A couple of days ago there appeared a paper by Freedman, Headrick, and Lawrence that I find highly original. It not just follows up on a number of other papers but actually answers a question that has been lurking around for quite a while but had not really been addressed so far (at least as far as I am aware of). I had asked myself the question before but attributed it to my lack of understanding of the field and never worried enough to try to work it out myself. At least, these gentlemen have and produced this beautiful paper.
It is set in the context of tachyon condensation (and this is of course where all this K-Theory stuff is located): You imagine setting up some arrangement of branes and (as far as this paper is concerned even more important as this is about closed strings) some spatial manifold (if you want with first fundamental form, that is the conjugated momentum to a spatial metric) with all the fields you like in terms of string theory and ask what happens.
In general, your setup will be unstable. There could be forces or you could be in some unstable equilibrium. The result is that typically your space-time goes BOOOOOOOOOOM as you had Planck scale energy densities all around but eventually the dust (i.e. gravitational and other radiation) settles and you ask: What will I find?
The K-Theory approach to this is to compute all the conserved charges before turning on dynamics and then predicting you will end up in the lowest energy state with the same value for all the charges (here one might worry that we are in a gravitational theory which does not really have local energy density but only different expansion rates but let's not do that tonight). Then K-Theory (rather than for example de Rham or some other cohomology) is the correct theory of charges.
The disadvantage of this approach is that it is potentially very crude and just knowing a couple of charges might not tell you a lot.
You can also try to approach the problem from the worldsheet perspective. There you start out with a CFT and perturb it by a relevant operator. This kicks off a renormalisation group flow and you will end up in some other CFT describing the IR fixed point. General lore tells you that this IR RG fixed point describes your space-time after the boom. The c-theorem tells you that the central charge decreases during the flow but of course you want a critical string theory before and after and this is compensated by the dilaton getting the appropriate slope.
The paper is addresses this lore and checks if it is true. The first concern is of course that proper space-time dynamics is expected to (classically) be given by some ordinary field equation in some effective theory with typically two time derivatives and time reversal symmetry where the beta functions play the role of force. In contrast, RG flow is a first order differential equation where the beta-functions point in the direction of the flow. And (not only because of the c-theorem) there is a preferred direction of time (downhill from UV to IR).
As it is shown in the paper, this general scheme is in fact true. And since we have to include the dilaton anyway, this also gets its equation of motion and (like the Hubble term in Friedman Robertson Walker cosmology) provides a damping term for the space-time fields. So, at least for large damping, the space-time theory is also effectively first order but at small (or negative which is possible and of course needed for time reversal) damping the dynamics is of different character.
What the two descriptions agree on is the set of possible end-points of this tachyon condensation, but in general the dynamics is different and because second order equations can overshoot at minima, the proper space-time dynamics can end up in a different minimum than predicted by RG flow.
All this (with all details and nice calculations) is in the paper and I can strongly recommend reading it!
Wednesday, October 26, 2005
Monday, October 24, 2005
Hamburg summary
So, finally, I am back on the train to Bremen and can write up my summary of the openening colloquium of the centre for mathematical physics in Hamburg.
As I reported earlier, this was a conference with an exceptionally well selected program. Not that all talks were in exactly on topics that I think about day and night but with very few exceptions, the speakers had something interesting to say and found good ways to present it. Well done, organisers! I hope your centre will be as successful as this colloquium!
The first physics talk on Thursday was Nikita Nekrasov who talked about Berkovtis' pure spinor approach. As you might know, this is an attempt to combine the advantages of the Green-Schwarz and the Ramond-Neveu-Schwarz formalism for superstrings and gives a covariant formulation with manifest supersymmetry in the target (amongst other things, Lubos has talked about this before). This is done by including not only the X and theta coordinates of the target superspace but also having an additional spinor lambda which obeys the "pure spinor" constraints lambda gamma^i lambda = 0 for all i. You can convince yourself that this equation describes the cone over SO(10)/U(5). This space has a conical singularity at the origin and Nikita asked the question if this can really give a consistent quantization. In particular, the beta-gamma-ghosts for the spinors have to be well defined not only in a patch but globally.
Nikita argued (showing quite explicitly how Cech-Cohomology arises) that this requires the first two Chern classes to vanish. He first showed how not to and then how to properly resolve the singularity of the cone and concluded that in the end, the pure spinor quantization is in fact consistent. However (and unfortunately my notes are cryptic at that point) he mentioned that there are still open problems when you try do use this approach for worldsheet of genus larger than two. Thus, even in this approach there might still be technical difficulties to define string amplitudes beyond two loops.
The next speaker was Roberto Longo. He is one of the big shots in the algebraic approach to quantum field theory and he talked about 2D conformal theories. As you know, the algebraists start from a mathematical definition of a quantum field theory (a Haag-Kastler net which is a refinement of the Wightman axioms) and then deduce general theorems with proofs (of mathematical standard) valid for large classes of QFTs. The problem however is to give examples of theories that can be shown to obey their definition. Free fields do but are a bit boring after a while. And perturbative descriptions on terms of Feynman rules are no good as long as the expansion can be shown to converge (which is probably wrong). You could use the lattice regularization to get a handle on gauge theories but there you have to show (and this hasn't been done despite decades of attempts in the constructive field theory community) Lorentz invariance, positivity of the spectrum and locality, all after the continuum limit has been taken. So you have a nice framework but you are not sure what theories it applies to (although there is little doubt that asymptotically free gauge theories should be in that class). Now Longo reviewed how you can cast the usual language of 2d CFTs into their language and thus have additional, interacting examples. He displayed several theorems that however sounded vaguely familiar to people that have some background in the BPZ approach to CFTs.
The last speaker of Thursday was Nikolai Reshetikhin. He started out with a combinatorial problem of certain graphs with two coloured vertices, transformed that into a dimer model and ended up getting a discrete version of a Dirac operator on graphs (in the sense that the adjacency matrix can give the Laplacian). He also mentioned a related version of Boson-Fermion-correspondence and a relation to the quantum foam of Vafa and collaborators but again my notes are too spares to be of any more use there.
Friday morning started with Philippe Di Francesco. He started out with a combinatorial problem again: Count 4-valent planar graphs with two external edges. He transformed this to rooted ternary trees with black and white leaves and always one more black than white leaves. This could be solved by giving a solvable recursion relation for the generating function. The next question was how many edges (in the first graph) have to be transversed to get from the outside to the face with the other external edge. Again there was a (now slightly more involved) generating function which he again solved and showed that the solution can be thought of as a one soliton solution in terms of a tau function.
