You may go to the main Strings 2008 page on this blog; that page includes the live webcast.

This Friday report includes a shortened version of Hiroši Ooguri's summary of all talks and David Gross' philosophical remarks and visions for the future. See also Jester's highlights at Resonaances.

**Boris Pioline**begins the Friday morning session with a review talk about BPS black holes from the topological string perspective; see e.g. Ooguri-Strominger-Vafa: his PDF here.

Microscopic explanation of black hole entropy is a hallmark test of every theory of quantum gravity and string theory succeeded marvelously. We were slowly going from large-charge BPS black holes to non-BPS and small black holes. He wants to cover OSV; multi-centered walls; MSW; Donaldson-Thomas invariants; improved OSV; 4D black holes and 3D instantons.

Boris reviews some basic type IIA setup in OSV, like in my article about it. He links the F-terms in the effective action with the (A-model) topological amplitudes, including terms with Gromov-Witten coefficients. Boris mentioned the M-theory lift to 5 dimensions. He explains the p,q,J charges and defines a second helicity supertrace, an index with J^2 in it (to swallow the fermionic zero modes), that is almost constant but jumps on lines of marginal stability.

Boris sketches some papers of him with Andy Neitzke and others: unfortunately too fast to catch something useful here. He instantly goes to attractor flows. If Z=0, a solution with a naked singularity should be dismissed. Maybe, a spherically asymmetric shape may become relevant here. The entropy when higher-derivative terms are relevant is given by Wald's formula; the entropy is the Legendre transform of the partition sum. He writes "topological free energy" and says that it was thus natural to write the OSV relation with the Laplace transform.

Questions: are both sides of the OSV well-defined, is it true everywhere, what do the holomorphic anomalies do, is it consistent with elmg. dualities, consistent nonperturbatively? Or, as Cumrun cares, does the formula lead us to a new interpretation of quantum mechanics? ;-)

Boris defines multi-centered solutions; see also attractor flow trees (Denef; Denef, Moore). They carry angular momentum. Infinite proper distances appear in various limits etc. Split attractor conjecture links the multi-centered solutions to the trees and there is a finite number of them. At lines of marginal stability, one loses BPS states. The jump of Omega is proportional to a product of two Omegas. Kontsevich and a collaborator generalized this formula profoundly; see later.

The MSW (Maldacena Strominger Witten) (0,4) SCFT is useful if the D6-brane charge vanishes. It comes from M5-brane on a 4-cycle, with a left-moving momentum. It is a kind of sigma-model. Cardy's formula applied to MSW reproduces the entropy, including the R^2 Wald correction. Some features must be avoided at this simple level, e.g. self-intersections.

To go to higher orders, consider the (0,4) elliptic genus (generalized index). It has to be a combination of theta functions, weighted by some modular forms H. The modular forms may be reproduced by SL(2,Z)-symmetrizing of its polar terms - see fairy tail. The polar terms come from states with dimensions between 0 and a multiple of central charge. The polar states may be described (Strominger Gaiotto Yin) by Landau levels near the poles of S2 (times AdS3). Another approach uses 2-centered solutions.

Bound states of D6 with D2,D0 in a ("very noncommutative") limit are described by ideal sheaves that are counted by Donaldson-Thomas (DT) invariants (related to GW invariants) and MacMahon's function appears here, too. The DT partition sum is written as a scary infinite product with two such factors; the first factor are "halos" while the second are "core states" (a bank I had in New Jersey, before it was bought by First Union and then by Wachowia). ;-)

Take this infinite product, it's a simple calculation, and up to genus 51 haha it can be shown that Omega grows like exp(lambda^3), confirming the 5D BH entropy. ;-) Extreme polar state conjecture says that only 1 D6 and 1 D6bar are needed. I missed the "swing states" but they may decide about Obama and McCain.

Some details are added to OSV now: D6 brane charge must be zero, t must go to i.infinity, DT partition sum must be cut off, a measure factor is added, exponentially small correction are estimated. The "entropy enigma" suggests that OSV can only work at stronger coupling, because of the dominance of multi-centered solutions.

In the last 10 minutes, he wants to use 3D instantóňs (the accents are supposed to be French here haha). He wants to replace Omega by something continuous at the lines of marginal stability, by some patchwork. His moduli space mimics the hypermultiplet but at finite radius, it receives some instanton corrections. Some hyperKähler cone comments are too tough for a blog, if not for me. ;-) But he has a compact way to encode a hyperKähler geometry. Conformal compensators appear, too. Finally he gets to Kontsevich and Soibelman - so you shouldn't be shocked that the lecture was heavily mathematical even a few minutes ago. ;-)

To conclude, a lot of progress in N=2 BPS black holes but not a complete progress. The instantons are promising. O,S,V are congratulated for a prize for the OSV conjecture. Good talk. Unfortunately, Boris didn't hire anyone to ask a question, so there was none. Everyone can join Kontsevich and Soibelman now.

