact that the energy of an object near a BH horizon is different when measured by an asymptotically far away observer. When this red-shifting effect is accounted for one can apparently consistently imagine the BH entropy being due to a (very) long string which is lying on the “stretched” event horizon in form of a random walk.

Of course, all this is nowhere near the technical sophistication of D5/D1 brane-system calculations. It is rather like a Bohr-atom model of quantum black holes.

### Experiments probing quantum gravity

#### Posted by Urs Schreiber

There still seems to be a problem with the newsserver at Harvard, from which the messages of sci.physics.strings emanate. For some reason whenever I take the time to write a longer message the server seems to decide that it has better things to do than propagating it to other newsservers. I hope we can solve this weird problem in the future (maybe it is just me not using the various programs correctly, somehow?) but for the time being I would like to make these lost messages available here at the Coffee Table, not the least because they are replies to questions that have been asked.

So here is the first one:

There are indeed a couple of interesting ways, in principle at least, to look for new physics without using ever more expensive particle accelerators:

Here is a set of slides by Pisin Chen from SLAC discussing some implications of possible astrophysical observations for new physics, see also this link.

[Update 06/23/04 A relatively new proposal is now getting a lot of attention. Polchinski and others have studied the effect of fundamental and D-strings of cosmic extension. They say the gravitational waves emitted by such ‘cosmic strings’ could be detected by the gravitational wave detector LIGO. See here for more details.]

People are looking in particular at the following effects:

1) Ultra High Energy Cosmic Rays

2) Lorentz violation on small scales

3) Detection of dark matter

4) Search for additional spatial dimensions and violations of the equivalence principle

5) Spacetime granularity

1) Ultra High Energy Cosmic Rays should on theoretical grounds be cut off at \$5x10^{19}\$ eV, the so-called Greisen-Zatzepin-Kuzmin (GZK) limit, due to the absorbtion of highly energetic rays by the cosmic microwave background.

But at least one experiment has claimed to have seen particles boyond the GZK limit, for a review see

Glennys R. Farrar, Tsvi Piran:
GZK Violation - a Tempest in a (Magnetic) Teapot?,
Phys.Rev.Lett. 84 (2000) 3527

I have heard that Smolin and some other people believe that they can explain the GZK violation with ‘doubly special relativity’, that this is furthermore somehow a prediction of LQG and that they are hoping that the GLAST experiment in 2006 will say something about this (but I am having
trouble finding more details on this speculation, in hep-th/0401087 Smolin vaguely talks about some such experiments).

Recently Edward Witten commented on this experiment where he said:

It is not clear at the moment that there is anything to explain. One of the two main experimental groups (AGASA) reported as of over a year ago that their high energy data are in accord with the expected GZK cutoff. The other group continues to report a discrepancy. We’ll see what happens as data improve.

(quote from the message at the above link).

2) People are apparently looking for violation of Lorentz invariance at small scales/large energies:

From some theories of quantum gravity one might expect such effects, probably not from string theory.

3) Nature of dark matter.

Several people are trying to determine the presence and maybe the nature of dark matter.

For instance the DAMA collaboration and the UK Dark Matter collaboration as briefly summarized in this message by John Baez.

Of course the true nature of dark matter may be a very important clue for how the true theory of quantum gravity looks like, but current experiments are of course very far from saying anything about this.

4) Search for additional spatial dimenions and violations of the equivalence principle

There are potentially very interesting high-precision measurements of the equivalence principle, most notably by the EotWash group.

Variation of the usual \$1/r^2\$ force law of gravity on small scales could be related to extra dimensions and or torsion, for instance.

5) Spacetime granularity

From quantum gravity some people expect that on extremely small scales
spacetime will show some sort of foamy structure, maybe being topologically non-trivial. An old idea by Percival and collaborators is that atom interferometry, e.g. the 2-slit experiment with heavy stuff such as Buckminster Fullerenes (Nature Vol. 401, No. 6754, p. 680 (1999).) as done by Zeilinger’s group, may be sensitive to such a spacetime grabularity.

I had mentioned that in the past from time to time.
giving some references.

I am not sure what string theory really says about ‘quantum spacetime foam’ at small scales.

When we discussed Smolin’s paper it was argued by some people that string theory (on Minkowski space, say) predicts smooth spacetime
down to all scales. But this is not correct, as everybody knows and as I have tried to argue in more detail in this and this message once based on the discussion in Fedele Lizzi anmd Richard Szabo, Duality Symmetries and Noncommutative Geometry of String Spacetimes (1997) because as you try to probe the smooth background with highly energetic strings you will eventually see stringy effects and not be able to resolve the smoothness of the classical background at all, so that effectively it is not smooth on small scales.

So: Might stringy physics effect the phase of atoms/molecues used in matter interferometry ever so slightly?

## June 7, 2004

### Sigh.

#### Posted by Urs Schreiber

Jacques Distler and Peter Woit have already commented on the latest attempt of Bogdanov & Bogdanov to convince laymen of their genius. One should really stop talking about this issue, since apparently the whole purpose of the exercise is to make everybody mention B&B, no matter in which sense. Apparently in lack of experts willing to support their ideas they rephrase comments of critics in such a way that it sounds like approvements.

A while ago I had written this summary of their mistakes (reproduced below). They thanked me personally for this ‘very accurate’ description of their work and said that I was among the very few who really understood the ‘big picture’ of their work. Well, that’s nice, because it allows me to use this certified authority to say in all clarity that the ideas summarized at the above link are indeed based on elementary misconceptions and invalid conclusions and hence make no sense.

Unfortunately it is precisely this little detail that surprisingly did not survive the adaption of my little text in their new book. Everybody who reads some french and knows what a topological field theory really is can compare my original summary of B&B’s mistakes with the respective excerpts I, II from their book to convince himself once and for all that their concern is not science.

I think I am allowed to say that. After all, I am among the few who understands them. ;-)

Posted at 10:58 AM UTC | Permalink | Followups (5)