After that, he talked about the six-vertex-model and treated it with similar means, showed a beautiful movie of how the transfer matrix acts and suddenly was right in the middle of Peron-Frobenius eigenvectors, Temperley-Lieb algebras and Yang-Baxter equation. Amazing!
Then came Tudor Ratiu who gave quite a dramatic presentation but I have to admit I did not get much out of it. It was on doing the symplectic treatment of symmetries in the infinite dimensional case and how to deal with the functional analysis issues coming up there (in general what would be a Hamiltonian vector field is not a vector field etc.)
John Cardy discussed Stochastic Loewner Evolution: Take as an example the 2D Ising model on a hexagonal lattice and instead of the spins view the phase boundary as your fundamental observable. Then you can ask about its statistics and again in the continuum limit this should be given in terms of a conformal field theory. He focussed on a phase boundary that runs from boundary to boundary. The trick is to parametrise it by t and consider it only up to a certain t1. If the domain before was the disk it is now a disk with a path that wiggles from the boundary somewhere into the interior. By the uniformisation theorem there is a function that maps the complement of the path again onto the unit disk, call it g_t1. Instead of looking at the propagation of the path you can ask how g_t1 varies if you change t1. Cardy derived a differential equation for g_t1 and argued that all the information about the CFT is encoded in the solution to this equation with the appropriate boundary conditions.
The afternoon was started by Robbert Dijkgraaf. He reviewed the connection of black hole entropy (including quantum corrections as computed by Cardoso, de Wit and Mohaupt) and wave functions in topological string theory. He did not give much details (which was good given the broad audience) but one thing I had not yet heard about is to understand why the entropy (after the Legendre transform to electric charges and chemical potential that Vafa and friends discovered to simplify the CdEM result) has to be treated like a wave function while the topological string partition function appears like a probability. Dijkgraaf proposed that the fact that Omega, the holomorphic volume form varies over a SLAG in the complex structure moduli space could be a key to understand this as a Lagrangian submanifold is exactly where a wave function lives after quantization (it only depends on position and not on momenta!). Furthermore, he displayed the diagram for open-closed string duality that can be viewed as a loop of an open string stretched between to D-branes or the D-branes exchanging a closed string at tree level. He interpreted this as an index theorem: The open string loop looked like Tr((-1)^F D-slash) with trace for the loop while the closed string side is given by the integral over ch(E1) ch(E2) A-roof(R) where E1/2 are bundles on the D-brane. He argued that the right hand side looked like scalar product with the Chern classes as wave function and the A-roof-genus as measure. He went on discussing YM on a 2-torus and free fermions (via the eigenvalues of the holonomy). Those are non-relativistic and thus have two Fermi surfaces one for each sign of the square root in the dispersion relation. Perturbations are then about creating holes in these Fermi surfaces and 1/N (N being interpreted as the difference between the two surfaces) effects appear when a hole makes it through the energy band to the other Fermi surface. This again can be computed via a recursion relation and Dijkgraaf ended by interpreting it as being about a grand canonical ensemble of multi black holes rather than a single one.
Then came Bob Wald who reviewed thirty years of quantum field theory on curved backgrounds. If you leave Minkowsky space you have to give up many things that are quite helpful in the flat space approach: Poincare invariance, a preferred vacuum state, the notion of particles (as irreps of the Poincare group), a Fourier transform to momentum space, Wick rotation, the S-Matrix. Wald gave an overview of how people have learnt to deal with these difficulties and which more general concepts replace the flat space one. In the morning, the lecture room was quite cool and more and more people put on their coats. In contrast in the afternoon, the heating worked properly however at the expense of higher levels of carbon dioxide that in my case overcame the effects of lots of coffee from the coffee breaks. So for this lecture I cannot tell you anymore.
Last speaker before the banquet was Sasha Zamolodchikov. He again admitted to mainly live in two dimensions and discussed behaviour of the mass gap and free energy close to criticality. Those are dominated by the most relevant operator perturbing the CFT and usually are well understood. He however wanted to understand the sub-leading contributions and gave a very general argument (which I am unfortunately unable to reproduce) of why the expectation value of the L_(-2) L-bar_(-1) descendant of the vacuum (which is responsible for these sub-leading effects) is given by the energy density.
The last day started out (half an hour later as Friday as I only found out by being the only one at the lecture hall) with Martin Zirnbauer. As he mentioned many different systems (atomic nuclei, disordered metallic grains, chaotic billiards, microwaves in a cavity, acoustic modes of vibration of solids, quarks in non-abelian gauge theory (?) and the zeros of the Riemann zeta function) show similar spectral behaviour: When you plot the histogram of energy differences between levels you do not get a Poisson distribution as you would get if the energy levels are just random but a curve that starts of with a power law and later decays exponentially. There are three different power laws and the universality classes are represented by Gaussian matrix models with either hermitian, real symmetric or quaternion self-dual matrices. This has been well known for decades. Zirnbauer now argued that you will get 11 classes if you allow for super-matrices. He mentioned a theorem of his that showed that any Hamiltonian quadratic in fermionic creation and annihilation operators is in one of those classes (although I did not understand the relevance of this result for what he discussed before). He went on and claimed (again not convincing to me) that the physics of the earlier systems would be described by a non-linear sigma model with these 11 supermatrix spaces as targets. He called all this supersymmetry but to me it sounded as at best this was about systems with both bosons and fermions. In the discussion he had to admit that although he has supergroups, the Hamiltonian is not an element of these and thus the crucial relation H={Q,Q} that gives us all the nice properties of really supersymmetric theories does not hold in his case.
Then came Matthias Staudacher who gave a nice presentation of integrability properties in the AdS/CFT correspondence in particular in spin chains and rotating strings. Most of this we have heard already several times but new to me was the detailed description of how the (generalised) Bethe ansatz arises. As you know, the results about spin-chains and strings do not agree anymore at the three loop level. This is surprising as they agreed up to two loops but on the other hand you are doing different expansions in the two cases so this does not mean that the AdS/CFT correspondence is in trouble. This is pretty much like the situation in M(atrix)-model vs. supergravity. There are certain amplitudes that work (probably those protected by susy) and certain more complicated ones that do not. Matthias summarised this by making the statement "Who cares about the correspondence if you have integrability?"