**Ashoke Sen**also discusses extremal black hole microstates - in a particular context of the AdS2/CFT1 correspondence: PDF here. The talk is based on two co-his 2008 papers, with a few earlier papers as background. Ashoke explains the entropy in terms of microstates and the area. To go to smaller black holes, he needs to learn how to deal with the higher-derivative corrections on the gravity side and with more accurate statistical methods on the microstate side.

The Mac presentation software collapsed for a while but it's back now.

The microstate side is well understood (see Boris above) so he will focus on the higher-derivative (Wald-like) issues on the macroscopic side. For BPS black holes, Wald's formula simplifies to "entropy function formalism". It starts with a partially proven proposition that extremal black holes have an SO(2,1) symmetry - i.e. an AdS_2 factor - near the horizon. The remaining dimensions are compact and fibered over AdS_2.

From string theory on this near-product space, he focuses on gravity and U(1) fields, and writes their configuration (metric plus gauge field) that preserves the SO(2,1) symmetry, determined up to a scaling of metric and the field strength. He defines "script E", by a transform of the Lagrangian (2.pi times (e.q - v.L)). Near-horizon quantities are obtained by extremizing E with respect to v,epsilon. And the value of E at the extremum is the entropy itself!

He will now try to generalize this formalism by adding higher-derivative corrections properly. SUSY won't be explicitly used. Semiclassically, his treatment will resemble the Euclidean black hole reasoning. He will assume no multi-centered states spoil his single-centered calculation - he can assure it by setting the attractor values at infinity, too. Ashoke writes the Wald entropy draft "Z in AdS_2" as an OSV-like Laplace transform of d(micro). The entropy is the log of d(micro) and the rules are supposed to coincide with the AdS/CFT expectations.

The SO(2,1)-invariant solution is Euclidean-continued. The gauge field must vanish at some complex positions. To check his formalism, he wants to reproduce Wald's formula. The partition function Z goes like exp(-A) where A is the Euclidean action. The latter needs an IR cutoff of "r" to avoid a divergence.

In "A", the boundary part is linear in "r0", no absolute piece. He keeps the nice, r0-independent part of "A"; imagine that the cutoff-dependent one is removed by a boundary counterterm. So he gets a rather simple proposal. The exponential of -2.pi.v.Lag_2(e) equals the sum of d(micro) multiplied by exp(-2.pi.e.q). Wald's entropy is then equal to ln(d(micro)).

So far he worked with AdS2 and the Z partition sum was properly called this way. To see why CFT1 matters, he first renames it to Z(CFT1). ;-) He changes the convention for UV/IR cutoffs, making all scales "r0" times longer than usual (only the ratios matter). Frankly speaking, I completely missed how he used anything about CFT1 in the following discussion: it really looks like he only renamed things to "CFT1" but that's probably because I didn't listen as carefully as needed.

Some simple measure factors should be calculated now, including quantum corrections. I thought that none of them should be there according to Andy et al. To summarize, he proposed an OSV-like formula for extremal black hole entropy, claiming that it has an AdS2/CFT1 spirit. Cumrun was too far so I didn't hear his questions too well but Ashoke says that there can be various boundary terms. Hiroši continues with a similar question about finite-size effects. Ashoke says that these come from the AdS interior, making his favorite term unambiguous in the limit. Shiraz raises a conjecture. It's confirmed: the Hamiltonian has a discrete spectrum, and they found something by looking at limits of the BTZ black hole.

Strings 2009 will be in Rome. Some more workshops such as New Perspectives in String Theory are announced.

After the coffee break,

**Greg Moore**talks about developments in BPS wall-crossing - his work with Gaiotto and Neitzke (Davide is on a karate-picture, claimed to struggle with a paradox). After Greg reviews wall crossing, he will talk on the Kontsevich-Soibelman-related stuff. That will be hard. Kontsevich and Soibelman showed that Greg's related 2007 talk about wall crossing in Madrid was far from complete.

He begins with an N=2 theory with a moduli space and symplectic lattice of elmg. charges. He recalls Boris' second helicity supertrace Omega, talks about walls of marginal stability and BPS bound states of other BPS states. For example, an orange, a lemon, and an egg form a bound state that looks like an orange but it has a lemon inside and an egg inside it (with a bird). ;-) Denef-Moore's formula for the jump of the index only applies when at least one decay product has an elementary electric charge.

