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Lubos great "kennedy landscape" essay
Lubos Motl has written a great and timely essay on his blog
it shows his notable gift with words, especially the bit about the keys in the ocean (near the end).
I am mortified that such talent should be on the far right of the political and environmental spectrum. Also it shows that physicists can usually write better than mathematicians or at least funnier----mathematicians are just doing drawing-room comedy with sometimes quaint self-parody for excitement, but by contrast physicists are doing full-blown and outrageous farce. Probably it is because they are morally more serious and having feet on the ground allows them scope. I will bold some interesting parts so your eye can find them:
------quote-----
Kennedy's landscape
Frederik Denef (Rutgers U.) was explaining how to build a better racetrack (with Bogdan Florea and Michael Douglas), i.e. how to construct particular examples of the numerous KKLT anti de Sitter vacua - the mathematical constructions that are used to argue that the anthropic principle is needed in string theory. The talk today was actually based on a newer paper with Douglas, Florea, Grassi, and Kachru; sorry for an incorrect reference, and thanks for Frederik's correction. Nevertheless I will keep examples from the older paper, too. This stuff is impressive geometry. A really high-brow mathematics, even if it happens to be just recreational mathematics.
Nevertheless, the most illuminating idea was the following variation of Kennedy's famous quote due to Abdus Salam:
My fellow physicists, ask not what your theory can do for you: ask what you can do for your theory.
This could become the motto of the landscape research. Suddenly it's not too important whether a theory teaches us something new about the real world - either predicts new unknown phenomena or previously unknown links between the known phenomena and objects. It's more important that such an unpredictive scenario might be true and we should all work hard to show that the scenario is plausible because we should like this scenario, for some reasons that are not clear to me.
It's slowly becoming a heresy not to believe the anthropic principle - but it already is heresy to think that even the question whether the anthropic reasoning is the explanation of the details of our universe is not the most interesting question, at least among the scientific ones. Even if some numbers in Nature - such as the particles masses - are random historical coincidences, we will never know for sure.
Let me remind you about the basic framework of the Kachru-Kallosh-Linde-Trivedi (KKLT) construction - the most frequently mentioned technical result to justify the anthropic principle in string theory. String theory often predicts many massless scalar fields that are unacceptable because they would violate the equivalence principle and we could already have detected them.
They must be destroyed - i.e. they must acquire masses. The potential energy as a function of the scalar fields must have a finite or countable number of minima. The scalar fields then sit at these minima - we say that the moduli (scalar fields) are stabilized which is a good thing and one of the unavoidable tasks. Moduli stabilization was only the main goal of Frederik's talk.
KKLT start with F-theory (a formally 12-dimensional theory due to Cumrun Vafa) compactified on an elliptically fibered (=interpretable as an elliptic curve, i.e. a two-torus, attached to every point of a lower-dimensional base space) Calabi-Yau four-fold (an eight-dimensional manifold) to give you a four-dimensional theory with a negative cosmological constant and all moduli stabilized. Then they add some non-supersymmetric objects (D3-branes) to create a de Sitter space (with the observationally correct, positive cosmological constant and broken supersymmetry) out of the original anti de Sitter space (AdS).
The talk today focused on the AdS, supersymmetric part of the task.