The conference was rounded off by Nigel Hitchin who gave an overview of generalised geometry. Most of this is beautifully explained in Gualtieri's thesis, but there are a few points to note: Hitchin only talked about generalised metrics (given in terms of generalisations of the graph of g in TM+T^*M he did not mention generalised complex structure (except than in the Q&A period). He showed how to write the Levi-Civita connection (well, with torsion given by +/- H) in terms of the Lie- and the Courrant-bracket and the generalised metric (actually g+/-B) given in terms of maximal rank isotropic subbundles. What was new to me was how to carry out generalised Hamiltonian reduction of a group action (which he said was related to gauged WZW-models): The important step is to lift the Hamilton vector field X to X + xi_a where a labels the coordinate patch under consideration. It is important that under changes of coordinates xi changes as xi_b - xi_a = i_X dA_ab where A_ab is the 1-form that translates the two B-fields B_a and B_b. Then one can define L_X (Y+eta_a) = Lie_X (Y+eta_a) -i_Y dxi_a in terms of the Lie derivative Lie. This is globally defined as it works across patches. Now if you have a symmetry, take K to be the bundle of its Hamilton vector fields and K-perp its orthogonal bundle (containing K). Then what you want is the bundle E-bar = (K-perp / K)/G. You have the exact sequence 0->T*(M/G)->E-bar->T(M/G)->0 with non-degenerate inner product and the Courrant bracket descends nicely but it is not naturally a direct sum. Furthermore, you can define the 'moment form' c = i_X B_a - xi_a which makes sense globally. We have dc = i_X H and on the quotient g(X,Y) = i_Y c. Note that even when dB=0 on M before, it we can have H non-vanishing in cohomology on M/G because the horizontal vector bundle can have a curvature F and in fact downstairs one computes H=cF. Again, as always in this generalised geometry context, I find this extremely beautiful!
Update: After arriving at IUB, I see that Urs has reported from Nikita's talk.
Update: Giuseppe Policastro has pointed out a couple of typos that I corrected.
As I reported earlier, this was a conference with an exceptionally well selected program. Not that all talks were in exactly on topics that I think about day and night but with very few exceptions, the speakers had something interesting to say and found good ways to present it. Well done, organisers! I hope your centre will be as successful as this colloquium!
The first physics talk on Thursday was Nikita Nekrasov who talked about Berkovtis' pure spinor approach. As you might know, this is an attempt to combine the advantages of the Green-Schwarz and the Ramond-Neveu-Schwarz formalism for superstrings and gives a covariant formulation with manifest supersymmetry in the target (amongst other things, Lubos has talked about this before). This is done by including not only the X and theta coordinates of the target superspace but also having an additional spinor lambda which obeys the "pure spinor" constraints lambda gamma^i lambda = 0 for all i. You can convince yourself that this equation describes the cone over SO(10)/U(5). This space has a conical singularity at the origin and Nikita asked the question if this can really give a consistent quantization. In particular, the beta-gamma-ghosts for the spinors have to be well defined not only in a patch but globally.
Nikita argued (showing quite explicitly how Cech-Cohomology arises) that this requires the first two Chern classes to vanish. He first showed how not to and then how to properly resolve the singularity of the cone and concluded that in the end, the pure spinor quantization is in fact consistent. However (and unfortunately my notes are cryptic at that point) he mentioned that there are still open problems when you try do use this approach for worldsheet of genus larger than two. Thus, even in this approach there might still be technical difficulties to define string amplitudes beyond two loops.
The next speaker was Roberto Longo. He is one of the big shots in the algebraic approach to quantum field theory and he talked about 2D conformal theories. As you know, the algebraists start from a mathematical definition of a quantum field theory (a Haag-Kastler net which is a refinement of the Wightman axioms) and then deduce general theorems with proofs (of mathematical standard) valid for large classes of QFTs. The problem however is to give examples of theories that can be shown to obey their definition. Free fields do but are a bit boring after a while. And perturbative descriptions on terms of Feynman rules are no good as long as the expansion can be shown to converge (which is probably wrong). You could use the lattice regularization to get a handle on gauge theories but there you have to show (and this hasn't been done despite decades of attempts in the constructive field theory community) Lorentz invariance, positivity of the spectrum and locality, all after the continuum limit has been taken. So you have a nice framework but you are not sure what theories it applies to (although there is little doubt that asymptotically free gauge theories should be in that class). Now Longo reviewed how you can cast the usual language of 2d CFTs into their language and thus have additional, interacting examples. He displayed several theorems that however sounded vaguely familiar to people that have some background in the BPZ approach to CFTs.
The last speaker of Thursday was Nikolai Reshetikhin. He started out with a combinatorial problem of certain graphs with two coloured vertices, transformed that into a dimer model and ended up getting a discrete version of a Dirac operator on graphs (in the sense that the adjacency matrix can give the Laplacian). He also mentioned a related version of Boson-Fermion-correspondence and a relation to the quantum foam of Vafa and collaborators but again my notes are too spares to be of any more use there.
Friday morning started with Philippe Di Francesco. He started out with a combinatorial problem again: Count 4-valent planar graphs with two external edges. He transformed this to rooted ternary trees with black and white leaves and always one more black than white leaves. This could be solved by giving a solvable recursion relation for the generating function. The next question was how many edges (in the first graph) have to be transversed to get from the outside to the face with the other external edge. Again there was a (now slightly more involved) generating function which he again solved and showed that the solution can be thought of as a one soliton solution in terms of a tau function.
After that, he talked about the six-vertex-model and treated it with similar means, showed a beautiful movie of how the transfer matrix acts and suddenly was right in the middle of Peron-Frobenius eigenvectors, Temperley-Lieb algebras and Yang-Baxter equation. Amazing!
Then came Tudor Ratiu who gave quite a dramatic presentation but I have to admit I did not get much out of it. It was on doing the symplectic treatment of symmetries in the infinite dimensional case and how to deal with the functional analysis issues coming up there (in general what would be a Hamiltonian vector field is not a vector field etc.)