The Kontsevich et al. formula is meant to generalize it to a pair of non-elementary charges in the lattice. A picture of BPS rays is "as close as we get to the LHC in this talk". ;-) Instead, a product over all symplectomorphisms over a whole set of rays may hint that it would be a good idea to give up the detailed content. ;-) At any rate, a product compensates changes of the jump in the index, to remain constant across the wall.

Seiberg-Witten theory is formulated as lattices fibered over the u-plane, with monodromies. He adds toroidal fibers to write the effective action. The G=SU(2) case is shown explicitly, with the two monopole and dyon singularities in the u-plane. At this moment, I am confused about a basic point - where he lost gravity (at the beginning, we were thinking about black holes, didn't we?).

Greg wakes up the audience because he wants to explain the structure of a space. The quantum corrections depend on the BPS spectrum. The wall crossing formula - the Kontsevich et al. formula - now can be derived from the continuity of the metric across the wall. The metric is continuous but it is not a trivial constraint because of the dependence of the quantum corrections in it on the spectrum.

To actually show this derivation, he considers a twistor space. Fibers appear in almost every sentence and almost all letters in the formulae are Greek or script-letters or they have at least heavy math-cultural accents such as asterisks. ;-) Why I am bothering you with all this? Because someone has a canonical symplectic form! Semiflat holomorphic Fourier modes, whatever they exactly are, help him.

On a point in the moduli space, he chooses a basis to reorganize a calculation as a weakly-coupled one. At one-loop level, he's led to a periodic Taub-NUT space, with some Bessel function in it. Suddenly a differential equation for some twistor modes enters the scene.

Chi_inst, one of the two factors in chi, is proportional to the exponential of an integral with some poles in it (two months of probing every possible mistake). As a function of zeta, Chi is therefore discontinuous across the wall.

He wants to go beyond one loop (to see all BPS states) but effective field theory is not usable here. Instead, they solve a Riemann-Hilbert (RH) problem. Instantons are written explicitly as a sum over trees. Again, KS WCF comes from continuity of the metric. How do they check that their hyperKähler metric is physically the right one? The RH problem is equivalent to some differential equations - and they physically correspond to R-symmetry, scale symmetry, holomorphy.

In Stokes' phenomena here, factors are u,R-independent. End of proof, summary: they constructed HK metric for Seiberg-Witten on circle and other things said above. The work is related to various work of Ceccotti, Kapustin, Hitchin, and others. In future, singularities at superconformal points should be studied, besides integrability on motivic version of WCF. Finally, generalization to SUGRA (that's what I thought was necessary but it was not). What is the relation to work of Joyce and others? Compute some moduli spaces etc. Thank you.

Cumrun asks whether the continuity is related to the continuity of something else in an infinite-dimensional case of Cumrun, and Greg gives a mixed answer. Someone said that duality in 2D will prevent him from generalizations. Cumrun says a few more things I can't hear well but Greg talks about many isomonodromic deformations that can arise. The last question is about the extension from regions in the u-plane to compact manifolds etc.

**Davide Gaiotto**contributes a talk about his recent 3-paper work with Edward Witten about S-duality and boundary conditions in N=4 SYM; he's written many other recent cool papers, too: see his PDF here.

There are many boundary conditions - jungle - even if you impose symmetry constraints. He motivates the work by everything about N=4 being interesting, and by a generalization of Wilson lines and surface operators to the co-dimension one case, as well as by the Geometric Langlands program in maths.

Davide reviews the symmetries and parameters of N=4 SYM and its 10D SYM possible origin. Boundary conditions break some translational and therefore super- symmetries, too. OSp(4|4,R) is the maximum preserved supergroup of PSU(4|2,2). For example, Neumann boundary conditions set F_{3i} to zero, much like some components of scalars and derivatives of others; a gauge symmetry survives at the boundary.

I wonder whether it would be more natural to study these things as orientifold planes rather than strict boundaries.

The generalized Neumann conditions allow some fields to diverge (as 1/r) while d_3(X) can be related to epsilon [X,X]. New boundary conditions may be obtained by S-dualizing the known ones. Moduli spaces will probably be found and nontrivial theories may live inside the boundaries.

He obtains new complex boundary conditions as "bound states" of known domain walls and simpler boundary conditions (or flowing their parallel arrangement into the IR, as he says). Some braneworld realization with D3, D5, NS5 intersecting in various directions follow. D5-branes add 3-5 strings while NS5-branes split D3-branes and create new 3-3 strings.

Among other simple example, he claims that Neumann and Dirichlet boundary conditions are S-dual to each other (electromagnetic duality). When charge 1 hypermultiplet is added to the Neumann ones, one obtains S-self-dual conditions, as can be seen from a Hanany-Witten stringy construction of the setup. The result is nontrivial in field theory, analogous to particle-vortex dualities in 3D.

