In the last TH colloquium Johannes Walcher was talking about topological strings. I was intrigued by the brave juxtaposition in the title that began with Topological Strings confront Reality,... From the little I heard about topological strings, it's a more abstract version of string theory. Or, something that Vafa understands but nobody understands Vafa. So I was expecting to learn about some surprising connections. The remaining part of the title, ...Tadpole Cancellation, and Background Independence should trigger a warning light that Johannes' reality may be more Platonic than mine. The reality however surpassed all expectations.
Johannes said: Fluxes Tadpoles Modulus Annulus Mobius Anus 3-Point Function On The Sphere Two Point Function On The Disc Open Strings Riemann Surfaces Anomalies Boundary Of Moduli Space BPS Black Holes Rational Curves Generic Quintic Holomorphic Anomaly Equation Tadpole OSV Holomorphic Polarization D6/D4/D2/D0 Branes Pandharipande A-Model B-Model 3-cycles O4-planes-Holomorphic Curve all in one breath. If Google could index spoken language, all string theory related searches would link to our seminar room. I'm only slightly disappointed that i missed what topological string theory is; I believe it might be interesting. Fortunately, there is this surprisingly good entry on Wikipedia which sheds some light.
No slides available.
Particle Physics Blog
Thursday, 31 January 2008
Dark Universe at CERN
This week Rocky Kolb gave a series of lectures on cosmology. Those who ever attended Rocky's lectures know very well that they are difficult to bear -- you're in the constant danger of laughing your head off. Still you may learn a lot while gasping for air between laughing fits.Rocky gave a review of the Lambda-CDM model at a non-specialist level. The first lecture is an overview of modern cosmology. The second is about inflation. The third is about the observational evidences for dark matter. The last one is about dark energy and recent actions of Dark Energy Task Force.
The lectures are especially recommended for those unfamiliar with recent developments in cosmology. If you know it all, you might still learn
- why Big Bang is more than just a theory,
- what can be done with duct tape and WD-40,
- why neutrinos may threaten your university,
and more.
Video and slides from all the lectures are available here. Enjoy.
Saturday, 26 January 2008
AdS/CFT and Integrability
As I indicated, I'm not able to give a comprehensive account of the past week string school at CERN. I can't help that fluxes make me sick in my stomach, or that counting BH microstates has a similar effect on me as counting sheep. Nevertheless, I listened with somewhat unexpected pleasure to the lectures on Integrability and AdS/CFT by Nick Dorey. This is a cute topic in mathematical physics that I had known nothing about before. Nick's lectures gave me a smattering of idea of what's going on, and I'm sharing a few bits and pieces that have made their way to my long-term memory.
4D maximally supersymmetric Yang-Mills theory is dual to 10D IIB superstrings on AdS5xS5, Maldacena dixit. While many aspects of these two theories are fixed by their powerful symmetries, there is still a lot to learn about the dynamics. Some help may come from the integrable structures that have recently been discovered on both sides of the duality.
Integrability is a very non-generic feature of classical or quantum systems that there are as many conserved charges as there are degrees of freedom. In classical mechanics, this would mean that the system can be fully solved by quadratures. Quantum mechanics is more tricky, but there still exists a method called the Bethe ansatz for finding the exact solutions.
The relevance of integrability in the context of SU(N) super Yang-Mill was pointed out in the paper by Minahan and Zarembo. Integrable structures pop out in the process of computing correlation functions of certain operators in perturbation theory. For example, we can compute gauge invariant local correlators of the scalars that are present in the theory. We pick up two of the three scalars, W and Z, and compute the conformal dimension of the operator
$\langle Z^{L-M} W^M \rangle $
or similar ones with different permutations of Z and W under the trace. The classical scaling dimension of this operator is L (the length of the chain), but there are divergent loop corrections that introduce an anomalous dimension. The additional complication is that loop corrections mix operators with various M, so that we have to deal with a matrix of anomalous dimensions that has to be diagonalized. The eigenvectors correspond to operators with definite scaling dimensions.