John Cardy discussed Stochastic Loewner Evolution: Take as an example the 2D Ising model on a hexagonal lattice and instead of the spins view the phase boundary as your fundamental observable. Then you can ask about its statistics and again in the continuum limit this should be given in terms of a conformal field theory. He focussed on a phase boundary that runs from boundary to boundary. The trick is to parametrise it by t and consider it only up to a certain t1. If the domain before was the disk it is now a disk with a path that wiggles from the boundary somewhere into the interior. By the uniformisation theorem there is a function that maps the complement of the path again onto the unit disk, call it g_t1. Instead of looking at the propagation of the path you can ask how g_t1 varies if you change t1. Cardy derived a differential equation for g_t1 and argued that all the information about the CFT is encoded in the solution to this equation with the appropriate boundary conditions.
The afternoon was started by Robbert Dijkgraaf. He reviewed the connection of black hole entropy (including quantum corrections as computed by Cardoso, de Wit and Mohaupt) and wave functions in topological string theory. He did not give much details (which was good given the broad audience) but one thing I had not yet heard about is to understand why the entropy (after the Legendre transform to electric charges and chemical potential that Vafa and friends discovered to simplify the CdEM result) has to be treated like a wave function while the topological string partition function appears like a probability. Dijkgraaf proposed that the fact that Omega, the holomorphic volume form varies over a SLAG in the complex structure moduli space could be a key to understand this as a Lagrangian submanifold is exactly where a wave function lives after quantization (it only depends on position and not on momenta!). Furthermore, he displayed the diagram for open-closed string duality that can be viewed as a loop of an open string stretched between to D-branes or the D-branes exchanging a closed string at tree level. He interpreted this as an index theorem: The open string loop looked like Tr((-1)^F D-slash) with trace for the loop while the closed string side is given by the integral over ch(E1) ch(E2) A-roof(R) where E1/2 are bundles on the D-brane. He argued that the right hand side looked like scalar product
Then came Bob Wald who reviewed thirty years of quantum field theory on curved backgrounds. If you leave Minkowsky space you have to give up many things that are quite helpful in the flat space approach: Poincare invariance, a preferred vacuum state, the notion of particles (as irreps of the Poincare group), a Fourier transform to momentum space, Wick rotation, the S-Matrix. Wald gave an overview of how people have learnt to deal with these difficulties and which more general concepts replace the flat space one. In the morning, the lecture room was quite cool and more and more people put on their coats. In contrast in the afternoon, the heating worked properly however at the expense of higher levels of carbon dioxide that in my case overcame the effects of lots of coffee from the coffee breaks. So for this lecture I cannot tell you anymore.
Last speaker before the banquet was Sasha Zamolodchikov. He again admitted to mainly live in two dimensions and discussed behaviour of the mass gap and free energy close to criticality. Those are dominated by the most relevant operator perturbing the CFT and usually are well understood. He however wanted to understand the sub-leading contributions and gave a very general argument (which I am unfortunately unable to reproduce) of why the expectation value of the L_(-2) L-bar_(-1) descendant of the vacuum (which is responsible for these sub-leading effects) is given by the energy density.
The last day started out (half an hour later as Friday as I only found out by being the only one at the lecture hall) with Martin Zirnbauer. As he mentioned many different systems (atomic nuclei, disordered metallic grains, chaotic billiards, microwaves in a cavity, acoustic modes of vibration of solids, quarks in non-abelian gauge theory (?) and the zeros of the Riemann zeta function) show similar spectral behaviour: When you plot the histogram of energy differences between levels you do not get a Poisson distribution as you would get if the energy levels are just random but a curve that starts of with a power law and later decays exponentially. There are three different power laws and the universality classes are represented by Gaussian matrix models with either hermitian, real symmetric or quaternion self-dual matrices. This has been well known for decades. Zirnbauer now argued that you will get 11 classes if you allow for super-matrices. He mentioned a theorem of his that showed that any Hamiltonian quadratic in fermionic creation and annihilation operators is in one of those classes (although I did not understand the relevance of this result for what he discussed before). He went on and claimed (again not convincing to me) that the physics of the earlier systems would be described by a non-linear sigma model with these 11 supermatrix spaces as targets. He called all this supersymmetry but to me it sounded as at best this was about systems with both bosons and fermions. In the discussion he had to admit that although he has supergroups, the Hamiltonian is not an element of these and thus the crucial relation H={Q,Q} that gives us all the nice properties of really supersymmetric theories does not hold in his case.
Then came Matthias Staudacher who gave a nice presentation of integrability properties in the AdS/CFT correspondence in particular in spin chains and rotating strings. Most of this we have heard already several times but new to me was the detailed description of how the (generalised) Bethe ansatz arises. As you know, the results about spin-chains and strings do not agree anymore at the three loop level. This is surprising as they agreed up to two loops but on the other hand you are doing different expansions in the two cases so this does not mean that the AdS/CFT correspondence is in trouble. This is pretty much like the situation in M(atrix)-model vs. supergravity. There are certain amplitudes that work (probably those protected by susy) and certain more complicated ones that do not. Matthias summarised this by making the statement "Who cares about the correspondence if you have integrability?"
The conference was rounded off by Nigel Hitchin who gave an overview of generalised geometry. Most of this is beautifully explained in Gualtieri's thesis, but there are a few points to note: Hitchin only talked about generalised metrics (given in terms of generalisations of the graph of g in TM+T^*M he did not mention generalised complex structure (except than in the Q&A period). He showed how to write the Levi-Civita connection (well, with torsion given by +/- H) in terms of the Lie- and the Courrant-bracket and the generalised metric (actually g+/-B) given in terms of maximal rank isotropic subbundles. What was new to me was how to carry out generalised Hamiltonian reduction of a group action (which he said was related to gauged WZW-models): The important step is to lift the Hamilton vector field X to X + xi_a where a labels the coordinate patch under consideration. It is important that under changes of coordinates xi changes as xi_b - xi_a = i_X dA_ab where A_ab is the 1-form that translates the two B-fields B_a and B_b. Then one can define L_X (Y+eta_a) = Lie_X (Y+eta_a) -i_Y dxi_a in terms of the Lie derivative Lie. This is globally defined as it works across patches. Now if you have a symmetry, take K to be the bundle of its Hamilton vector fields and K-perp its orthogonal bundle (containing K). Then what you want is the bundle E-bar = (K-perp / K)/G. You have the exact sequence 0->T*(M/G)->E-bar->T(M/G)->0 with non-degenerate inner product and the Courrant bracket descends nicely but it is not naturally a direct sum. Furthermore, you can define the 'moment form' c = i_X B_a - xi_a which makes sense globally. We have dc = i_X H and on the quotient g(X,Y) = i_Y c. Note that even when dB=0 on M before, it we can have H non-vanishing in cohomology on M/G because the horizontal vector bundle can have a curvature F and in fact downstairs one computes H=cF. Again, as always in this generalised geometry context, I find this extremely beautiful!