Neumann with several charge 1 hypermultiplets is dual to Neumann coupled to a strongly coupled 3D theory on the boundary. Quiver picture of it is here. More importantly, they seem to have a prescription to construct S-duals of the boundary conditions. SL(2,Z) is generated by S,T. T adds a Chern-Simons coupling at the boundary and they can check that (ST)^3=1 and S^2=1.

He may view their procedures as a tool to engineer new 3D theories (on the boundary) and find new mirror pairings between them. The holographic dual of the boundary conditions should be found, he says, and people may try to generalize the conditions to lower SUSY. Hiroši asks a question but I cannot extract any useful information from the answer, sorry. Cumrun asks what happens for various gauge groups, especially the exceptional ones. Somewhat surprisingly, no interesting answer.

Now I am looking at mail. Andrei Starinets wrote an amusing correction to my Thursday text (I have roughly 100+ visitors a day from cern.ch these days haha). He confirms that what he cares about is the ALICE detector, as he correctly said, of course. But he incorrectly used a picture of the ATLAS detector, and someone correctly told him about the mistake. ;-) He doesn't like when their AdS/CFT treatment of heavy ion physics is referred to as a part of AdS/QCD. Sorry, but I will probably keep my imprecise vocabulary.

**Simeon Hellerman**ends the morning session by cosmological unification of string theories - his work also including Ian Swanson about the interpolation between string theories of very different kinds (e.g. super and bosonic or supercritical ones): PDF here.

Simeon began with the hexagon duality network for M-theory and says that when SUSY is broken, we don't know much but we know his papers. ;-) As I explained in several blog articles about these issues, Ian and Simeon work in the light cone gauge, adding nice profiles of tachyon and dilaton that lead to fully conformal theories.

The theory in their picture is typically conformal for a simple reason - all the Feynman vertices are "outgoing" so you can't even construct loops such as those for the beta-function. Their first interesting cosmology looks like a bubble of nothing - an expanding hell with a tachyon vev that expels all strings.

Now, he allows the tachyon to (quadratically) depend on a transverse dimension X2, too. The coefficient grows with time. X2 is massive and physically disappears. All X2-excited modes are killed in the bulk of the bubble. It looks like he is reducing the critical central charge. Well, now the theory does have 1-loop diagrams, but not higher. I had to look outside now, but I guess that he explained that the central charge is transmitted to a linear dilaton?

Now, he looks at the interpolation between type 0 and type II. The interpolation allows you to say that SUSY is spontaneously, not explicitly broken in type 0. Finally, Simeon interpolates type 0 and bosonic string theory, extending an old Berkovits-Vafa construction to the level of a full cosmological solution. In the context of E8 heterotic strings, he wants to follow the fate of the gravitons, use the insights of Hořava and Fabinger, and other things: they construct a new non-SUSY E8 string and he presents its spectrum.

His next goal is to connect the N=2 superstring (with the D=4 real critical dimension). To conclude, his "big picture" connecting the theories looks like complicated ladders - box diagrams - showing the transitions of supercritical strings to other strings etc. Time-dependence is essential to connect them and many other links may be waiting to be discovered. A very good talk.

A question asks about some UV/IR links, the tachyon storing the information about the high-dimension spectrum etc. The answer is a bit ambiguous but Simeon proposes a precedent. Simeon is sad that no one asked about phenomenology because he would show that their preliminary graphs are promising (the graph was not shown to the camera, too bad). ;-)

In the afternoon,

**Johannes Walcher**talks about the tadpole cancellation in topological string theory; see his paper. To start, topological strings are both toy models for string theory as well as a tool to study BPS-protected stringy quantities (F-terms). He will look at top. strings on compact Calabi-Yaus with both D-branes and orientifolds.

In type I physical string theory, anomaly cancellation is one of the things that initiated the 1984 revolution. On the worldsheet, it comes from a cancellation between boundaries and crosscaps: tadpole cancellation (vanishing of the 1-point function of the RR 0-form). Now, yes, topological strings also exhibit tadpole cancellation: A-model and B-model can only be decoupled from one another if the total charge is canceled between the D-branes and orientifolds. Otherwise, the loop amplitudes would suck. SUGRA/spacetime interpretation is not yet known.

Open/closed topological string theories for noncompact, local manifolds were solved in this century by Vafa and various collaborators (with Dijkgraaf: via matrix model; with Aganagič and others: using the topological vertex). To define topological strings, he twists - interprets various supercharges "G" of the physical string as BRST operators "Q" and/or antighosts "b". Mirror symmetry exchanges the A- and B-models. He also introduces anti-A and anti-B models, CPT duals of A and B. They're probably equivalent to A,B and the role of the new concepts is not quite clear to me now.

