Workshop Programme
for period 10  14 December 2007
Integrability and the Gauge/String Correspondence
10  14 December 2007
Timetable
Monday 10 December  
08:3009:55  Registration  
09:5510:00  Welcome  David Wallace  
10:0011:00  Staudacher, M (AEI Potsdam)  
Exact results for twist operators in planar N=4 SYM  Sem 1  
We present some new exact results for twist operators in planar N=4 SYM, based on Bethe ansatz techniques. 

11:0011:30  Coffee  
11:3012:30  Beisert, N (AEI Potsdam)  
Spectral curve for the Heisenberg ferromagnet and AdS/CFT  Sem 1  
Highly spinning classical strings on R x S^3 are described by the LandauLifshitz model or equivalently by the Heisenberg ferromagnet in the thermodynamic limit. The spectrum of this model can be given in terms of spectral curves. In this talk we study the issue of stability, i.e. whether any given admissible spectral curve can actually be realized as a solution to the discrete Bethe equations. In particular, what about those classical string solutions which have unstable excitation modes? Are they physical or not? We also find and explore the general twocut solution whose modulispace shows a surprisingly rich structure. 

12:3013:30  Lunch at Wolfson Court  
14:0014:20  Klose, T (Princeton)  
Factorised worldsheet scattering in nearflat AdS$_5$xS$^5$  Sem 1  
We show the factorization of the threeparticle worldsheet Smatrix of AdS_5 x S^5 superstring theory in the nearflatspace limit at one loop order. This is done by computing various scattering amplitudes from Feynman diagrams in the worldsheet theory. The knowledge of certain highest weight amplitudes allows us to fix all the freedom in the threeparticle Smatrix, which we argue to be constrained up to four scalar functions due to the symmetries of the model. We demonstrate that these amplitudes are given by corresponding products of the known twoparticle Smatrix elements, from which it follows that the scattering of any three worldsheet excitations factorizes. This provides an explicit and direct check of the quantum integrability of string theory in nearflat AdS_5 x S^5 as it renders evidence for the existence of higher conserved charges. By computing further amplitudes we also obtain an indirect confirmation of the supersymmetries of the nearflatspace model. 

14:2014:40  Mann, N (Chicago)  
Finite Size Corrections to the Bethe Ansatz in N = 4 SYM: A Diagrammatic Approach  Sem 1  
I discuss the problem of wrapping corrections to the BES Bethe Ansatz. The problem can be phrased as a perturbation on the BES Bethe Ansatz, which assumes long operators. I discuss applying this philosophy to the SU(2) Konishi operator, and present work in progress on this calculation. I predict that the trancendentality degree of the fourloop anomalous dimension o the Konishi operator will be five, in contradiction to previous proposals. 

15:0015:30  Tea  
15:3016:10  Grigoriev, M (Imperial)  
Pohlmeyer reduction of AdS$_5$ x S$^5$ superstring sigma model  Sem 1  
Motivated by a desire to find a useful 2d Lorentzinvariant reformulation of the $AdS_5 \times S^5$ superstring worldsheet theory in terms of physical degrees of freedom we investigate a Pohlmeyerreduced version of the corresponding sigma model. The Pohlmeyer reduction procedure involves several steps. Starting with a coset space string sigma model in the conformal gauge and writing the classical equations in terms of currents one can fix the residual conformal diffeomorphism symmetry and kappasymmetry and introduce a new set of variables (related locally to currents but nonlocally to the original string coordinate fields) so that the Virasoro constraints are automatically satisfied. The resulting gaugefixed equations can be obtained from a Lagrangian of a nonabelian Toda type: a gauged WZW model with an integrable potential coupled also to a set of 2d fermionic fields. The final form of the Pohlmeyerreduced theory can be found by integrating out the 2d gauge field of the gauged WZW model. Its smallfluctuation spectrum is that of 8 bosonic and 8 fermionic degrees of freedom with equal masses. We show that in the special case of the $AdS_2 \times S^2$ superstring model the reduced theory is supersymmetric: it is equivalent to the (2,2) supersymmetric extension of the sineGordon model. Related Links


16:1016:50  Kristjansen, C (Niels Bohr Institute)  
Quasilocality of splitting/joining strings from coherent states  Sem 1  
Using the coherent state formalism we calculate matrix elements of the oneloop nonplanar dilatation operator N=4 SYM between operators dual to folded FrolovTseytlin strings and observe a curious scaling behavior. We comment on the qualitative similarity of our matrix elements to the interaction vertex of a string field theory. In addition, we present a solvable toy model for string splitting and joining. The scaling behaviour of the matrix elements suggests that the contribution to the genus one energy shift coming from semiclassical string splitting and joining is small. 