Update: After arriving at IUB, I see that Urs has reported from Nikita's talk.
Update: Giuseppe Policastro has pointed out a couple of typos that I corrected.
Friday, October 21, 2005
No news is good news
You might have noticed that I haven't reported from the Hamburg opening colloquium, yet. First of all this is due to the fact that there is no wlan in the lecture hall (or at least no wlan that I can log into) and second, and that really is the good news, that the talks are so interesting that I am far too busy listening and taking notes to turn on my laptop. The organisers really have done a great job in selecting not only prominent speakers but at the same time people who know how to give good talks. Thanks a million!
However, you my dear readers will have to wait until Monday for me to give you a conference summary.
However, you my dear readers will have to wait until Monday for me to give you a conference summary.
Wednesday, October 19, 2005
More conference reporting
I just found that the weekly quality paper "Die Zeit" has an interview with Smolin on the occation of Loops '05. Probably no need to learn German for this, nothing new: String theory doesn't predict anything because there are 10^500 String theories (they lost the ^ somewhere), Peter W. can tell you more about this, stringy people have lost contact to experiment, LQG people do better because they predict a violation of the relativistic dispersion relation for light (is this due to the 3+1 split of their canonical formalism?) and Einstein would have been suppressed today because he was an independant thinker and not part of the (quantum mechanics) mainstream.
I was told, "Frankfurter Allgemeine Sonntagszeitung" also had a report on Loops '05. On their webpage, the article costs 1.50 Euros and I am reluctant to pay this. Maybe one of my readers has a copy and can post/fax it to me?
Tomorrow, I will be going to Hamburg where for three days they are celebrating the opening of the centre for mathematical physics. This is a joint efford of people from the physics (Louis, Fredenhagen sen., Samtleben, Kuckert) and math (Schweigert) departments of Hamburg university and the DESY theory group (Schomerus, Teschner). This is only one hour away and I am really looking forward to having a stringy critical mass coming together in northern Germany. Speakers of the opening colloquium include Dijkgraaf (hopefully he will make it this time), Hitchin, Zamolodchikov, Nekrassov, Cardy and others.
If there is some reasonable network connection, there will be some more live blogging, Urs in now a postdoc in Christoph Schweigert's group, I assume he will be online as well.
I was told, "Frankfurter Allgemeine Sonntagszeitung" also had a report on Loops '05. On their webpage, the article costs 1.50 Euros and I am reluctant to pay this. Maybe one of my readers has a copy and can post/fax it to me?
Tomorrow, I will be going to Hamburg where for three days they are celebrating the opening of the centre for mathematical physics. This is a joint efford of people from the physics (Louis, Fredenhagen sen., Samtleben, Kuckert) and math (Schweigert) departments of Hamburg university and the DESY theory group (Schomerus, Teschner). This is only one hour away and I am really looking forward to having a stringy critical mass coming together in northern Germany. Speakers of the opening colloquium include Dijkgraaf (hopefully he will make it this time), Hitchin, Zamolodchikov, Nekrassov, Cardy and others.
If there is some reasonable network connection, there will be some more live blogging, Urs in now a postdoc in Christoph Schweigert's group, I assume he will be online as well.
Monday, October 17, 2005
Classical limit of mathematics
The most interesting periodic event at IUB is the mathematics colloquium as the physicists don't manage to get enough people together for a regular series. Today, we had G. Litvinov who introduced us to idempotent mathematics. The idea is to build upon the group homomorphism x-> h ln(x) for some positive number h that maps the positive reals and multiplication to the reals with addition.
So we can call addition in R "multiplication" in terms of the preimage and we can also define "addition" in terms of the pre-image. The interesting thing is what becomes of this when we take the "classical limit" of h->0: Then "addition" is nothing but the maximim and this "addition" is idempotent: a "+" a = a.
This is an example of an idempotent semiring and in fact it is the generic one: Besides idempotency, it satisfies many of the usual laws: associativity, distributional law, commutativity. Thus you can carry over much of the usual stuff you can do with fields to this extreme limit. Other examples of this structure are Boolean algebras or compact convex sets where "multiplication" is the usual sum of sets and "addition" is the convex hull (obviously, the above example is a special case). Another example are polynomials with non-negative coefficients and for these the degree turns out to be a homomorphism! The obvious generalization of the integral is the supremum and the Fourier transform becomes the Legendre transform (you have to work out what the characters of the addition are!).
This theory has many applications, it seems especially strong for optimization problems. But you can also apply this limiting procedure to algebraic varieties under which they are turned into Newton polytopes.
I enjoyed this seminar especially because it made clear that many constructions can be thought of extreme limits of some even more common, linear constructions.
But now for something completely different: When I came back to my computer, I had received the following email:
I have no idea what he is talking about but maybe one of my readers has. I googled for a passage from the question and found that exactly the same question has also been posted in the comment sections of the Coffee Table and Lubos's blog.
So we can call addition in R "multiplication" in terms of the preimage and we can also define "addition" in terms of the pre-image. The interesting thing is what becomes of this when we take the "classical limit" of h->0: Then "addition" is nothing but the maximim and this "addition" is idempotent: a "+" a = a.
This is an example of an idempotent semiring and in fact it is the generic one: Besides idempotency, it satisfies many of the usual laws: associativity, distributional law, commutativity. Thus you can carry over much of the usual stuff you can do with fields to this extreme limit. Other examples of this structure are Boolean algebras or compact convex sets where "multiplication" is the usual sum of sets and "addition" is the convex hull (obviously, the above example is a special case). Another example are polynomials with non-negative coefficients and for these the degree turns out to be a homomorphism! The obvious generalization of the integral is the supremum and the Fourier transform becomes the Legendre transform (you have to work out what the characters of the addition are!).
This theory has many applications, it seems especially strong for optimization problems. But you can also apply this limiting procedure to algebraic varieties under which they are turned into Newton polytopes.
I enjoyed this seminar especially because it made clear that many constructions can be thought of extreme limits of some even more common, linear constructions.