BCOV 1993 shows that B-model amplitudes depend on CS moduli nonholomorphically: that's the holomorphic anomaly, arising from the boundary of the Riemann surface moduli space. BCOV showed that closed string amplitudes don't depend on the "wrong" moduli.

Witten 1993 included D-branes to topological strings: A-branes are Lagrangian submanifolds with flat bundle; B-branes are holomorphic branes with holomorphic bundles. The corresponding charge is carried by the "other" model: Ooguri-Oz-Yin 1996 (no, it's not Xi Yin who was in the kindergarten in 96 and was only starting with the A-model).

Back to the tadpole cancellation. He must extend the holomorphic anomaly equation first. The integration constants must be determined. For the quintic, it's computed up to genus 51. So the normal BCOV holomorphic anomaly - one for closed strings - must be extended to the open string. The moduli space talks about Riemann surfaces with boundaries, so it has new (codimension one) boundaries itself.

By now, we've included D-branes but orientifolds are needed to cancel the tadpoles. Strings themselves become unoriented. Klein surfaces have genus g, h boundaries, c crosscaps. 2 crosscaps can be traded for 1 handle (as far as at least 1 crosscap is left), with chi=2-2g-n-c. He says he doesn't know a textbook saying that, and I am convinced that even Polchinski does. These surfaces may also be viewed as Z2 quotients of closed oriented manifolds.

He's summing the amplitude over the number of crosscaps 0,1,2. The sum obeys the extended holomorphic anomaly equation. Now he jumps to some old stuff - Gopakumar-Vafa 1998 interpretation of the topological amplitudes as BPS state counting. RHS = sum of powers of sinh. Open/unoriented amplitudes of this kind go back to Ooguri-Vafa 2000 (and Walcher 2007). The expansion coefficients are integers iff you cancel the tadpoles between orientifolds and crosscaps (not an obvious relationship). It's natural for him to say what real top. amplitudes are counting - well, real enumerative invariants (Welschinger, Solomon etc.).

There are two ways to embed an A-model setup to type IIA, one with O6 and the other with O4. I didn't quite get this: they seem to be geometrically different, even in the 6 compact dimensions, don't they? For the O4/D4 he gets a better picture, with the topological tadpole cancellation linked to a local cancellation in the physical string.

Speculations: Witten interpreted the hol. anomaly as evidence of background independence of the top. string. The translation of the anomaly in terms of infinitesimal Bogolyubov transformation is given. As Cumrun Vafa likes to say, the partition sum is not a function, it's a "wave function".

Some Psi doesn't coincide with the closed Psi; why should it? There was the first speculation 1 here, not sure what it exactly was. The speculation 2 says that top. string is only OK if it is possible to effectively reduce the number of relevant states to a finite number. No questions.

**Marcos Mariňo**is going to remodel the topological string, both perturbatively and nonperturbatively. He has a black T-shirt but I can't say whether it is Che Guevara on it or another comrade. Optimistically, it looks like a revolutionary after counter-revolution. The work is based on work with Bouchard, Klemm, Pasquetti, and others (previous work).

Marcos believes that topological strings are less complex than superstrings but more complex than noncritical strings (the latter is not clear to me). A goal is to compute everything, at once if you can - which you can with the matrix models. Mirror symmetry together with the B-model is helpful, too. Pichard-Fuchs equations encode what you need.

The holomorphic anomaly etc. can be used to solve the equations recursively but the boundary conditions are unknown and make it complicated, except for cases when they're known, e.g. gap conditions of Huang and Klemm.

Marcos is reviewing the Dijkgraaf-Vafa 2002 solution of the B-model. On a particular geometry, the partition sum is obtained from a large-N matrix model with a potential related to the functions that determine the shape of the local Calabi-Yau. Unfortunately, these B-backgrounds have no A-mirrors.

He wants more general manifolds, e.g. toric manifolds. They have fun moduli spaces, rich enumerative content, large N (CS-like) duals, mirrors in terms of algebraic curves. They're helpful for N=2 engineering. A toric picture (triangle with 3 external legs) and its thickening is explained. They can be also encoded in toric A-branes with some open moduli.

How would you remodel the B-model on the background? I would first like to know Why I would be remodelling it. Did someone mismodel it? ;-) I might misunderstand what he means by remodelling. Under the remodelling title, he writes some schematical recursive formulae - a complicated closed surface equals another surface with one thin bridge plus two surfaces connected by a thin bridge. These amplitudes solve the loop equations and give a 1/N expansion of the matrix model.