Now, the scalars W,Z form a doublet under the SU(2) subgroup of the SO(6) R-symmetry so we can call them spin up and spin down. It looks more fashionable to represent the operators as spin chains, for example
$\langle WWZWWZ\rangle \to |up,up,down,up,up,down>$
It turns out that this analogy is more far reaching. One-loop computations simplify in the large N limit of SU(N) because the planar diagrams can only "flip one spin". One finds that the matrix of anomalous dimensions is given by
$\frac{\lambda}{8 \pi^2}\Sigma_1^L(1 - P_{l,l+1})$
where $\lambda$ is the t'Hooft coupling and P is an operator that exchanges the neighboring spins. A trained eye recognizes in the above the Hamiltonian of the Heisenberg spin chain with nearest neighbor interactions. One can see that a vector with all spins up (the ferromagnetic vacuum) is an eigenvector, but simple vectors with one spin flipped to down are not. Nevertheless, the full spectrum of this system can be found exactly and the eigenvalue problem was solved in the 1930s by Bethe with the help of the Bethe ansatz (the connection to integrability was made much later by Faddeev). The whole spectrum can be constructed out of the combinations of vectors with one spin down, the so-called magnons.
The story is continues on the string theory side duality, as shown in the paper by Hofman and Maldacena. But I stop here, since all these intricate connections make my head spinning.
The video and transperencies should be available via the school's web page. But they are not. A commenter pointed out that there are some technical problems to which string theory has no solution for the moment.
4D maximally supersymmetric Yang-Mills theory is dual to 10D IIB superstrings on AdS5xS5, Maldacena dixit. While many aspects of these two theories are fixed by their powerful symmetries, there is still a lot to learn about the dynamics. Some help may come from the integrable structures that have recently been discovered on both sides of the duality.
Integrability is a very non-generic feature of classical or quantum systems that there are as many conserved charges as there are degrees of freedom. In classical mechanics, this would mean that the system can be fully solved by quadratures. Quantum mechanics is more tricky, but there still exists a method called the Bethe ansatz for finding the exact solutions.
The relevance of integrability in the context of SU(N) super Yang-Mill was pointed out in the paper by Minahan and Zarembo. Integrable structures pop out in the process of computing correlation functions of certain operators in perturbation theory. For example, we can compute gauge invariant local correlators of the scalars that are present in the theory. We pick up two of the three scalars, W and Z, and compute the conformal dimension of the operator
Now, the scalars W,Z form a doublet under the SU(2) subgroup of the SO(6) R-symmetry so we can call them spin up and spin down. It looks more fashionable to represent the operators as spin chains, for example
It turns out that this analogy is more far reaching. One-loop computations simplify in the large N limit of SU(N) because the planar diagrams can only "flip one spin". One finds that the matrix of anomalous dimensions is given by
$\frac{\lambda}{8 \pi^2}\Sigma_1^L(1 - P_{l,l+1})$
where $\lambda$ is the t'Hooft coupling and P is an operator that exchanges the neighboring spins. A trained eye recognizes in the above the Hamiltonian of the Heisenberg spin chain with nearest neighbor interactions. One can see that a vector with all spins up (the ferromagnetic vacuum) is an eigenvector, but simple vectors with one spin flipped to down are not. Nevertheless, the full spectrum of this system can be found exactly and the eigenvalue problem was solved in the 1930s by Bethe with the help of the Bethe ansatz (the connection to integrability was made much later by Faddeev). The whole spectrum can be constructed out of the combinations of vectors with one spin down, the so-called magnons.
The story is continues on the string theory side duality, as shown in the paper by Hofman and Maldacena. But I stop here, since all these intricate connections make my head spinning.
The video and transperencies should be available via the school's web page. But they are not. A commenter pointed out that there are some technical problems to which string theory has no solution for the moment.
Monday, 21 January 2008
String School @ CERN
This week the RTN Winter School on Strings, Supergravity and Gauge Theories takes place here at CERN. On a deeply philosophical level, I noticed that running a blog for more than one year implies that some events are recurring. Maybe because they happen annually?
This year we have the following five series of lectures:
- Black hole microstates by Jan de Boer
- Supersymmetry phenomenology for string theorists by Gian Giudice
- Integrability and AdS/CFT by Nick Dorey
- Flux compactifications from gauged supergravities by Henning Samtleben
- Pure spinor formulation of string theory by Yaron Oz
The topics are more formal than usual; Beckenbauer obviously a bit of a surprise there. For this reason, you should not expect an extensive coverage this year.
Video recordings of all lectures are posted on school's web page.
This year we have the following five series of lectures:
- Black hole microstates by Jan de Boer
- Supersymmetry phenomenology for string theorists by Gian Giudice
- Integrability and AdS/CFT by Nick Dorey
- Flux compactifications from gauged supergravities by Henning Samtleben
- Pure spinor formulation of string theory by Yaron Oz
The topics are more formal than usual; Beckenbauer obviously a bit of a surprise there. For this reason, you should not expect an extensive coverage this year.
Video recordings of all lectures are posted on school's web page.
