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July 2, 2004

Strings 2004: a Day Late and a Dollar Short

Wednesday morning was Cosmic Strings Morning. Rob Myers gave a very pretty review of his work with Copeland and Polchinski. Nick Jones talked about the evolution of cosmic F- and D-string networks. They compute the reconnection probability for colliding F- and D-strings (which get folded into existing computations of the evolution of string networks). They claim that the reconnection probablility can be much smaller (P10 3P\sim 10^{-3} for F-F and P.1P\sim .1 for D-D strings) than the standard Nielsen-Olesen strings (P1P\sim 1). I’m not sure why these results differ from the gauge theory answer (presumably because the quantum corrections to the latter have never been properly taken into account) but, in any case, the most interesting features of these string networks is that the collision of general (p,q)(p,q) strings produces 3-string junctions (they also compute the probability for this process) and the existence of “baryons” (D3-branes wrapped on cycles with nn-units of F (3)F_{(3)} flux) on which nn fundamental strings can end. These change the evolution of these string networks in ways that cannot be captured by simply taking existing simulations of string networks and changing the reconnection probablility, PP.

Probably the most exciting talks so far have been about a cluster of work by Vafa and collaborators on topological string theory and its relationship with other aspects of string theory.

Strominger talked about the surprising relation between black hole entropy in N=2N=2 supergravity and the topological A-model. If you take Type IIA string theory compactified on a Calabi-Yau manifold, and look for supersymmetric blackhole solutions, you find the well-known attractor mechanism where, whatever the values of the Kähler moduli, X ΛX^\Lambda, of the Calabi-Yau at spatial infinity, as you approach the horizon, they are attracted to the locus

(1)Re(CX Λ)=p Λ,Re(CF 0Λ)=q Λ Re(C X^\Lambda)= p^\Lambda, \qquad Re(C F_{0\Lambda}) = q_\Lambda

where (q Λ,p Λ)(q_\Lambda,p^\Lambda) are the electric and magnetic charges of the blackhole, F 0Λ=F 0/X ΛF_{0\Lambda}= \partial F_0/\partial X^\Lambda and F 0F_0 is the prepotential (related to the genus-zero topological string vacuum amplitude). The Kahler form is

(2)K=log(iX¯ ΛF 0ΛX ΛF¯ 0Λ) K = - \log( i \overline{X}^\Lambda F_{0\Lambda}- X^\Lambda \overline{F}_{0\Lambda})

and the attractor equations fix CC and the moduli, X ΛX^\Lambda up to a Kähler transformation. Lopes, Cardoso, de Wit and Mohaupt, found a beautiful formula for the corrections to the area law expression for entropy of the blackhole. Define

(3)F(X,T 2)= h=0 F h(X)T 2h F(X,T^2) = \sum_{h=0}^\infty F_h(X) T^{2h}

where F hF_h is proportional to the genus-hh topological string amplitude, and T μνT_{\mu\nu} is the (anti-self dual part of the) graviphoton field strength. At the horizon, the exact attractor equation is

(4)C 2T 2=256,Re(CX Λ)=p Λ,Re(CF Λ(X,T 2))=q Λ C^2 T^2 =256,\qquad Re(C X^\Lambda)= p^\Lambda, \qquad Re(C F_{\Lambda}(X,T^2)) = q_\Lambda

and the blackhole entropy

(5)S BH=iπ2[C¯X¯ Λq Λp ΛC¯F¯ Λ] attr+128πi[F¯T¯ 2FT 2] attr S_{\text{BH}} = \frac{i\pi}{2}\left[\overline{C}\overline{X}^\Lambda q_\Lambda - p^\Lambda \overline{C}\overline{F}_\Lambda \right]_{\text{attr}} + 128\pi i \left[ \frac{\partial \overline{F}}{\partial\overline{T}^2}-\frac{\partial F}{\partial T^2}\right]_{\text{attr}}

where the first term is, essentially, the area law. This formula can be recast in terms of a mixed canonical/microcanonical partition function

(6)S BH=logZ BH(ϕ Λ,p Λ)ϕ Λϕ ΛlogZ BH S_{\text{BH}}= \log Z_{\text{BH}}(\phi^\Lambda,p^\Lambda) - \phi^\Lambda \frac{\partial}{\partial \phi^\Lambda} \log Z_{\text{BH}}

The stunning result is

(7)Z BH(ϕ Λ,p Λ)=|Z top(t A,g top)| 2 Z_{\text{BH}}(\phi^\Lambda,p^\Lambda) = | Z_{\text{top}}(t^A, g_{\text{top}})|^2

where

(8)t A=p A+iϕ A/πp 0+iϕ 0/π,g top=±4πip 0+iϕ 0/π t^A =\frac{p^A +i\phi^A/\pi}{p^0 +i\phi^0/\pi},\qquad g_{\text{top}}= \pm \frac{4\pi i}{p^0 +i\phi^0/\pi}

and logZ top= h=0 g top 2h2F h(top)(t A)\log Z_{\text{top}}= \sum_{h=0}^\infty g_{\text{top}}^{2h-2} F_{h(\text{top})}(t^A) is the topological A-model partition function.