A conjecture wants to compute B-model TS amplitudes from the mirror curve - residue amplitudes on it - in a format resembling matrix models. The Dijkgraaf-Vafa backgrounds obey it automatically but even if you don't have an explicit matrix model, you can calculate. Motivation comes from string field theory of the B-model (Kodaira-Spencer theory) for the noncompact case. He gets a general theorem for all toric CYs but what the theorem exactly is is not too clear. But it's probably the picture formula for the amplitude written in the middle. It follows that these amplitudes obey loop equations. An application to a local P^2 is shown. A test against some explicit orbifold calculations works. Another test passes: a comparison with Wilson loop vevs in a dual large N Chern-Simons theory (on a Z_p orbifold of S^3), for local F_p. With a nontrivial modular transformation, one can interpolate weak and strong 't Hooft coupling regimes.

In the last 5 minutes, he talks about non-perturbative physics of topological strings. In the matrix models, non-perturbative effects include tunneling of eigenvalues. However, these terms are contour-dependent, implying a non-perturbative ambiguity (theta-angle-like). In the double scaling limit, they become spacetime instantons due to ZZ branes.

As usual, these instantons control the large-genus behavior of the perturbative series. One can derive something about the singularities in the Borel plane. One large-order test worked out well. These instanton effects become domain wall in the full physical string. He is running overtime now. Oh, I see: Z_{MM} is matrix model. I thought it was Marcos Mariňo.

Nontrivial tests convince you that the holographic dual is really the CS theory on the lens space. He wants to get a background independent topological partition sum by summing over all non-perturbative effects, a statement whose precise meaning is unfortunately not transparent to me.

OK, so they generalized the matrix model calculational framework, which also generalized special geometry to higher genus that avoids holomorphic ambiguities. Also, there should exist a "special" brane analogous to the ZZ brane in 2D gravity. Thank you very much.

Someone asked what happened with an old well-known non-perturbative ambiguity of matrix models. Marcos says that it's fixed in one model but not sure about other models. Cumrun Vafa asks his first question about the existence of matrix models and/or matrix integrals in some cases. Marcos can write the integral but sees no reason why the matrix model should be the correct one. Cumrun's second question is surprise that the result is always a sum over saddle points. Marcos says that it is universal for his backgrounds. Applause again.

Before

**Hiroši Ooguri**summarizes the conference, the list of organizers is read. Applause of grateful participants follows. Younger generation was more exposed and great things are going to happen. They're going to live under pressure - seeing experimental discoveries for the first time in their life.

Hiroši thanks again. It's his 2nd summary, after 2004. Four years ago, Gross told Hiroši there are two ways to give the summary: summary talk or vision talk. The first one requires him to attend all the talks and he already did it in 2004. So the other method is the vision talk, which is why he will give the traditional summary talk.

(1) String theory is a candidate - a picture of Obama and McCain. It should govern everything - a childish picture of galaxies and everything.

(2) String theory is also a model (I thought an attractive woman would appear here, not...). It is already a good approximation of the real world and explains new things - for example how the Hawking's seemingly robust argument for information loss could have gone wrong.

(3) String theory is also a tool to study many things.

(4) String theory is a language, giving us new concepts (instead of normal geometry etc.) for the Planck regime etc. E.g. Freddie showed that one doesn't need Lagrangians (even though it was not exactly the most stringy talk).

Among the 1,2,3,4 values, important work often plays several roles. But he has to divide the talks into the four categories arbitrarily, anyway.

In (1), he talks about string phenomenology. Realistic models may be rare and predictive. Ibáňez reviewed the MSSM landscape, Weigand included orientifolds, Donagi looked at the High Country - one best model they have, Vafa offered his bottom-up, comparably unique F-theory alternative. He has to ask: is there any problem left or should they be all fired? Shih solves general gauge mediation and Verlinde looks at mediation holographically. Kallosh reviewed future cosmological tests of stringy models. There is already tension: chaotic inflation is hard.

In (2), Hiroši mostly means - even though doesn't explicitly say - generic features of quantum gravity, as seen in string theory. Veneziano studies high-energy scattering, including trapped surfaces. It could be useful to see something about information non-loss. Strominger talked about the chiral 3D gravity, possibly relevant for 4D extremal Kerr black holes.

Hiroši summarized Polyakov's talk better than I understood it: particle production screens cosmological constant in de Sitter space much like Schwinger effect screens magnetic field. Hellerman presented cool things about time dependence, decays into nothing and completely different types of string theory (bosonic, supercritical, non-supersymmetric). Simeon is employed by IPM, a Japan-funded institute meant to take over the Universe. Murayama is the boss and Susanne Reffert's picture is also shown to make the slide smarter and prettier. They have up to USD 200 million for a decade.

In (3), it's a tool. Here we talk about applied string theory. Gubser and Starinets talked about quark-gluon plasma, Hiroši says. The transport coefficients are now calculated, heavy quarks are calculated, with nicely working links to RHIC, LHC, lattice QCD, and others. Hiroši asks how to quantify theoretical errors. Minwalla showed the equivalence between Einstein and generalized Navier-Stokes equations from hydrodynamics. Starinets proposed a new strongly correlated liquid with a bulk dual.