Friday, 18 January 2008
Non-Gaussian CMB
A discussion at this week's CosmoCoffe revealed to me that there is a claim of observing non-Gaussianity in the Cosmic Microwave Background. Inflation produces approximately Gaussian primordial fluctuations if the inflaton fluctuations around the background behaved roughly as a free field, so that its different frequencies oscillate independently. This is what happens in simplest models of slow-roll inflation. Non-Gaussianity appears when the dynamics of inflaton fluctuations is affected by sizable non-linearities.
The CMB plot we see most often, that with the acoustic peaks, is the power spectrum of the 2-point correlation function. If the primordial fluctuations were Gaussian, all higher-point correlation functions would be determined by the two-point one. Searching for non-Gaussianity thus requires studying the 3-point correlation function of the CMB. In principle, non-Gaussianity should be described by a function of momenta, but it is often simply parametrized by a number called $f_{NL}$. Single field slow-roll inflation predicts $f_{NL} < 1$, (see for example this cute paper by Maldacena). Such a small effect would be overshadowed by cosmic variance and thus not observable. There exist, however, many inflationary models that can accommodate larger non-Gaussianity.
The WMAP 3-year analysis quotes the constraint
$-54 < f_{NL} < 114$. A month ago, Amit Yadav and Benjamin Wandelt presented their own analysis that yields $27 < f_{NL} < 147$ at 95% confidence level. This is 3 sigma away from zero! If confirmed, this observation would rule out single field slow-roll inflation and point to a more complicated mechanism of inflation.
Of course, the sigmas are not the same in cosmology so no reasons to get excited yet. But it's worth to keep an eye on future developments, since the implications of non-Gaussianity would be quite profound. The nice thing is that the Planck satellite, who will be launched this autumn, is sensitive down to $f_{NL} \sim 5$. If the effect is really up there, we shall know it soon.
The CMB plot we see most often, that with the acoustic peaks, is the power spectrum of the 2-point correlation function. If the primordial fluctuations were Gaussian, all higher-point correlation functions would be determined by the two-point one. Searching for non-Gaussianity thus requires studying the 3-point correlation function of the CMB. In principle, non-Gaussianity should be described by a function of momenta, but it is often simply parametrized by a number called $f_{NL}$. Single field slow-roll inflation predicts $f_{NL} < 1$, (see for example this cute paper by Maldacena). Such a small effect would be overshadowed by cosmic variance and thus not observable. There exist, however, many inflationary models that can accommodate larger non-Gaussianity.
The WMAP 3-year analysis quotes the constraint
$-54 < f_{NL} < 114$. A month ago, Amit Yadav and Benjamin Wandelt presented their own analysis that yields $27 < f_{NL} < 147$ at 95% confidence level. This is 3 sigma away from zero! If confirmed, this observation would rule out single field slow-roll inflation and point to a more complicated mechanism of inflation.
Of course, the sigmas are not the same in cosmology so no reasons to get excited yet. But it's worth to keep an eye on future developments, since the implications of non-Gaussianity would be quite profound. The nice thing is that the Planck satellite, who will be launched this autumn, is sensitive down to $f_{NL} \sim 5$. If the effect is really up there, we shall know it soon.
Thursday, 10 January 2008
DGP inside DGP
The first thing that happened this year here at CERN was Cosmo Coffee. TH Cosmo Coffee is similar to Amsterdam coffeshops in that you should not expect coffee being served. Last Wednesday, Michele Redi was talking about modified gravity theories from the DGP family.
One way to tackle the cosmological constant problem is to modify gravity at large distances. What one could hope to achieve by that is either accelerated expansion without a cosmological term in the lagrangian or filtering out the deadly effects of a large cosmological constant. Gravity, however, is not easily manipulated. In those instances when obvious inconsistencies are avoided, there always remain problematic features. For example, giving a mass to the graviton results in a strong coupling of its longitudinal polarization at some scale and, at distances smaller than that scale, non-linear effects become important. It pays to have a non-linear formulation in order to study the physical consequences of such modified gravities
The DGP model is an attempt along these lines that has a generally covariant non-linear formulation. DGP is the minimal 5D gravity with the 5th dimension infinite on one side and cut off by a 4D brane on the other side. The 4D brane carries a large gravity kinetic term (that is just the usual 4D Einstein-Hilbert term). Because of that brane kinetic term, gravity approximately obeys the 4D Newton law at small distances, while at large distances it switches to a 5D behaviour. One may hope that the weakening of gravity at large distances could help in solving some problems. Yet the DGP model is not quite satisfying in this respect. Self-accelerated solutions, although existing, are plagued by ghosts. Furthermore, the cosmological constant problem is not solved: loop effects would generate a cosmological term and destabilize the set-up.
