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.

## 8 comments:

"In the discussion he had to admit that although he has supergroups, the Hamiltonian is not an element of these"

That's a language issue which I have tried to discuss several times in the past, not with M. Zirnbauer himself, but with other members of that SFB on "Universality" which he is the head of. Following certain textbooks on "susy methods in statistical mechanics" some people in non-hep fields tend to use the word 'supersymmetry' whenever they encounter any graded vector space.

Mostly, all they do is to re-express matrix determinants as Berezin integrals over exponentials of bilinears in Grassmann variables. Hence their use of graded algebra is more like in Fadeev-Popov or BRST than in SUSY.

But there is a general confusion of terms. Because even when you are not dealing with susy proper, you may still work with honest supermanifolds.

"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)."

Is this right? So Feynmann graph expansion fits with the axioms as long as the expansion diverges??

Got me, one 'not' too much. Of course nothing that diverges fits the axioms. If your perturbative expansion does not converge it does not really define what you mean by your theory so there is nothing concrete that you could even envision to check the axioms agains.

Thank you for a very nice summary. It seems like a fun conference. Do you know if the talks will be available online?

The talk by Hitchin is already online.

I guess all speakers have been asked to provide transparancies, which should appear on that site.

I'd have one more comment.

Robert wrote:

"Nikita argued [...] that this requires the first two Chern classes to vanish."

I have begun looking at some literature on this, which in particular goes under the keyword "chiral deRham algebra" or "chiral deRham complex". This is a sheaf of conformal superalgebras on some variety X.

There is something closely related, namely a sheaf of graded vertex algebras, also called a "chiral structure sheaf".

While the chiral deRham complex exists for arbitrary X, the chiral structure sheaf is apparently well known to be obstructed by the second Chern class of X.

Moreover, it is apparently well known that the existence of a globally defined Virasoro field (stress-energy tensor) in this context is precisely obstructed by the first Chern class of X.

I don't know if Nikita Nekrasov is aware of this, but it sure seems as if he deals with a special case of this general theorem.

This is discussed for instance in

Gorbunov, Malikov, Schechtmann: "Gerbes of chiral differential operators I, II, III", math.AG/9906117, math.AG/0003170, math.AG/0005201 .

See right the first page of part I.

Giuseppe gave me the hint that Nikita's talk is closely related to Witten's latest paper which I started to read on the plane to Cambridge (I am currently in a cafe at Stansted airport).

Yes, see also Jacques' latest post

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