In (4), strings are a language. Hiroši believes that the S-matrix theory is returning. Schwinger is quoted as saying that the complex plane is the top discovery of particle physics. It may be the twistor plane. Dixon found some new structures in the helicity formalism. Berkovits explained fermionic T-duality. Sokatchev showed it works for a weak 't Hooft coupling while Alday did the same thing, using AdS/CFT, at the strong coupling. In both limits, one can see the exponentiation.

String theory and QFT (boy and girl) are making progress hand in hand and we may see the full answers soon.

Cachazo showed a remarkable convergence of N=8 SUGRA at large complex momentum. The S-matrix is determined by the leading singularities. The Lagrangian becomes unnecessary for the S-matrix: a yet new language to formulate quantum field theory. Green showed how dualities and string theory expansions constraint the amplitudes in SUGRA. In Hiroši's opinion, the finiteness of N=8 SUGRA is an open question. Stieberger showed remarkable simplifications coming from SUSY Ward identities.

Steps have been made to prove the AdS/CFT correspondence. The planar Yang-Mills should be reproduced from the worldsheet. Integrability has been helpful for these things, to check the anomalous dimensions of operators etc. Staudacher considered a more general limit L/log(M) is fixed, where L is the number of derivatives and M is a number of letters.

There has been a discrepancy between AdS and CFT at four loops and Hiroši is surprised that no blogger has used it to "disprove AdS/CFT". I think that Hiroši heavily overestimates numerous capitalist imperialist pigs and Woits, among similar dense stuff. (By Woits, I primarily mean a chap called Mr Lee Smolin who dedicates a nearly whole chapter of his dumb book to speculations that AdS/CFT is invalid, but if you want to see that Mr Peter Woit also considers AdS/CFT "damn conjectural" and even ill-defined, see here.) They couldn't find what's going on at the four-loop level! Maybe if the paper was called "String theory sucks and unravels at four loop level". ;-) I've known about the discrepancy but it was always clear to me that it's due to a subtle term neglected in the spin chains because the spin chains were simply not quite identical to the exact theory. Janik removed the discrepancy, anyway.

Hiroši talks about the membrane minirevolution, a possible new kind of Chern-Simons-like theories that could have analogous to N=4 SYM from many viewpoints. Lambert reviewed the maximally supersymmetric theory, Maldacena reviewed the N=6 theory. Now Hirosi switched to the squashed sphere and included Tomasiello's talk about the CP3 AdS stuff.

Pioline reviewed the OSV conjectures from the microscopic viewpoint - the more abstract mathematical one - and Sen was adding derivatives in the macroscopic, black hole description. Mariňo essentially gave the mirrors of toric manifolds, derivable from Kodaira-Spencer theory, and a method to calculate non-perturbative effects (summing over branes, in the top. case: sum over Calabi-Yau geometry). Hiroši thinks it's quite radical and should be compared with existing knowledge about OSV.

Moore discussed how the counting jumps on the walls, giving a nice derivation of the Kontsevich-Soibelman formula. Gaiotto discussed how S-duality maps a large class of 1/2 BPS boundary conditions. String theory is an extreme sport and he enjoys to look at the parade of beautiful ideas. Hiroši shows Gross' transparency in 1998, right after AdS/CFT began. Hiroši wonders whether he forgot a talk. Let's see. He says: AdS/CFT has a huge range of validity and he wonders whether there's place for loop quantum gravity in it.

Sorry, Carlo, but Hiroši did forget, after all. ;-) Or didn't he?

403 people from 36 countries participated. The numbers are summarized under the Olympic circles. The most active countries have already hosted the conference, which is why they should do it again. Hiroši enumerates a few German, British, and U.S. cities that he would apparently like to visit. ;-) Hiroši thanks local secretaries and other organizers. Applause!

Someone asked how many string theorists there are, comparing with 200 loop quantum gravity magicians. Hiroši has no idea.

**David Gross**looks in the future and he was the optimal choice for a public talk. He starts by thanking the organizers for a wonderful conference, especially for the great review talks in the morning. Gross won't thank that he was asked to speak.

Gross decided to be completely revolutionary and he won't use any PowerPoint, blackboard etc. (Also because he had no time to prepare them - but he is still a revolutionary!) Someone else was able to speak about planets without any tools: it was a tour de force. Gross says that others like Elias know Gross as the last grandfather who remembers interactions of theory and experiment. ;-)

David recalls that 40 years ago, CERN was a center of theory while the centers of experiments were elsewhere, e.g. SLAC. David wrote one of the first papers to generalize Veneziano's formula. The paper had a contrived name and it was a failure. It's nice to see that CERN and string theory survived for 40 years and are both flourishing. It's amazing and a lot of things have happened.