Robert Dijkgraaf talked about a 7-dimensional field theory based on work of Hitchin on G 2G_2 structures, which might be called (the spacetime theory of) topological M-theory. When defined on a manifold of the form M CY×S 1M_{\text{CY}}\times S^1, it provides a derivation of the proposed S-duality between the topological A-model and the topological B-model (whose spacetime theories are Kodaira-Spencer theory and Kähler gravity, respectively). There’s also an 8-dimensional theory, based on Spin(7)Spin(7) structures.

Both Seiberg and Rastelli gave beautiful talks about c<1c\lt 1 noncritical string theory, and progress in understanding them from the point of view of D-branes (whose collective field theory is none other than the Matrix model).

Posted by distler at July 2, 2004 4:52 AM

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11 Comments & 1 Trackback

Re: Strings 2004, a Day Late and a Dollar Short

Thanks for updating - esp since the slides do not seem to be available
on the web. See you here at the CERN post-meeting next week !

-Wolfgang

Posted by: WL on July 2, 2004 5:08 AM | Permalink | Reply to this

Re: Strings 2004, a Day Late and a Dollar Short

Seriously, what’s up with the no slides? Very odd.

So, is there any new news on the twistor front?

Posted by: Aaron Bergman on July 2, 2004 3:30 PM | Permalink | Reply to this

Re: Strings 2004, a Day Late and a Dollar Short

As far as I could tell, there wasn’t anything in Witten’s twistor talk that is not contained in the recent papers on which he was an author. (It was basically a summary talk – pretty good summary, though!) His was the only talk on the subject. I didn’t hear much interesting twistor scuttlebutt, either, although I’m sure I didn’t talk to everyone who might be working on them.

Posted by: Andy Neitzke on July 3, 2004 2:44 PM | Permalink | Reply to this

Re: Strings 2004, a Day Late and a Dollar Short

Of course, a small surge for occurrences of “twistor” in hep-th is already apparent: 30 matches for 2004 versus 17-9-13-16-11-… in previous years 2003, 2002, 2001, …

I remember the duality talk, also in Paris in Sorbonne, 1995 (a satellite to IAMP congress?). And then people went out of the talk very excited about having heard the key development of next years. From your comment, I guess it is not the case, this time.

Posted by: alejandro rivero on July 5, 2004 12:04 AM | Permalink | Reply to this

Re: Strings 2004: a Day Late and a Dollar Short

Hi Jacques,

Thanks for keeping us all posted. I think that the probability is 1 exactly for the gauge strings (your colleague Matzner discovered this), as long as g sg_s is small enough that the classical description is valid: it’s deterministic, not probabilistic. I don’t know how to estimate corrections to ‘1’ but they are probably like e 1/g s 2e^{-1/g_s^2}, so really small if g sg_s is small.

Joe

Posted by: Joe Polchinski on July 2, 2004 7:28 PM | Permalink | Reply to this

Reconnection

Classically, the reconnection probability is 1. Quantum mechanically, there’s some probability for the strings to tunnel past one another. As you say, naively one would expect that effect to be exponentially suppressed at weak coupling. [I took the liberty of changing the text filter on your comment so that the TeX got processed.]

But I don’t see why that argument would be any different between gauge theory and (classical) open string field theory. So it’s a bit of a surprise if the reconnection probability at weak string coupling turns out to be vastly different in string theory.

Is there an intuition for why reconnection is so suppressed in the string case, even at weak string coupling?

Posted by: Jacques Distler on July 3, 2004 2:03 AM | Permalink | PGP Sig | Reply to this

Re: Strings 2004: a Day Late and a Dollar Short

A good question. For FF it is clear why it is small, because reconnection is the basic string interaction and so the amplitude is proportional to g sg_s. But you undoubtedly mean DD, because this is the case where there is an OSFT description. In this case we did find that the basic probability was O(1), but we identified two suppression effects, which cover much of the parameter space.