16:5017:30  Maillet, JM (ENS Lyon \& CNRS)  
Correlation functions of quantum integrable models  Sem 1  
We will review recent advances in the computaion of correlation functions of quantum integrable models with a particular interest for spin chains. 

17:3018:30  Welcome Wine Reception  
18:4519:30  Dinner at Wolfson Court (Residents only) 
Tuesday 11 December  
09:3010:10  Roiban, R (Penn State)  
Anomalous dimensions at strong coupling: NNLO and beyond  Sem 1  
We discuss the expansion of the anomalous dimensions of N=4 SYM in inverse string tension. We explicitly show that the NNLO calculation of the anomalous dimension of the cusp Wilson loop is free of logarithmic divergences and reproduces the results of the BES equation. Features of the higher order corrections are also discussed. 

10:1510:55  Serban, D (SPhT Saclay)  
Strong coupling limit of Bethe ansatz equations  Sem 1  
I will discuss the strong coupling limit of the equations which are supposed to reproduce the spectrum of anomalous dimensions of the N=4 SYM theory. The different regimes contributing to strong coupling can be easily identified using a version of the elliptic parametrization of Janik for the spectral parameter. This parametrization can be used for a systematic expansion of the Bethe ansatz equations in the regime of strong coupling. 

10:5511:30  Coffee  
11:3012:30  Berkovits, N (IFTUNESP, Sao Paulo)  
Gauged linear sigma model for the AdS$_5$xS$^5$ superstring  Sem 1  
Using the pure spinor formalism, a gauged linear sigma model is constructed for the superstring on AdS_5xS^5. In the zero radius limit, the Coulomb branch of this sigma model is interpreted as Dbrane holes which are related to gauge invariant N=4 d=4 superYangMills operators. As in the GopakumarVafaOoguri derivation of openclosed duality for ChernSimons theory, this construction may lead to a worldsheet derivation of the Maldacena conjecture. 

12:3013:30  Lunch at Wolfson Court  
14:0014:20  Mansson, T (MPI, Golm)  
The LeighStrassler deformation and the quest of integrability  Sem 1  
We will consider a particular deformation of N=4 SYM theory, the LeighStrassler deformation. First we will find the parameter values for which the dilatation operator in the full scalar field theory satisfies Reshetikhins criteria for integrability. Then we will have a look at a particular three state subsector of the qdeformed theory with su_q(3) symmetry which preserves integrability. And in the end we will discuss higher loop integrability for the qdeformed su(2) sector. 

14:2014:40  Cicuta, G (Parma)  
Dilatation operator and Cayley graphs  Sem 1  
Simple symmetry properties and graph techniques are useful to evaluate many new eigenvalues of the dilaton operator in the scalar sector of the maximally supersymmetric Y.M. theory in the large N limit, at one loop. 

14:4015:00  Ferreira, LA (IFSC/USP Brazil)  
Solitons and integrable field theories in any dimension  Sem 1  
We present an overview of the Lorentz invariant field theories that can be studied using a generalization, to a spacetime of any dimension, of the usual zero curvature condition of 1+1 integrable field theories. That generalization involves flat connections on loop space, and allows the construction of infinite number of conservation laws, and in some cases of exact soliton solutions. 

15:0015:30  Tea  
15:3016:10  Fendley, P (Oxford/Virginia)  
Collapsing the Bethe ansatz  Sem 1  
I discuss a system where the Bethe ansatz "collapses", meaning that the nested Bethe equations end up requiring only that the momenta are those of free fermions. The model describes two freefermion chains coupled via strong attractive nearestneighbor interactions, akin to a ferromagnet. Despite the collapse of the Bethe ansatz, the system is not free: the spectrum consists of zeroenergy fermion pairs (Cooper pairs), and particles obeying mutual exclusion statistics. The extensive degeneracies are described by a (spacetime) supersymmetric generalization of the Yangian algebra. Related Links


16:1016:50  Ferretti, G (Chalmers)  
Applying the BES trick to QCD  Sem 1  
I will discuss some preliminary results concerning the application of the Beisert, Eden, Staudacher (BES) "analytic continuation trick" to the two point function of gauge invariant operators in Minkowski space. I will start by briefly reviewing the status of integrability in large N QCD and showing that it necessarily fails at higher loops. The disappointment is somewhat mitigated by the fact that, for confining theories, the anomalous dimension is not the most interesting quantity entering the two point functions. Rather, they are controlled by an infinite sum of zerowidth resonances that can be expanded at strong coupling. I will motivate the BES trick in this context and apply it to the correlation functions. After some "adventurous" mathematical manipulations, one is led to what looks like a new series of sum rules that might eventually allow one to make contact with perturbation theory. 