The next thing to try is 6D gravity. However, a codimension-2 brane (here, a 4D brane embedded in 6D) is too singular. This means that divergences in the graviton propagator appear already at the classical level. Furthermore, codimension-2 brane typically leads to a ghost - the effective theory has a scalar excitation with a negative sign of the kinetic term.
In a recent paper, Michele et many al tried a funny regularization inspired by Russian folklore.
A 4D brane with a gravity kinetic term is embedded into a 5D brane with a gravity kinetic term who, in turn, lives inside a 6D bulk. The model interpolates between 6D gravity at very large distances and 4D gravity at small distances, with a possible 5D intermediate region. This construction avoids any divergences at the classical level. The ghost persists in general, but it can be cured by adding a cosmological term on the 4D brane. A cosmological constant on a codimension-two brane has this peculiar feature that it does not curve the spacetime, so that the flat spacetime remains a solution to the Einstein equations.
At this moment it is not clear if this matroshka solves any problems or if it does not create problems of its own. Certainly, it is another example of a non-linear formulation of IR modified gravity. The interesting thing about this particular model is that, at the linearized level, the effective lagrangian for the graviton does not have the Pauli-Fierz structure. Such gravity theories have not been much studied before.
One way to tackle the cosmological constant problem is to modify gravity at large distances. What one could hope to achieve by that is either accelerated expansion without a cosmological term in the lagrangian or filtering out the deadly effects of a large cosmological constant. Gravity, however, is not easily manipulated. In those instances when obvious inconsistencies are avoided, there always remain problematic features. For example, giving a mass to the graviton results in a strong coupling of its longitudinal polarization at some scale and, at distances smaller than that scale, non-linear effects become important. It pays to have a non-linear formulation in order to study the physical consequences of such modified gravities
The DGP model is an attempt along these lines that has a generally covariant non-linear formulation. DGP is the minimal 5D gravity with the 5th dimension infinite on one side and cut off by a 4D brane on the other side. The 4D brane carries a large gravity kinetic term (that is just the usual 4D Einstein-Hilbert term). Because of that brane kinetic term, gravity approximately obeys the 4D Newton law at small distances, while at large distances it switches to a 5D behaviour. One may hope that the weakening of gravity at large distances could help in solving some problems. Yet the DGP model is not quite satisfying in this respect. Self-accelerated solutions, although existing, are plagued by ghosts. Furthermore, the cosmological constant problem is not solved: loop effects would generate a cosmological term and destabilize the set-up.
The next thing to try is 6D gravity. However, a codimension-2 brane (here, a 4D brane embedded in 6D) is too singular. This means that divergences in the graviton propagator appear already at the classical level. Furthermore, codimension-2 brane typically leads to a ghost - the effective theory has a scalar excitation with a negative sign of the kinetic term.
In a recent paper, Michele et many al tried a funny regularization inspired by Russian folklore.
A 4D brane with a gravity kinetic term is embedded into a 5D brane with a gravity kinetic term who, in turn, lives inside a 6D bulk. The model interpolates between 6D gravity at very large distances and 4D gravity at small distances, with a possible 5D intermediate region. This construction avoids any divergences at the classical level. The ghost persists in general, but it can be cured by adding a cosmological term on the 4D brane. A cosmological constant on a codimension-two brane has this peculiar feature that it does not curve the spacetime, so that the flat spacetime remains a solution to the Einstein equations.At this moment it is not clear if this matroshka solves any problems or if it does not create problems of its own. Certainly, it is another example of a non-linear formulation of IR modified gravity. The interesting thing about this particular model is that, at the linearized level, the effective lagrangian for the graviton does not have the Pauli-Fierz structure. Such gravity theories have not been much studied before.
Tuesday, 8 January 2008
Best of 2007
CERN is waking up slowly, but it'll take a while till something starts happening. I can use this time to recall my favourite bits of 2007. Like every year, the past year arXiv papers that I enjoyed the most are awarded the prestigious Narrow Resonaance awards. This year's Narrow Resonaances go to (drums) :
1. G.Dvali, S.Hofmann, J.Khoury,
Degravitation of the cosmological constant and graviton width,
arXiv:hep-th/0703027
I wrote about it the other day. The idea is to solve the cosmological constant problem by modifying gravity at long distances such that large wavelength sources, like the cosmological constant, are effectively filtered out. A complete, non-linear model that could achieve it does not exist yet. Gia et al. just propose a phenomenological description of massive or resonance gravity theories with the desired long distance behaviour.
