# Workshop Programme

## for period 10 - 14 December 2007

### Integrability and the Gauge/String Correspondence

10 - 14 December 2007

Timetable

 Monday 10 December 08:30-09:55 Registration 09:55-10:00 Welcome - David Wallace 10:00-11:00 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. 11:00-11:30 Coffee 11:30-12:30 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 Landau-Lifshitz 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 two-cut solution whose moduli-space shows a surprisingly rich structure. 12:30-13:30 Lunch at Wolfson Court 14:00-14:20 Klose, T (Princeton) Factorised world-sheet scattering in near-flat AdS$_5$xS$^5$ Sem 1 We show the factorization of the three-particle world-sheet S-matrix of AdS_5 x S^5 superstring theory in the near-flat-space limit at one loop order. This is done by computing various scattering amplitudes from Feynman diagrams in the world-sheet theory. The knowledge of certain highest weight amplitudes allows us to fix all the freedom in the three-particle S-matrix, 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 two-particle S-matrix elements, from which it follows that the scattering of any three world-sheet excitations factorizes. This provides an explicit and direct check of the quantum integrability of string theory in near-flat 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 near-flat-space model. 14:20-14:40 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 four-loop anomalous dimension o the Konishi operator will be five, in contradiction to previous proposals. 15:00-15:30 Tea 15:30-16:10 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 Lorentz-invariant reformulation of the $AdS_5 \times S^5$ superstring world-sheet theory in terms of physical degrees of freedom we investigate a Pohlmeyer-reduced 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 kappa-symmetry and introduce a new set of variables (related locally to currents but non-locally to the original string coordinate fields) so that the Virasoro constraints are automatically satisfied. The resulting gauge-fixed equations can be obtained from a Lagrangian of a non-abelian Toda type: a gauged WZW model with an integrable potential coupled also to a set of 2d fermionic fields. The final form of the Pohlmeyer-reduced theory can be found by integrating out the 2d gauge field of the gauged WZW model. Its small-fluctuation 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 sine-Gordon model. Related Links http://xxx.lanl.gov/abs/0711.0155 - e-print 16:10-16:50 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 one-loop non-planar dilatation operator N=4 SYM between operators dual to folded Frolov-Tseytlin 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 semi-classical string splitting and joining is small. 16:50-17:30 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. 17:30-18:30 Welcome Wine Reception 18:45-19:30 Dinner at Wolfson Court (Residents only)
 Tuesday 11 December 09:30-10:10 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. 10:15-10:55 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. 10:55-11:30 Coffee 11:30-12:30 Berkovits, N (IFT-UNESP, 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 D-brane holes which are related to gauge invariant N=4 d=4 super-Yang-Mills operators. As in the Gopakumar-Vafa-Ooguri derivation of open-closed duality for Chern-Simons theory, this construction may lead to a worldsheet derivation of the Maldacena conjecture. 12:30-13:30 Lunch at Wolfson Court 14:00-14:20 Mansson, T (MPI, Golm) The Leigh-Strassler deformation and the quest of integrability Sem 1 We will consider a particular deformation of N=4 SYM theory, the Leigh-Strassler 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 q-deformed theory with su_q(3) symmetry which preserves integrability. And in the end we will discuss higher loop integrability for the q-deformed su(2) sector. 14:20-14:40 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. 14:40-15:00 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 space-time 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. 15:00-15:30 Tea 15:30-16:10 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 free-fermion chains coupled via strong attractive nearest-neighbor interactions, akin to a ferromagnet. Despite the collapse of the Bethe ansatz, the system is not free: the spectrum consists of zero-energy fermion pairs (Cooper pairs), and particles obeying mutual exclusion statistics. The extensive degeneracies are described by a (space-time) supersymmetric generalization of the Yangian algebra. Related Links http://arxiv.org/abs/cond-mat/0612270 - paper by Fendley and Schoutens 16:10-16:50 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 zero-width 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. 16:50-17:30 Braden, HW (Edinburgh) On the construction of monopoles Sem 1 The construction of su(2) BPS monopoles is reviewed with new results presented. 18:45-19:30 Dinner at Wolfson Court (Residents only)
 Wednesday 12 December 09:30-10:10 Plefka, J (Humboldt) The quantum AdS$_5$xS$^5$ superstring in the light cone gauge Sem 1 10:15-10:55 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 sine-Gordon model, or, more generally, the a-series 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. 10:55-11:30 Coffee 11:30-12:30 Kazakov, V (Ecole Normale Superieure) From characters to quantum (super) spin chains via fusion Sem 1 We give an elementary proof of the Bazhanov-Reshetikhin determinant formula for rational transfer matrices of the twisted quantum super-spin chains associated with the gl(K|M) 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 Jacobi-Trudi formula for characters to its quantum analogue using certain group derivatives. 12:30-13:30 Lunch at Wolfson Court 14:00-14:20 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 four-gluon 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 one-loop and two-loop confirmation of this result. We also analyse the conformal Ward identities satisfied by the Wilson loop and propose their all-order form. We show that the Bern-Dixon-Smirnov ansatz for the all-order 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. 14:20-14:40 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 spin-chain. Another example are some recently discovered supersymmetric Wilson loops whose expectation value seems to be related to that of Wilson loops in 2-dimensional YM, which is an exactly-soluble model. 14:40-15:00 Heslop, P (QMUL) MHV amplitudes in N=4 super Yang-Mills 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 Yang-Mills. In the four-point 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 one-to-one mapping between certain Wilson loop diagrams and (finite) two-mass easy box functions. We make some comments on the two loop calculation. 15:00-15:30 Tea 15:30-16:10 Alday, LF (Utrecht) Scattering amplitudes via AdS/CFT Sem 1 We describe how to compute gluon scattering amplitudes of planar maximally super-symmetric Yang-Mills 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. 18:45-19:30 Dinner at Wolfson Court (Residents only)
 Friday 14 December 09:30-10:10 Belitsky, A (Arizona State) Baxter equations for super Yang-Mills 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 gauge-invariant composite operators in supersymmetric gauge theories. 10:15-10:55 Mikhailov, A (Caltech) Perturbative study of the transfer matrix on the string worldsheet in AdS$_5$xS$^5$ Sem 1 Quantum non-local charges are central to the quantum integrability of a sigma-model. In this paper we study the quantum consistency and UV finiteness of non-local charges of string theory in AdS(5)xS(5). We use the pure spinor formalism. We determine the near-flat space expansion of the transfer matrix and calculate the one-loop 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. 10:55-11:30 Coffee 11:30-12:30 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. 12:30-13:30 Lunch at Wolfson Court 14:00-14:20 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 q-bosons. After briefly recalling these features, I will discuss a new application of our approach to planar field theories in two dimensions. 14:20-14:40 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 hep-th/ 0307031. I also discuss the analytical properties of the dressing phase. 14:40-15:00 Stefanski, B (MIT/Imperial) On the continuum limit of spin-chains Sem 1 I will discuss the continuum limit of spin chains in the context of AdS/CFT with particular emphasis on super-symmetric results. 15:00-15:30 Tea 15:30-16:10 Frolov, S (Trinity College Dublin) On bound states of the light-cone AdS5 x S5 superstring Sem 1 We discuss the apparent bound states of the light-cone 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^{M-1} solutions of the M-particle 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. 16:15-17:15 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~Yang-Mills theory and use it to determine the strong coupling expansion of the cusp anomalous dimension. (Based on the work arXiv:0708.3933[hep-th] done in collaboration with Benjamin Basso and Jan Kotanski). 18:45-19:30 Dinner at Wolfson Court (Residents only)