Both survival stories are fascinating. CERN is pretty much the only lab that survived. Also, only one theoretical umbrella survived in theoretical physics and it has eaten everyone else. So we can see people like string theorist Andrei Linde here, string theorist Lance Dixon, ...

The only discipline that string theory has not yet eaten is loop quantum gravity but Gross is not sure whether we want to! ;-) Needless to say, crackpots are appalled by Gross' comments! :-) String theory will of course never be buried because it already includes the Standard Model and it is a working model of quantum gravity and a unifying theory, much like a tool for nuclear physics, condensed matter physics, mathematics. No one expected that 40 years ago when it was a half-baked theory to describe the nuclear force.

Much like other string meetings, it's been extraordinarily exciting. Unlike other conferences, there have been no great revolutions. There have been a couple of minirevolutions, or as Jeff says, insurrections.

Someone can't understand how they can do experiments with 2,000 collaborators, and Gross is not certain either. But it's because it's not one experiment but one world of experiments. And string theory is kind of similar.

Gross was also excited to see the LHC that will run in two weeks and feed us with new data. When it comes to string theory, the progress in the calculation of gauge-theory quantities is remarkable. They were beautifully reviewed and discussed and the progress both for QFT and string theory was enormous. Gross was really impressed by fermionic T-duality explanation for the dual superconformal symmetry. Ah! It's so obvious when we see it.

Together with the unbelievable advances in integrability of the closed sector of the N=4 theory, we are gaining evidence that the planar limit will be solved. We may be close to it. Janik's calculation of the Konishi operator anomaly is just stunning.

And the membrane BLG-ABJM is just beautiful and will lead to hundreds of papers in the two years. A lot of tools and applications. Progress in string phenomenology is beautiful. It's great to see Cumrun transformed into a phenomenologist - a bottom-up one. ;-)

Things were not discussed here: there was no talk on string field theory and the only anthropic talk was moved after a dinner. So he doesn't have to talk about them and he won't. ;-) But let's ask: what do we really want to learn from the LHC? We will discover SUSY because much like your humble correspondent, Gross has a lot of bets on it, too. ;-) SUSY will be extremely important to learn about but it won't be enough to crack the deep questions or to prove that string theory is right as the "candidate".

We might be lucky to produce black holes (whispering) but it is very unlikely. Nature owes us something - something totally unexpected. Of course, we will abruptly realize that the unexpected thing was an obvious prediction of string theory. Condensed matter physicists found the Hall effect "obvious" once it was seen but even at that time, they were not visionary enough to predict the fractional one.

Gross is going to be critical. String theorists in Hiroši's 1,2,3,4 were following the lamppost strategy. There are thousands of lampposts. They're really lasers, not old-fashioned lampposts, and not enough work is dedicated to connecting the lasers and build new lampposts. Gross was disappointed not to hear about three kinds of things. (Young people, go to the darkness!)

Gross wanted to hear an alternative to the anthropic principle, which we know is wrong, right. (Yes, of course, :-) David!) But to do better, we must explain the cosmological constant differently. Gross wanted to hear more about time dependence, besides Simeon, including string cosmology. Stationary/BPS lampposts are safer and good cosmological lampposts are not available.

Third, what is string theory? Everyone is testing AdS/CFT. Maybe it's time to admit that it's true. There's no value of further tests. No one has any doubt that it is true. When someone says "this is another test", people should ask "what is it good for?" Maybe, can you use the tests to answer What is string theory?

AdS/CFT is the equivalence between two rich, but not impossible, structures. That makes it priceless. We have other, less useful dualities where one side is trivial and the other is hard. For example, there's a duality between perturbative string theory (for 4 decades) which is easy and classically solved (Veneziano) and a theory - Matrix theory - that's impossibly hard to calculate.

If we fully solve the planar limit in a few years, have we solved string theory? No, only classical string theory. If you manage to go to finite N, is it a solution to string theory? Well, if you made it independent of the background and go e.g. to the flat space, by adding higher-dimension operators (moving you away from the horizon). It seems impossible right now to allow the motion in all directions. Non-commutative theories perhaps give us hints of some safe directions where you can move.

This was about the question What is string theory? What fundamental equations govern all of it? These are the future topics he would love to hear. But Gross doesn't want to end up with a pessimistic mood. It's, in fact, fantastic that we have such great questions. The history of the field assures us that it will remain rich and dynamic for many years. And we should be assured to get some hints, and maybe crucial ones, from Nature.

Just amazing thinker and speaker! Flowers for two key ladies among the organizers and thanks to all the speakers (and participants).

Monday, Tuesday, Wednesday, Thursday, Friday

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