The F-theory vacuum on a four-fold may be re-interpreted as a type IIB vacuum with orientifold planes (both O3 and O7 where 3 and 7 count the spatial dimensions along the fixed planes). Moreover, there are some fluxes of the three-forms over three-cycles (both the NS-NS as well as the R-R field strengths). The integral
int (H3 wedge F3) + #(D3)
must vanish due to a tadpole cancellation which constrains the fluxes H3 and F3 (numerical constants ignored). In terms of the four-fold, the vanishing quantity may be written as
L = 1/2 (int G4 wedge G4) = chi (X) / 24 - #(D3)
where you may think about M-theory on a four-fold instead of F-theory (a dual description for finite areas of the elliptic fiber), and G4 is the standard M-theoretical four-form field strength (its integral over one of the two 1-cycles of the toroidal fiber gives you the NS-NS and R-R three-form field strengths, respectively). Such a cancellation condition still allows for a huge spectrum of possible choices of the integer-valued fluxes: as Bousso and Polchinski estimated 5 years ago, if there are 300 three-cycles and each of them can carry a flux roughly between 0 and 30, then there are 30^{300} or so possible universes. The light scalar fields that we need to stabilize are
the dilaton/axion
the complex structure moduli, the shape parameters of the four-fold
the Kahler moduli, the areas of topologically non-trivial two-dimensional manifolds (2-cycles)
The former two categories are stabilized perturbatively by the Gukov-Vafa-Witten superpotential
W = int (Omega wedge G3)
where Omega is the holomorphic three-form and G3 is the complexified three-form field strength that includes both the NS-NS and R-R components (with "tau" as the relative coefficient, which makes "tau" also stabilized). This perturbative superpotential handily stabilizes the dilaton/axion and the complex structure moduli at some values that are in principle calculable. Well, I should really write the 8-dimensional integral "int (Omega4 wedge G4)" from the M-theory or F-theory picture.
However, the Kahler moduli (the sizes of the two-cycles) are not stabilized by any perturbative effects. Such a fact is also known from other types of stringy models of reality, the so-called "no-scale supergravities" obtained e.g. by compactifying the heterotic strings on Calabi-Yau three-folds. These moduli are, however, stabilized by M5 (or "F5")-brane instantons wrapped on six-cycles of the four-fold. This can either be interpreted as D3-brane instantons in type IIB, or condensation of gauginos living on the D7-branes.
Note that we want to add new terms to the superpotential W that stabilize all the moduli. The precise value of the Kahler potential (not to be confused with the Kahler moduli although Mr. Kahler is of course identical in both cases; the Kahler potential is another function that determines the physics of four-dimensional supersymmetric theories) is not protected and it's always a source of controversies.
OK, these are the general rules - everything else is to look for more exact, particular examples. A goal is to stabilize the Kahler moduli at sufficiently large volumes of the internal space whatever the space exactly is. This (large volume) is something that can be marginally achieved (if you think that the number 20 is large), but the 2-cycles are never really large at the end. Instead, they are comparable to the string size.
The anthropic strategy is to pick as complicated Calabi-Yau manifolds as possible, to guarantee that there will be a lot of mess, confusion, and possibilities, and that no predictions will ever be obtained as long as all the physicists and their computers fit the observed Universe (which is an encouraging prediction that Frederik has also mentioned).
This means that you don't want to start with Calabi-Yaus whose Betti numbers are of order 3. You want to start, if one follows the 2004 paper, with something like F_{18}, a toric Fano three-fold. That's a 3-complex-dimensional manifold that is analogous to the two-complex-dimensional del Pezzo surfaces, in a sense. But you don't want just this simple F_{18}. You take a quadric Z in a projective space constructed from this F_{18} and its canonical bundle. OK, finally the Euler character of the four-fold X is 13,248. Great number and one can probably estimate the probability that such a construction has something to do with the real world. It becomes a philosophical question whether we should be distinguishing this probability from the number "zero" and how much this "zero" differs from the probability that loop quantum gravity describes quantum gravity at the Planck scale. One can also estimate the values of the scalar fields at the minima of the potential, and the number of vacua (some of their models only had a trillion, others have 10^{300} - of course, the Kennedy rule is that the more ambiguous and unpredictive the set of vacua is, the more attention physics should pay to them).
The example today, from the 2005 paper, was the resolved orbifold "T^6 / Z_2 x Z_2" which has 51 Kahler moduli and 3 complex structure moduli. The singularities were analyzed by a local model, and various toric diagrams shown were related by a flop (or a flip, as is now a more popular terminology). Sorry for neglecting the real model of this talk in the first version of this article.