16:5017:30  Braden, HW (Edinburgh)  
On the construction of monopoles  Sem 1  
The construction of su(2) BPS monopoles is reviewed with new results presented. 

18:4519:30  Dinner at Wolfson Court (Residents only) 
Wednesday 12 December  
09:3010:10  Plefka, J (Humboldt)  
The quantum AdS$_5$xS$^5$ superstring in the light cone gauge  Sem 1  
10:1510:55  Corrigan, E (York)  
Classical and quantum aspects of integrable shocks  Sem 1  
An integrable shock or discontinuity is a type of 'defect' allowed, in the sense of not destroying the property of integrability, within (at least some) integrable field theories. A main example is the sineGordon model, or, more generally, the aseries of affine Toda field theories. Such 'defects' have interesting properties in the classical and quantum domains especially with regard to their interactions with solitons  or indeed with each other, if there is more than one of them and they are allowed to move. This talk will present a survey of recent results and describe some open problems. 

10:5511:30  Coffee  
11:3012:30  Kazakov, V (Ecole Normale Superieure)  
From characters to quantum (super) spin chains via fusion  Sem 1  
We give an elementary proof of the BazhanovReshetikhin determinant formula for rational transfer matrices of the twisted quantum superspin chains associated with the gl(KM) algebra. This formula describes the most general fusion of transfer matrices in symmetric representations into arbitrary finite dimensional representations of the algebra and is at the heart of analytical Bethe ansatz approach. Our technique represents a systematic generalization of the usual JacobiTrudi formula for characters to its quantum analogue using certain group derivatives. 

12:3013:30  Lunch at Wolfson Court  
14:0014:20  Drummond, J (LAPTH Annecy)  
Hidden conformal symmetry of gluon amplitudes and Wilson loops  Sem 1  
I will summarise recent developments which indicate there is a hidden confomal symmetry of planar gluon scattering amplitudes in N=4 SYM. I will show that the momentum integrals appearing in the fourgluon scattering amplitude up to four loops are conformal when written in terms of dual coordinates. Then we will discuss a relation between gluon scattering amplitudes and Wilson loops orginally proposed by Alday and Maldacena at strong coupling. I will show evidence that this relation also holds at weak coupling and present a oneloop and twoloop confirmation of this result. We also analyse the conformal Ward identities satisfied by the Wilson loop and propose their allorder form. We show that the BernDixonSmirnov ansatz for the allorder form of the MHV amplitudes is the unique solution of the Ward identity for the Wilson loop (up to a function of conformal invariants which appear at six points and beyond) providing further evidence of a hidden conformal symmetry. 

14:2014:40  Drukker, N (Humboldt)  
Integrability and Wilson loops  Sem 1  
I will present some exact results for the expectation values of Wilson loops in N=4 gauge theory and in AdS. One example is when considering local insertions into a Wilson loop it gives an integrable open spinchain. Another example are some recently discovered supersymmetric Wilson loops whose expectation value seems to be related to that of Wilson loops in 2dimensional YM, which is an exactlysoluble model. 

14:4015:00  Heslop, P (QMUL)  
MHV amplitudes in N=4 super YangMills and Wilson loops  Sem 1  
We show how MHV amplitudes at 1 loop with an arbitrary number of external legs can be derived using Wilson loops. Our result is in precise agreement with the known expression for the infinite sequence of MHV amplitudes in N=4 super YangMills. In the fourpoint case, we are able to reproduce the expression of the amplitude to all orders in the dimensional regularisation parameter epsilon. This prescription disentangles cleanly infrared divergences and finite terms, and leads to an intriguing onetoone mapping between certain Wilson loop diagrams and (finite) twomass easy box functions. We make some comments on the two loop calculation. 