The first is that the quantum fluctuations of the strings can cause them to miss in the compact directions. Most field theory cosmic strings don’t have such degrees of freedom. You could probably contrive a field theory in which gauge strings had such extra very light quasi-collective coordinates in which case you would get the same effect, since you can always reverse-engineer (deconstruct) what the string examples are doing (the quantum effects are enhanced by a large log, which is why the classical picture breaks down), but I don’t think this happens in most GUTS.

The second suppression has a geometric origin shown in figure 5 of our paper. It was a surprise to us, that while DD interact with probability 1 the interaction leads to reconnection only with probability e 1/ge^{-1/g}. This has to do with the detailed shape of the field theory configuration space, which is different for OSFT and renormalizable gauge theory; I do not know any way to reverse-engineer figure 5.

By the way, even the (p,q) junctions can be reverse-engineered. So you probably can’t rule out a gauge theory (unless you can compare the full P(v,theta) with a string calculation, which is a long way off), but you can say that small P arises more naturally in string theory.

Posted by: Joe Polchinski on July 3, 2004 3:46 AM | Permalink | Reply to this

p.s.

Sorry if I’m boring you, Jacques, but you raised an interesting question. There is a surprising difference between the OSFT description of DD collisions and the field theory description of soliton string collisions. We found that the DD collision was necessarily a quantum effect: you can only pair produce open strings, because they are charged under the D-string gauge fields. So two D-strings passing through one another without any interaction whatsoever is a classical OSFT solution, since tadpoles are forbidden by symmetry. But obviously, due to the nonlinearities of the theory, the same cannot be true for gauge theory solitons (we’re not in 1+1 dimensions, and all SUSY is broken by the relative angle and velocity of the strings). This is why figure 5 cannot be reverse-engineered, and why we get probabilistic answers rather than deterministic ones.

Posted by: Joe Polchinski on July 3, 2004 2:24 PM | Permalink | Reply to this

Re: p.s.

Sorry if I’m boring you, Jacques, but you raised an interesting question.

Not bored at all … fascinated. The difference between OSFT and gauge theory is really striking. Clearly, one cannot truncate to the lowest-lying open string modes. One might have thought that, at low velocity (whatever the relative angle between the D-strings), that would have been a valid approximation.

Apparently not …

Posted by: Jacques Distler on July 3, 2004 5:45 PM | Permalink | PGP Sig | Reply to this

Re: p.s.

Dear Joe and Jacques, please allow me to post my view here on the reconnection probability of colliding DD. There is a way to reverse-engeneer the brane configuration of Figure 5 of Joe’s paper. It is given by turning on off-diagonal elements of an SU(2) YM electric field, supposing that effective field theory of gauge strings has a sector of off-diagonal YM components as in the usual DD effective field theory. The reason why this corresponds to the production of F-Fbar pair and also to reconnection was shown in my paper hep-th/0401043. The off-diagonal tachyon condensation of the 2d SU(2) YM realizes this process, and this is related to Sen’s conjecture. A sort of Seiberg-Witten map is used for checking the time-dependent bulk distribution of the charges and energy from the YM fields. So, I believe that reconnection probability of DD will be reproduced once a field theory model incorporating YM dinamycs as gauge vortex effective theories is adopted. (I agree that most of GUTS do not have this property.)

Namely, it is an issue of what kind of field theory (giving vortices) we are using. OSFT should give the YM effective field theory as tachyonic vortex D-strings, but two-D-brane sector has not been found yet in an OSFT of a single unstable D-brane. (Of course if one starts with two unstable D-brane OSFT then YM structure is already there as CP indices and thus vortex dynamics is given by effective YM.)

Koji

Posted by: Koji Hashimoto on July 18, 2004 9:47 PM | Permalink | Reply to this

On a different note

Dear Prof. Hashimoto -

I was hoping that maybe I would meet you at Strings04 in Paris, which would have been a pleasure for me, but unfortunately that didn’t work out.

Nice to see you online here, though!

Actually, I was hoping that maybe you could have given me some advise on an idea I was thinking about while in Paris, which turned out to be related to your (old) work on boundary state deformations in JHEP04(2000)023. I have considered a nonabelian generalization of this deformation with the aim to say something about nonabelian 2-forms and connections on loop space. After I had talked to some other people about it I have now put these ideas on the archive (hep-th/0407122).

I don’t mean to imply that you will necessarily find this interesting, but since I think it is an application of a generalization of the deformation technique in your above mentioned paper, I thought that maybe I could mention it to you.

In any case, I very much enjoy seeing your interesting comments on physics in general and on DD string reconnection in particular here!

Best regards,
Urs Schreiber

Posted by: Urs Schreiber on July 19, 2004 8:23 PM | Permalink | PGP Sig | Reply to this
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