2. E.Katz, T.Okui,
The 't Hooft Model As A Hologram,
arXiv:0710.3402
AdS/CFT is still a mystery mainly because, typically, we can calculate on the weakly coupled side only. For some toy models, however, the strongly coupled side can be solvable too, so that we could explicitly see how the AdS/CFT prescription arises. This paper maps 2D QCD at large N to 3D AdS gravity and relates the parton-x variable in QCD to the radial coordinate in AdS3.
3. H.~Georgi,
Unparticle Physics,
arXiv:hep-ph/0703260
The two previous papers have 3 citations (together), but I could not quite ignore the most cited paper of 2007. I had my doubts though, mainly because of the lemming rush that followed. The bulk of these unpapers could well have the abstract: ''We consider unparticles in whatever uncontext. You are encouraged to forget the paper as soon as soon as you add it to your citation list". Moreover, two recent papers, here and here, point out that an explicit construction of the conformal sector typically leads to a different phenomenology. Nevertheless, I must give the credit to Howard for drawing our attention to a whole wide class of collider signatures. Besides, I appreciate Howard's writing style. He is probably the last man on Earth who truly enjoys particle physics.
Consolation Bump goes to
S.Dubovsky, A.Nicolis, E.Trincherini, G.Villadoro,
Microcausality in Curved Space-Time
arXiv:0709.1483
A little neat thing about quantum field theory in curved backgrounds. They show that causality of the classical theory implies micro-causality of the quantum theory. Clear and illuminating.
1. G.Dvali, S.Hofmann, J.Khoury,
Degravitation of the cosmological constant and graviton width,
arXiv:hep-th/0703027
I wrote about it the other day. The idea is to solve the cosmological constant problem by modifying gravity at long distances such that large wavelength sources, like the cosmological constant, are effectively filtered out. A complete, non-linear model that could achieve it does not exist yet. Gia et al. just propose a phenomenological description of massive or resonance gravity theories with the desired long distance behaviour.
2. E.Katz, T.Okui,
The 't Hooft Model As A Hologram,
arXiv:0710.3402
AdS/CFT is still a mystery mainly because, typically, we can calculate on the weakly coupled side only. For some toy models, however, the strongly coupled side can be solvable too, so that we could explicitly see how the AdS/CFT prescription arises. This paper maps 2D QCD at large N to 3D AdS gravity and relates the parton-x variable in QCD to the radial coordinate in AdS3.
3. H.~Georgi,
Unparticle Physics,
arXiv:hep-ph/0703260
The two previous papers have 3 citations (together), but I could not quite ignore the most cited paper of 2007. I had my doubts though, mainly because of the lemming rush that followed. The bulk of these unpapers could well have the abstract: ''We consider unparticles in whatever uncontext. You are encouraged to forget the paper as soon as soon as you add it to your citation list". Moreover, two recent papers, here and here, point out that an explicit construction of the conformal sector typically leads to a different phenomenology. Nevertheless, I must give the credit to Howard for drawing our attention to a whole wide class of collider signatures. Besides, I appreciate Howard's writing style. He is probably the last man on Earth who truly enjoys particle physics.
Consolation Bump goes to
S.Dubovsky, A.Nicolis, E.Trincherini, G.Villadoro,
Microcausality in Curved Space-Time
arXiv:0709.1483
A little neat thing about quantum field theory in curved backgrounds. They show that causality of the classical theory implies micro-causality of the quantum theory. Clear and illuminating.
Saturday, 5 January 2008
CERN TH 2008
A.D. 2008. The year to come will be remembered forever since it's the year of the first collisions at the LHC... damn it, i have deja vu... didn't I write a similar thing a year ago?
Yeah, it's been a year since. I wrote the other day that CERN in early January was a dreary place. My psychoanalist advised me to start a blog as a means to fight off the depression. I expected it to be a short-lived prank but, to my surprise, the blog is still alive and kicking. The depression, likewise. During the past year Resonaances has become the central point of attention for the particle physics community...well...not really, it's just a small, local blog read mostly by other bloggers. At the beginning, the blog was advertised on the TH web page, but then it was filed under external(!?) links after protests from some past and future Noble prize winners. That's what they call an internal exile :-)
Me? Somehow, I prefer to stay in the shadow. In fact, the time when I was really anonymous was most fun. Unfortunately, here at CERN you cannot run or hide. Those who know me didnt have problems to identify. Those that I ridiculed, mistreated or slandered quickly found out too. So, if you dig dip enough you can also find out. But what's in a name?