Cumrun - who is not exactly a fan of the anthropic principle (unlike Nima, who tried to counter) - was extremely active during the talk and he argued for the existence of many new effects that were neglected. For example, there is new physics near a high-codimension singularity that is needed in one of these models. Cumrun argued that the fivebrane instantons could get destabilized - kind of unwrap from the singularity; that a lof of instanton corrections could arise from various cycles, and so forth. The expansions are never quite under control because they rely on some "small" numbers that can be as large as (4.pi/flux) where the "flux" is of order "ten" or "one hundred". Most estimates for the Kahler potential are unjustified, and so forth.
Their calculations required to draw a lot of toric diagrams (that's a representation of a manifold where toroidal fibers are attached to a region with boundaries on which some of the circles of the tori shrink to zero); determine various cycles and their triple intersection numbers (it's like counting how many holes a doughnut has, but in a more difficult 8-dimensional setup) which are needed for the volume; a lot of computer time. Do we really believe that by studying the orientifold of the weighted projective space CP^{4}_{[1,1,1,6,9]}, we will find something that will assure us (and others - and maybe even Shelly Glashow) that string theory is on the right track? I believe that the simplest compactifications, whatever the exact counting is, should be studied before the convoluted ones. If we deliberately try to paint the string-theoretical image of the real world as the most ambiguous and uncalculable one, I kind of feel that it's not quite honest.
When we study the harmonic oscillator and the Hydrogen atom, we want to understand their ground states (and low-lying states) first - where the numbers are of order one. Someone could study the "n=1836th" excited level of the Hydrogen atom, hoping that it is messy enough so that it could explain why the proton mass is so much larger than the electron mass. But it is a well-motivated approach? Some people used to blame string theorists that they were only looking for the keys (to the correct full theory) under the lamppost. It's unfortunately not the case anymore: most of the search for the keys is now being done somewhere in the middle of the ocean (on the surface). Maybe, someone will eventually show that the keys can't stay on the surface of the ocean, and we will return to the search for the keys in less insane contexts. But it's not easy to prove something about the middle of the ocean, especially if we don't yet understand the shape of the Manhattan island.
posted by Lumo at 1:47 PM
----end quote---
http://motls.blogspot.com/2005/04/kennedys-landscape.html
Lubos Motl has written a great and timely essay on his blog
it shows his notable gift with words, especially the bit about the keys in the ocean (near the end).
I am mortified that such talent should be on the far right of the political and environmental spectrum. Also it shows that physicists can usually write better than mathematicians or at least funnier----mathematicians are just doing drawing-room comedy with sometimes quaint self-parody for excitement, but by contrast physicists are doing full-blown and outrageous farce. Probably it is because they are morally more serious and having feet on the ground allows them scope. I will bold some interesting parts so your eye can find them:
------quote-----
Kennedy's landscape
Frederik Denef (Rutgers U.) was explaining how to build a better racetrack (with Bogdan Florea and Michael Douglas), i.e. how to construct particular examples of the numerous KKLT anti de Sitter vacua - the mathematical constructions that are used to argue that the anthropic principle is needed in string theory. The talk today was actually based on a newer paper with Douglas, Florea, Grassi, and Kachru; sorry for an incorrect reference, and thanks for Frederik's correction. Nevertheless I will keep examples from the older paper, too. This stuff is impressive geometry. A really high-brow mathematics, even if it happens to be just recreational mathematics.
Nevertheless, the most illuminating idea was the following variation of Kennedy's famous quote due to Abdus Salam:
My fellow physicists, ask not what your theory can do for you: ask what you can do for your theory.
This could become the motto of the landscape research. Suddenly it's not too important whether a theory teaches us something new about the real world - either predicts new unknown phenomena or previously unknown links between the known phenomena and objects. It's more important that such an unpredictive scenario might be true and we should all work hard to show that the scenario is plausible because we should like this scenario, for some reasons that are not clear to me.
It's slowly becoming a heresy not to believe the anthropic principle - but it already is heresy to think that even the question whether the anthropic reasoning is the explanation of the details of our universe is not the most interesting question, at least among the scientific ones. Even if some numbers in Nature - such as the particles masses - are random historical coincidences, we will never know for sure.