But now for something completely different: When I came back to my computer, I had received the following email:
Dear Mr. Helling
I would greatly appreciate your response.
Please what is interrelation mutually
fractal attractor of the black hole condensation,
Bott spectrum of the homotopy groups
and moduli space of the nonassociative geometry?
Thank you very much obliged.
[Sender]
I have no idea what he is talking about but maybe one of my readers has. I googled for a passage from the question and found that exactly the same question has also been posted in the comment sections of the Coffee Table and Lubos's blog.
Thursday, October 13, 2005
How to read blogs and how not to write blogs
I usually do not have articles here saying basically "check out these articles in other blogs I liked them". Basically this is because I think if you, dear reader, have found this blog you will be able to find others that you like as well, so no need for me to point you around the blogsphere. And , honestly, I don't think too many people read this blog, anyway. I don't have access to the server's log files and I do not have a counter (I must say, I hate counters because often they delay loading a page a lot). But it happens more and more often that I meet somebody in person and she/he tells me that she/he has read this or that in atdotde. So in the end, I might not write for the big bit bucket.
My reporting on Loops '05 was picked up in other places so that might have brought even more reader to my little place. I even got an email from a student in China asking me that he cannot read atdotde anymore (as well as for example Lubos' Reference Frame). Unfortunatly, I had to tell him that this was probably due to the fact that his government decided to block blogger.com from the Chinese part of the net because blogs are way to subversive.
So as a little service for some of my readers who not already now, here is a hint on how to read blogs: Of course you can if you have some minutes of boredom type your friends names into google and surf to their blogs every now and then. That is fine. Maybe at some point, you want to find out, what's new in all those blogs. So you go through your bookmarks (favourites in MS speak) and check I you've seen everything that you find there.
But that is cyber stone age! What you want is a "news aggregator". This is a little program that does this for you periodically and informs you about the new articles it found. You just have to tell it where to look. This comes in form of a URL called the "RSS feed". Often you find little icons in the sidebar of the blogs that link to that URL. In others like this you have to guess. All the blogs on blogger.com it is in the form URL_of_blog/atom.xml so for atdotde it is http://atdotde.blogger.com/atom.xml. You have to tell your news aggregator about this URL. In the simplest form, this is just your web browser. Firefox calls it "live bookmarks": You open the "manage bookmarks" window and select "new life bookmark" from the menu. I use an aggregator called liferea, that even opens a little window once it found anything new, but there are many others.
Coming back to the theme of the beginning, I will for once tell you which blogs I monitor (in no particular order):
Amelies Welt(in German) I know Amelie from a mailing list an like the variety of topics she writes about.
BildBlog They set straight the 'news' from the biggest German tabloid. And it's funny.
Bitch, PhD Academic and feminist blogger. I learned a lot.
String Coffee Table you can chat about strings while nipping a coffee.
Musings The first blog of a physicist I came across. Jacques still sets standards.
Die Schreibmaschine Anna is a friend from Cambridge, currently not very active because...
Broken Ankle Diary a few weeks ago she broke here ankle
Lubos Motl's Reference Frame Strong opinions on physics , global warming, politics.
hep-th The arxiv now also comes in this form but still I prefer to read it in the classic way.
Jochen Weller One of the Quantum Diaries. Jochen was in Cambrigde while I was there.
Preposterous Universe Sean's blog is dead now because he is part of...
Cosmic Variance Currently my best loved blog. Physics and everything else.
Not Even Wrong Peter Woit has one point of criticism of string theory that he keeps repeating. But he is a very reasonable guy.
Daily ACK I met Al on a dive trip in the English Channel. Some astronomy and some Apple and Perl news.
Physics Comments Sounded like a good idea but not really working at least in the hep-th area.
Have fun!
Now I should send trackback pings. This is such a pain with blogger.com...
Ah, I nearly forgot: This article and how academic blogs can hurt your job hunting scares me a lot! (I admit, I found it on Cosmic Variance.)
My reporting on Loops '05 was picked up in other places so that might have brought even more reader to my little place. I even got an email from a student in China asking me that he cannot read atdotde anymore (as well as for example Lubos' Reference Frame). Unfortunatly, I had to tell him that this was probably due to the fact that his government decided to block blogger.com from the Chinese part of the net because blogs are way to subversive.
So as a little service for some of my readers who not already now, here is a hint on how to read blogs: Of course you can if you have some minutes of boredom type your friends names into google and surf to their blogs every now and then. That is fine. Maybe at some point, you want to find out, what's new in all those blogs. So you go through your bookmarks (favourites in MS speak) and check I you've seen everything that you find there.
But that is cyber stone age! What you want is a "news aggregator". This is a little program that does this for you periodically and informs you about the new articles it found. You just have to tell it where to look. This comes in form of a URL called the "RSS feed". Often you find little icons in the sidebar of the blogs that link to that URL. In others like this you have to guess. All the blogs on blogger.com it is in the form URL_of_blog/atom.xml so for atdotde it is http://atdotde.blogger.com/atom.xml. You have to tell your news aggregator about this URL. In the simplest form, this is just your web browser. Firefox calls it "live bookmarks": You open the "manage bookmarks" window and select "new life bookmark" from the menu. I use an aggregator called liferea, that even opens a little window once it found anything new, but there are many others.
Coming back to the theme of the beginning, I will for once tell you which blogs I monitor (in no particular order):
Have fun!
Now I should send trackback pings. This is such a pain with blogger.com...
Ah, I nearly forgot: This article and how academic blogs can hurt your job hunting scares me a lot! (I admit, I found it on Cosmic Variance.)
Tuesday, October 11, 2005
IUB is noble
Coming back from Loops '05 I find a note in my mailbox that the International University Bremen has now an Ig-Noble Laurate amongst its faculty: V. Benno Meyer-Rochow has received the prize in fluid dynamics for his work on the pressure produced when penguins pooh.
More news on the others
9:30
The careful reader will have noticed that yesterday, my blogging got sparser and sparser. This was probably due to increasing boreodom/annoyance on my part. Often, I thought the organisers should have applied the charta of sci.physics.research that forbits contributions that are so vague and speculative that they are not even wrong. I could not stand it anymore and had to leave the room when the speaker (I am not going to say who it was) claimed that "the big bang is just a phase transition".