15:0015:30  Tea  
15:3016:10  Alday, LF (Utrecht)  
Scattering amplitudes via AdS/CFT  Sem 1  
We describe how to compute gluon scattering amplitudes of planar maximally supersymmetric YangMills at strong coupling, using the AdS/CFT correspondence. We explicitly show how the prescription works for the case of four gluons. Furthermore, by considering a particular configuration involving a large number of gluons, we test an iterative relation conjectured to hold between scattering amplitudes at different loop orders. 

18:4519:30  Dinner at Wolfson Court (Residents only) 
Thursday 13 December  
09:3010:10  Dorey, P (Durham)  
Encoding the Bethe ansatz in ordinary differential equations  Sem 1  
In the last few years it has been found that functional relations and Bethe Ansatz equations from the theory of integrable models also govern the properties of certain ordinary differential equations. This "correspondence" will be described, and some of its applications will be outlined. 

10:1510:55  Lipatov, L (Petersburg Nuclear Physics Institute)  
Anomalous dimensions of local operators and Pomeron in N=4 SUSY  Sem 1  
The BFKL approach to the high energy scattering in nonabelian gauge theories is reviewed. In the multicolor QCD the equations for composite states of reggeized gluons in the leading logarithmic approximation have properties of Moebius invariance, holomorphic separability, duality and integrability. In the nexttoleading approximation the eigenvalue of the BFKL kernel in N=4 SUSY is constructed from the most transcendental special functions. In the diffusion approximation the BFKL equation for the Pomeron wave function in the framework of the AdS/CFT correspondence coincides with the graviton Regge trajectory. With the use of the hypothesis of maximal transcedentality the anomalous dimension of twist2 operators is calculated up to 4 loops. The results are in an agreement with the BFKL prediction and with the doublelogarithmic resummation only if one would take into account wrapping effects. 

10:5511:30  Coffee  
11:3012:30  Janik, R (Jagellonian)  
Wrapping interactions at strong coupling  Sem 1  
I would like to derive generalized Luscher formulas for finite size corrections in a quantum field theory and apply them to the case of the worldsheet theory of the AdS_5xS^5 superstring. From the gauge theory point of view these corrections come from wrapping interactions which go beyond the asymptotic allloop Bethe ansatz. I show that one can recover the classical string computation of finite size corrections to the dispersion relation of the giant magnon. The agreement involves a Borel resummation of all evenloop orders of the BES/BHL dressing factor. 

12:3013:30  Lunch at Wolfson Court  
14:0014:20  Vieira, P (Ecole Normale Superieure)  
From quasiclassics to nested Bethe ansatze  Sem 1  
We shall explain how to quasiclassicaly quantize integrable models whose classical limit is described by some algebraic curve finite gap construction. As a particular example, we shall upgrade the classical equations of Beisert, Kazakov, Sakai and Zarembo into an integral equation encoding the proper quasiclassical quantization of the super string in AdS5xS5. This equation coincides precisely with the equation describing the finite size corrections to the BeisertStaudacher equations thus providing a very nontrivial check of the latter. 

14:2014:40  Gromov, N (CEA Sacley, Paris)  
From nested Bethe ansatze to quasiclassics  Sem 1  
Finite size corrections to generic Nested Bethe Ansatze are shown to be given by a universal integral equation. In particular the BeisertStaudacher equations are considered. A new type of duality in Bethe Ansatze will be used. 

14:4015:00  Rivelles, V (Sao Paulo)  
The Smatrix of the FaddeevReshetikhin model, diagonalisability and PT symmetry  Sem 1  
We consider the diagonalizability of the Hamiltonian for the FaddeevReshetikhin (FR) model in the two particle sector. We find that the Hamiltonian is not diagonalizable. We then look for the most general quartic interaction Hamiltonian that can be diagonalized. This includes the bosonic Thirring model as well as the bosonic chiral GrossNeveu model which we find share the same Smatrix. In addition, we find a general quartic interaction Hamiltonian, violating Lorentz invariance, that can be diagonalized with the same two particle Smatrix element as calculated by Klose and Zarembo for the FR model. This family of generalized interaction Hamiltonians is not Hermitian, but is PT symmetric, and belongs to a class of nonHermitian but PT symmetric Hamiltonians which lead to a unitary Smatrix. Related Links 

15:0015:30  Tea  
15:3016:10  Basso, B (LPT Orsay)  
Cusp anomalous dimension in N=4 SYM at strong coupling  Sem 1  
We construct an exact analytical solution to the integral equation which is believed to describe logarithmic growth of the anomalous dimensions of high spin operators in planar N=4 super~YangMills theory and use it to determine the strong coupling expansion of the cusp anomalous dimension. (Based on the work arXiv:0708.3933[hepth] done in collaboration with Gregory Korchemsky and Jan Kotanski). 