Let me remind you about the basic framework of the Kachru-Kallosh-Linde-Trivedi (KKLT) construction - the most frequently mentioned technical result to justify the anthropic principle in string theory. String theory often predicts many massless scalar fields that are unacceptable because they would violate the equivalence principle and we could already have detected them.
They must be destroyed - i.e. they must acquire masses. The potential energy as a function of the scalar fields must have a finite or countable number of minima. The scalar fields then sit at these minima - we say that the moduli (scalar fields) are stabilized which is a good thing and one of the unavoidable tasks. Moduli stabilization was only the main goal of Frederik's talk.
KKLT start with F-theory (a formally 12-dimensional theory due to Cumrun Vafa) compactified on an elliptically fibered (=interpretable as an elliptic curve, i.e. a two-torus, attached to every point of a lower-dimensional base space) Calabi-Yau four-fold (an eight-dimensional manifold) to give you a four-dimensional theory with a negative cosmological constant and all moduli stabilized. Then they add some non-supersymmetric objects (D3-branes) to create a de Sitter space (with the observationally correct, positive cosmological constant and broken supersymmetry) out of the original anti de Sitter space (AdS).
The talk today focused on the AdS, supersymmetric part of the task.
The F-theory vacuum on a four-fold may be re-interpreted as a type IIB vacuum with orientifold planes (both O3 and O7 where 3 and 7 count the spatial dimensions along the fixed planes). Moreover, there are some fluxes of the three-forms over three-cycles (both the NS-NS as well as the R-R field strengths). The integral
int (H3 wedge F3) + #(D3)
must vanish due to a tadpole cancellation which constrains the fluxes H3 and F3 (numerical constants ignored). In terms of the four-fold, the vanishing quantity may be written as
L = 1/2 (int G4 wedge G4) = chi (X) / 24 - #(D3)
where you may think about M-theory on a four-fold instead of F-theory (a dual description for finite areas of the elliptic fiber), and G4 is the standard M-theoretical four-form field strength (its integral over one of the two 1-cycles of the toroidal fiber gives you the NS-NS and R-R three-form field strengths, respectively). Such a cancellation condition still allows for a huge spectrum of possible choices of the integer-valued fluxes: as Bousso and Polchinski estimated 5 years ago, if there are 300 three-cycles and each of them can carry a flux roughly between 0 and 30, then there are 30^{300} or so possible universes. The light scalar fields that we need to stabilize are
the dilaton/axion
the complex structure moduli, the shape parameters of the four-fold
the Kahler moduli, the areas of topologically non-trivial two-dimensional manifolds (2-cycles)
The former two categories are stabilized perturbatively by the Gukov-Vafa-Witten superpotential
W = int (Omega wedge G3)
where Omega is the holomorphic three-form and G3 is the complexified three-form field strength that includes both the NS-NS and R-R components (with "tau" as the relative coefficient, which makes "tau" also stabilized). This perturbative superpotential handily stabilizes the dilaton/axion and the complex structure moduli at some values that are in principle calculable. Well, I should really write the 8-dimensional integral "int (Omega4 wedge G4)" from the M-theory or F-theory picture.
However, the Kahler moduli (the sizes of the two-cycles) are not stabilized by any perturbative effects. Such a fact is also known from other types of stringy models of reality, the so-called "no-scale supergravities" obtained e.g. by compactifying the heterotic strings on Calabi-Yau three-folds. These moduli are, however, stabilized by M5 (or "F5")-brane instantons wrapped on six-cycles of the four-fold. This can either be interpreted as D3-brane instantons in type IIB, or condensation of gauginos living on the D7-branes.
Note that we want to add new terms to the superpotential W that stabilize all the moduli. The precise value of the Kahler potential (not to be confused with the Kahler moduli although Mr. Kahler is of course identical in both cases; the Kahler potential is another function that determines the physics of four-dimensional supersymmetric theories) is not protected and it's always a source of controversies.