Today, I give it a new shot. And the plenary talks are promising. Currently, John Baez has been giving a nice overview on various incarnations of spin foam models (he listed Feynman diagrams, lattice gauge theory and topological strings among them although I am under the impression that in the last point he is misguided as topological strings in fact take into account the complex/holomorphic structure of the background). However, starting from the point "what kind of matter do we have to include to have a nice continuum limit" he digressed via a Witten anecdote (who answered the question if he thinks LQG is right said that he hoped not because he hoped (in the 90s) that there is only one unique matter content (ie strings) consistend with quantised gravity) to making fun of string theorists asking them to make their homework to check the 10^whatever vacua in the landscape.
The next speaker will be Dijkgraaf who hopefully will do a better job than did Theissen yesterday in presenting that stringy people have interesting, deep stuff to say about physics.
Unfortunately, electro-magnetism lectures back in Bremen require me to run off at 11:00 and catch the train so I will not be able to follow the rest of the conference closely.
9:51
Baez got back on track with a nice discussion of how Lorentzlian Triangulation fit into the scheme of things and what role the prescribed time slicing might have on large scale physics (introducting further terms than Lambda and R in the LEEA). He showed also a supposed to be spin foam version of it.
9:56
Oh no. They have grad students and young postdocs as chairpersons. Bianca Dittrich just announced "Our next speaker is Robbert Dijkgraaf" and nothing happened. It seems Dijkgraaf didn't make it here on the early morning plane. Now, I can fulfil the annonymous reader's wish and report on the presentation of Laurent Friedel, the next speaker.
10:20
Before I power down my laptop: Friedel looks at effects of quantum gravity on low energy particle actions. In order to do that he couples matter to the Ponzano Regge model and then will probably try to integrate out the gravitational degrees of freedom.
The careful reader will have noticed that yesterday, my blogging got sparser and sparser. This was probably due to increasing boreodom/annoyance on my part. Often, I thought the organisers should have applied the charta of sci.physics.research that forbits contributions that are so vague and speculative that they are not even wrong. I could not stand it anymore and had to leave the room when the speaker (I am not going to say who it was) claimed that "the big bang is just a phase transition".
Today, I give it a new shot. And the plenary talks are promising. Currently, John Baez has been giving a nice overview on various incarnations of spin foam models (he listed Feynman diagrams, lattice gauge theory and topological strings among them although I am under the impression that in the last point he is misguided as topological strings in fact take into account the complex/holomorphic structure of the background). However, starting from the point "what kind of matter do we have to include to have a nice continuum limit" he digressed via a Witten anecdote (who answered the question if he thinks LQG is right said that he hoped not because he hoped (in the 90s) that there is only one unique matter content (ie strings) consistend with quantised gravity) to making fun of string theorists asking them to make their homework to check the 10^whatever vacua in the landscape.
The next speaker will be Dijkgraaf who hopefully will do a better job than did Theissen yesterday in presenting that stringy people have interesting, deep stuff to say about physics.
Unfortunately, electro-magnetism lectures back in Bremen require me to run off at 11:00 and catch the train so I will not be able to follow the rest of the conference closely.
9:51
Baez got back on track with a nice discussion of how Lorentzlian Triangulation fit into the scheme of things and what role the prescribed time slicing might have on large scale physics (introducting further terms than Lambda and R in the LEEA). He showed also a supposed to be spin foam version of it.
9:56
Oh no. They have grad students and young postdocs as chairpersons. Bianca Dittrich just announced "Our next speaker is Robbert Dijkgraaf" and nothing happened. It seems Dijkgraaf didn't make it here on the early morning plane. Now, I can fulfil the annonymous reader's wish and report on the presentation of Laurent Friedel, the next speaker.
10:20
Before I power down my laptop: Friedel looks at effects of quantum gravity on low energy particle actions. In order to do that he couples matter to the Ponzano Regge model and then will probably try to integrate out the gravitational degrees of freedom.
Monday, October 10, 2005
The Others
I sneaked into the Loops '05 Conference at the AEI at Potsdam. So, I will be able to give you live blogging for today and tomorrow. After some remarks by Nicolai and Thiemann and the usual impedence mismatch between laptops and projectors, Carlo Rovelli has started the first talk. He is on slide 2, and still reviews recent and not so recent devellopments of LQG.
9:55
Rovelli talked about his paper on the graviton propagator. If you like he wants to recover Newton's law from his model. The obvious problem of course is that any propagator g(x,y) cannot depend on x or y if everything is diffeomorphism invariant (at least in these people's logic). So he had to include also a dependence on a box around the 'detector' and introduce the metric on the box as boundary values. He seems to get out of this problem by in fact using a relational notion as you would of course have to in any interesting background independent theory (measure not with respect to coordinates but with respect to physical rulers). Then there was a technical part which I didn't quite get and in the end he had something like g(x,y)=1/|x-y|^2 on his slide. This could be interesting. I will download the paper and read it on the train.
11:15
Next is Smolin. Again computer problems, this time causing an unscheduled coffee break. Smolin started out talking about problems of bckground independent approaches including unification and the nature of anomalies. Then, however, he decided to focus on another one: How does macroscopic causality arise? He doesn't really know, but looked at some simple models where macro causality is usually destroyed my some non-local edges (like in a small world network). Surprisingly, he claims, these non-local connection do not change macroscopic physics (critical behaviour) a lot and thus they are not really detectable.
Even more, these non-local "defects" could, according to Smolin, play the role of matter. Then he showed another model where instead of a spin network, the physics is in twisted braided ribbon graphs. There, he called some configurations "quarks" and asigned the usual quantum numbers and ribbon transformations for C, P and T. Then it got even better, next slides mentioned the problem of small power in low l modes in the CMB ("scales larger than 1/Lambda"), the Poineer anomly and the Tully Fisher relation that is the empirical observation behind MOND. I have no idea what his theory as to do with all these fancy open probelms. Stefan Theissen next to me makes interesting noises of astonishment.
12:15
Next speaker is John Barrett. This talk sounds very solid. He presents a 3+0 dimensional model which to me looks much like a variant of a spin network (a graph with spin labels and certain weight factors for vertices, links, and tetrahedra). He can do Feynman graph like calculations in this model. Further plus: A native speaker of British English.
13:00
Last speaker of the forenoon is Stefan Theissen. He tries to explain how gravity arises from string theory to the LQG crowd. Many have left before he started and so far he has only presented string theory as one could have done this already 20 years ago: Einstein's equation as consistency requirement for the sigma model and scattering amplitudes producing the vertices of the Einstein Hilbert action. Solid but not really exciting.