16:1016:30  Spill, F (Imperial)  
Algebraic aspects of the AdS/CFT S matrix  Sem 1  
The symmetry algebra behind the AdS/CFT S matrix will be discussed. 

16:3016:50  Vicedo, B (Cambridge)  
Semiclassical finitegap strings  Sem 1  
I will discuss the semiclassical quantisation of finitegap solutions to the string equations of motion from first principles. 

19:3023:00  Conference Dinner at Trinity Hall 
Friday 14 December  
09:3010:10  Belitsky, A (Arizona State)  
Baxter equations for super YangMills theories  Sem 1  
We give an overview of recent results related to construction and solution of Baxter equations which encode spectra of anomalous dimensions of gaugeinvariant composite operators in supersymmetric gauge theories. 

10:1510:55  Mikhailov, A (Caltech)  
Perturbative study of the transfer matrix on the string worldsheet in AdS$_5$xS$^5$  Sem 1  
Quantum nonlocal charges are central to the quantum integrability of a sigmamodel. In this paper we study the quantum consistency and UV finiteness of nonlocal charges of string theory in AdS(5)xS(5). We use the pure spinor formalism. We determine the nearflat space expansion of the transfer matrix and calculate the oneloop divergences. We find that the logarithmic divergences cancel. This gives strong support to the quantum integrability of the full string theory. We develop a calculational setup for the renormalization group analysis of Wilson line type of operators on the worldsheet. 

10:5511:30  Coffee  
11:3012:30  Minahan, JA (Uppsala)  
Finite size effects for giant magnons on physical strings  Sem 1  
In this talk we consider finite size effects when more than one magnon is present. By taking a certain limit we can compare our results to those from the Heisenberg spin chain. 

12:3013:30  Lunch at Wolfson Court  
14:0014:20  Wosiek, J (Jagellonian)  
From planar quantum mechanics to planar quantum field theory  Sem 1  
Recently proposed simple quantum mechanical system  a distant cousin of reduced SYM_2  has been found to have many intriguing properties analogous to the phenomena appearing in the applications of the AdS/CFT correspondence. They include: 1) solubility 2) phase transition and 3) equivalences with the Heisenberg chain and a gas of qbosons. After briefly recalling these features, I will discuss a new application of our approach to planar field theories in two dimensions. 

14:2014:40  Volin, D (SPhT, Saclay)  
Investigation of dressing phase at strong coupling  Sem 1  
Investigation of the dressing phase at strong coupling is important for the verifying AdS/CFT conjecture. I review some recent progress on this subject, part of which is published in hepth/ 0307031. I also discuss the analytical properties of the dressing phase. 

14:4015:00  Stefanski, B (MIT/Imperial)  
On the continuum limit of spinchains  Sem 1  
I will discuss the continuum limit of spin chains in the context of AdS/CFT with particular emphasis on supersymmetric results. 

15:0015:30  Tea  
15:3016:10  Frolov, S (Trinity College Dublin)  
On bound states of the lightcone AdS5 x S5 superstring  Sem 1  
We discuss the apparent bound states of the lightcone AdS5 x S5 superstring and its mirror model, and show that depending on a choice of the physical region there are one, two or 2^{M1} solutions of the Mparticle bound state equations sharing the same conserved charges. For very large but finite values of J, most of these solutions, however, exhibit various signs of pathological behavior. In particular, they might receive finite J correction to their energy which is complex, or the energy correction might exceed corrections arising due to finite J modifications of the Bethe equations thus making the asymptotic Bethe ansatz inapplicable. 

16:1517:15  Korchemsky, G (LPT, Paris XI)  
Planar gluon amplitudes/Wilson loop duality in N=4 SYM  Sem 1  
We construct an exact analytical solution to the integral equation which is believed to describe logarithmic growth of the anomalous dimensions of high spin operators in planar N=4 super~YangMills theory and use it to determine the strong coupling expansion of the cusp anomalous dimension. (Based on the work arXiv:0708.3933[hepth] done in collaboration with Benjamin Basso and Jan Kotanski). 

18:4519:30  Dinner at Wolfson Court (Residents only) 