OK, these are the general rules - everything else is to look for more exact, particular examples. A goal is to stabilize the Kahler moduli at sufficiently large volumes of the internal space whatever the space exactly is. This (large volume) is something that can be marginally achieved (if you think that the number 20 is large), but the 2-cycles are never really large at the end. Instead, they are comparable to the string size.
The anthropic strategy is to pick as complicated Calabi-Yau manifolds as possible, to guarantee that there will be a lot of mess, confusion, and possibilities, and that no predictions will ever be obtained as long as all the physicists and their computers fit the observed Universe (which is an encouraging prediction that Frederik has also mentioned).
This means that you don't want to start with Calabi-Yaus whose Betti numbers are of order 3. You want to start, if one follows the 2004 paper, with something like F_{18}, a toric Fano three-fold. That's a 3-complex-dimensional manifold that is analogous to the two-complex-dimensional del Pezzo surfaces, in a sense. But you don't want just this simple F_{18}. You take a quadric Z in a projective space constructed from this F_{18} and its canonical bundle. OK, finally the Euler character of the four-fold X is 13,248. Great number and one can probably estimate the probability that such a construction has something to do with the real world. It becomes a philosophical question whether we should be distinguishing this probability from the number "zero" and how much this "zero" differs from the probability that loop quantum gravity describes quantum gravity at the Planck scale. One can also estimate the values of the scalar fields at the minima of the potential, and the number of vacua (some of their models only had a trillion, others have 10^{300} - of course, the Kennedy rule is that the more ambiguous and unpredictive the set of vacua is, the more attention physics should pay to them).
The example today, from the 2005 paper, was the resolved orbifold "T^6 / Z_2 x Z_2" which has 51 Kahler moduli and 3 complex structure moduli. The singularities were analyzed by a local model, and various toric diagrams shown were related by a flop (or a flip, as is now a more popular terminology). Sorry for neglecting the real model of this talk in the first version of this article.
Cumrun - who is not exactly a fan of the anthropic principle (unlike Nima, who tried to counter) - was extremely active during the talk and he argued for the existence of many new effects that were neglected. For example, there is new physics near a high-codimension singularity that is needed in one of these models. Cumrun argued that the fivebrane instantons could get destabilized - kind of unwrap from the singularity; that a lof of instanton corrections could arise from various cycles, and so forth. The expansions are never quite under control because they rely on some "small" numbers that can be as large as (4.pi/flux) where the "flux" is of order "ten" or "one hundred". Most estimates for the Kahler potential are unjustified, and so forth.
Their calculations required to draw a lot of toric diagrams (that's a representation of a manifold where toroidal fibers are attached to a region with boundaries on which some of the circles of the tori shrink to zero); determine various cycles and their triple intersection numbers (it's like counting how many holes a doughnut has, but in a more difficult 8-dimensional setup) which are needed for the volume; a lot of computer time. Do we really believe that by studying the orientifold of the weighted projective space CP^{4}_{[1,1,1,6,9]}, we will find something that will assure us (and others - and maybe even Shelly Glashow) that string theory is on the right track? I believe that the simplest compactifications, whatever the exact counting is, should be studied before the convoluted ones. If we deliberately try to paint the string-theoretical image of the real world as the most ambiguous and uncalculable one, I kind of feel that it's not quite honest.
When we study the harmonic oscillator and the Hydrogen atom, we want to understand their ground states (and low-lying states) first - where the numbers are of order one. Someone could study the "n=1836th" excited level of the Hydrogen atom, hoping that it is messy enough so that it could explain why the proton mass is so much larger than the electron mass. But it is a well-motivated approach? Some people used to blame string theorists that they were only looking for the keys (to the correct full theory) under the lamppost. It's unfortunately not the case anymore: most of the search for the keys is now being done somewhere in the middle of the ocean (on the surface). Maybe, someone will eventually show that the keys can't stay on the surface of the ocean, and we will return to the search for the keys in less insane contexts. But it's not easy to prove something about the middle of the ocean, especially if we don't yet understand the shape of the Manhattan island.
posted by Lumo at 1:47 PM
----end quote---
http://motls.blogspot.com/2005/04/kennedys-landscape.html
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