14:55
In the afternoon, there are parallel sessions. I chose the "seminar room". Here, Markopoulou presents her idesa that dynamics in some (quantum gravity?) theory has formal similiarities to quantum information processing. In some Ising type model she looks at the block spin transformation and reformulates the fact that low energy fields only talk to the block spins and not to the high frequency fields. With some fancy mathematical machinery, she relates this to error correction where the high frequency fields play the role of noise.
15:25
Next is Olaf Dreyer. Very strange. He proposes that quantum mechnics should be deterministic and non-linear. Most of what he says are philosophical statements (and I do by far not agree with all of them) but what seems to be at the core of it is that he does not want macroscopic states that are superpositions of elementary states. I thought that was solved by decoherence long ago...
15:35
At least Rovelli asks "[long pause] maybe I didn't understand it. you make very general statements. But where is the physics?"
15:56
The next speaker is Wang who expands a bit on what Smolin said in the morning. It's really about Small World Networks (TM). If you have such a network with gauge flux along the edges then in fact a non-local random link looks locally as a charged particle. This is just like in Wheeler's geometrodynamics. The bulk of the talk is about the Ising model on a lattice with a small number of additional random links. The upshot is that the critical temperature and the heat capacity as well as the correltations at criticality do not much depend on the existence of the additional random links.
16:25
Martinetti reminds us that time evolution might have a connection with temperature. Concretely, he wants to take the Tomito-Takesaki unitary evolution as time evolution and build a KMS-state out of it. There is a version of the Unruh effect in the language of KMS states and Martinetti works out the corretion to the Unruh temperature from the fact that the observer might have a finite life time. This correction turns out to be so small that by uncertainty, one would have to measure longer than the life time to detect the difference in tempaerture.
18:15
I stopped reporting on the afternoon talks as I did not get much out of those. Currently, Rüdiger Vaas, a science journalist, is the last speaker of the day. He at least admits that his talk is on philosophy rather than physics. His topic are the philosophical foundations of big bang physics.
9:55
Rovelli talked about his paper on the graviton propagator. If you like he wants to recover Newton's law from his model. The obvious problem of course is that any propagator g(x,y) cannot depend on x or y if everything is diffeomorphism invariant (at least in these people's logic). So he had to include also a dependence on a box around the 'detector' and introduce the metric on the box as boundary values. He seems to get out of this problem by in fact using a relational notion as you would of course have to in any interesting background independent theory (measure not with respect to coordinates but with respect to physical rulers). Then there was a technical part which I didn't quite get and in the end he had something like g(x,y)=1/|x-y|^2 on his slide. This could be interesting. I will download the paper and read it on the train.
11:15
Next is Smolin. Again computer problems, this time causing an unscheduled coffee break. Smolin started out talking about problems of bckground independent approaches including unification and the nature of anomalies. Then, however, he decided to focus on another one: How does macroscopic causality arise? He doesn't really know, but looked at some simple models where macro causality is usually destroyed my some non-local edges (like in a small world network). Surprisingly, he claims, these non-local connection do not change macroscopic physics (critical behaviour) a lot and thus they are not really detectable.
Even more, these non-local "defects" could, according to Smolin, play the role of matter. Then he showed another model where instead of a spin network, the physics is in twisted braided ribbon graphs. There, he called some configurations "quarks" and asigned the usual quantum numbers and ribbon transformations for C, P and T. Then it got even better, next slides mentioned the problem of small power in low l modes in the CMB ("scales larger than 1/Lambda"), the Poineer anomly and the Tully Fisher relation that is the empirical observation behind MOND. I have no idea what his theory as to do with all these fancy open probelms. Stefan Theissen next to me makes interesting noises of astonishment.
12:15
Next speaker is John Barrett. This talk sounds very solid. He presents a 3+0 dimensional model which to me looks much like a variant of a spin network (a graph with spin labels and certain weight factors for vertices, links, and tetrahedra). He can do Feynman graph like calculations in this model. Further plus: A native speaker of British English.
13:00
Last speaker of the forenoon is Stefan Theissen. He tries to explain how gravity arises from string theory to the LQG crowd. Many have left before he started and so far he has only presented string theory as one could have done this already 20 years ago: Einstein's equation as consistency requirement for the sigma model and scattering amplitudes producing the vertices of the Einstein Hilbert action. Solid but not really exciting.
14:55
In the afternoon, there are parallel sessions. I chose the "seminar room". Here, Markopoulou presents her idesa that dynamics in some (quantum gravity?) theory has formal similiarities to quantum information processing. In some Ising type model she looks at the block spin transformation and reformulates the fact that low energy fields only talk to the block spins and not to the high frequency fields. With some fancy mathematical machinery, she relates this to error correction where the high frequency fields play the role of noise.
15:25
Next is Olaf Dreyer. Very strange. He proposes that quantum mechnics should be deterministic and non-linear. Most of what he says are philosophical statements (and I do by far not agree with all of them) but what seems to be at the core of it is that he does not want macroscopic states that are superpositions of elementary states. I thought that was solved by decoherence long ago...
15:35
At least Rovelli asks "[long pause] maybe I didn't understand it. you make very general statements. But where is the physics?"
15:56
The next speaker is Wang who expands a bit on what Smolin said in the morning. It's really about Small World Networks (TM). If you have such a network with gauge flux along the edges then in fact a non-local random link looks locally as a charged particle. This is just like in Wheeler's geometrodynamics. The bulk of the talk is about the Ising model on a lattice with a small number of additional random links. The upshot is that the critical temperature and the heat capacity as well as the correltations at criticality do not much depend on the existence of the additional random links.
16:25
Martinetti reminds us that time evolution might have a connection with temperature. Concretely, he wants to take the Tomito-Takesaki unitary evolution as time evolution and build a KMS-state out of it. There is a version of the Unruh effect in the language of KMS states and Martinetti works out the corretion to the Unruh temperature from the fact that the observer might have a finite life time. This correction turns out to be so small that by uncertainty, one would have to measure longer than the life time to detect the difference in tempaerture.
18:15
I stopped reporting on the afternoon talks as I did not get much out of those. Currently, Rüdiger Vaas, a science journalist, is the last speaker of the day. He at least admits that his talk is on philosophy rather than physics. His topic are the philosophical foundations of big bang physics.
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