Contents -- Preface -- I. Lie Theory -- Twisted Modules over Lattice Vertex Algebras B . Bakalov and V.G. Kac -- 1. Introduction -- 2. Preliminaries on Vertex Algebras -- 2.1. Local Fields -- 2.2. Vertex Algebras -- 2.3. Conformal Vertex Algebras -- 3. Twisted Modules over Vertex Algebras -- 3.1. Definition in Terns of Borcheds Identity -- 3.2. Consequences of the Definition -- 3.3. Associativity of Twisted Fields -- 3.4. Definition in Terms of Associativity -- 4. Twisted Modules over a Lattice Vertex Algebra -- 4.1. Lattice Vertex Algebras -- 4.2. Twisted Vertex Operators -- 4.3. The Heisenberg Pair (h G ) -- 4.4. The Groups G and G -- 4.5. Representations of (h G) -- 4.6. Classification of - Twisted V-Modules -- Acknowledgments -- References -- Structure Theory of Finite Lie Conformal Superalgebras D. Fattori, V.G. Kac and A . Retakh -- 1. Introduction -- 2. Basic definitions and structures -- 2.1. Formal distributions and conformal algebras -- 2.2. Structural terminology -- 2.3. Conformal modules -- 2.4. Derivations -- 2.5. Minimal ideals -- 3. Differentiably simple conformal superalgebras -- 3.1. Centroid -- 3.2. Conformal derivations and the centroid -- 3.3. Constructing a chain of ideals -- 3.4. Constructing the maximal ideal -- 3.5. Centroid structure -- 3.6. More on differential simplicity -- 3.7. Splitting R -- 4. Simple Lie conformal superalgebras and their derivations -- 4.1. Simple confomal supemlgebms -- 4.2. Conformal derivations of simple Lie conformal superalgebras -- 5. Conformal derivations of differentiably simple Lie conformal superalgebras -- 5.1. Conformal centroid -- 5.2. Conformal derivations of a tensor product -- 6. Semisimple Lie conformal superalgebras -- 6.1. Proof of Theorem 6.1 -- 6.2. Conformal derivations -- 7. Physical Virasoro pairs -- 7.1. Definitions -- 7.2. Classification results.

7.3. Physical conformal superalgebras -- 8. Representations of Solvable Lie Conformal Superalgebras -- 8.1. Preliminaries -- 8.2. Rank 1 modules -- 8.3. Induced modules -- Acknowledgments -- References -- On Characters and Dimension Formulas for Representations of the Lie Superalgebra gl(m/n) E.M. Moens and J. Van der Jeugt -- 1. Introduction -- 2. Covariant modules of the Lie superalgebra gI(mln) -- 3. Supersymmetric Schur polynomials -- 4. The t-dimension of a covariant module -- References -- Matching Conditions for Invariant Eigendistributions on Some Semisimple Symmetric Spaces S. Aoki and S. Kato -- 1. Matching Conditions for IED's -- 2. Case of X = GL(4, R ) / (GL(2, R) X GL(2, R)) -- 2.1. Cartan Subspaces and Weyl Groups -- 2.2. Radial Parts of Invariant Differential Operators -- 2.3. Attached Symmetqic Subspaces -- 2.4. Local Matching Conditions -- 2.5. Explicit Form of IED's -- References -- II. Field Theory -- Rational Conformal Correlation Functions of Gauge Invariant Local Fields in Four Dimensions N.M. Nikolov, Y.S. Stanev and I.T. Todorov -- 1. Introduction -- 2. Implications of GCI. General 4-point function -- 2.1. Strong locality and energy positivity imply rationality -- 2.2. General truncated 4-point function of a GCI scalar field -- 3. OPE in terms of bilocal fields -- 3.1. Fixed twist fields. Conformal partial wave expansion -- 3.2. Symmetrized contribution of twist 2 (conserved) tensors -- 3.3. The concept of a minimal model -- 4. Is there a non-trivial gauge field theory model? -- 4.1. Restrictions on the parameters in the 4-point function -- 4.2. A composite bilocal field reproducing J1 -- 4.3. Elementary contributions to Wt2n, symmetric under Z 2 X Zn -- 4.4. Concluding remarks -- Acknowledgments -- References -- Renormalisation of Noncommutative Scalar Field Theories H. Grosse and R . Wulkenhaar -- 1. Introduction.

2. Field theory on noncommutative RD in momentum space -- 3. Renormalisation group approach to noncommutative scalar models -- 3.1. Matrix representation -- 3.2. The Polchinski equation for matrix models -- 3.3. 4-theory o n noncommutative R2 -- 3.4. 4-theory on noncommutative R4 -- Acknowledgement -- References -- On the Insertion-Elimination Lie Algebra of Feynman Graphs K. Ebrahimi-Fard, D. Kreimer and I. Mencattini -- 1. Introduction -- 2. Pre-Lie Algebra of Feynman Graphs -- 3. Two Representations as Derivations on the Hopf Algebra of Feynman Graphs -- 4. The Ladder case -- 5. Conclusions and Outlook -- Acknowledgments -- References -- Superconformal Kinematics and Dynamics in the AdS/CFT Correspondence E. Sokatchev -- 1. The role of BPS operators in the AdS/CFT correspondence -- 2. BPS short multiplets -- 2.1. The superconformal algebm PSU(2,2/4) -- 2.2. Realization on superfields in real superspace -- 2.3. Chiml superspace -- 2.4. Gmssmann-analytic or BPS superfields -- 3. N = 4 SYM and 1/2 BPS operators -- 3.1. N = 4 SYM on shell -- 3.2. Composite BPS operators -- 4. Correlators of 1/2 BPS operators -- 4.1. General properties of two-, three- and four-point functions -- 4.2. Two- and three-point functions -- 4.3. Four-point functions -- 4.4. Superconformal Ward identities -- 5. Dynamical input from the insertion formula -- 5.1. The insertion formula -- 5.2. (4 + 1)-point nilpotent invariants -- 5.3. Restrictions on the quantum part of the amplitude -- 5.3.1. Three points -- 5.3.2. FOUT points, extremal and subextremal -- 5.3.3. Four points, ki = 2 -- 5.3.4. Four points, ki = 3 and ki = 4 -- 5.4. Large N degeneracy in perturbation theory -- 6. Confronting CFT with AdS -- 6.1. Effective action on AdS x S5 -- 6.2. Main results -- 7. Conclusions -- Acknowledgments -- References -- Renormalons and Fractional Instantons C. Gomez -- 1. Introduction.

2. Divergences in Quantum Field Theory: Renormalons and Fractional Instantons -- 3. The Effective Action for N = 1 super Yang Mills -- 4. Some Comments on the Mass Gap -- Acknowledgments -- References -- III. String Theory -- The Hagedorn/Deconfinement Phase Transition in Weakly Coupled Large N Gauge Theories O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk -- 1. Introduction -- 2. Partition function for free Yang-Mills theory on a compact space -- 2.1. Two matrix harmonic oscillators give Hagedom behavior for large N -- 2.2. Exact partition function for N = -- 2.3. Exact partition function for free Yang-Mills theory -- 2.4. U ( N ) gauge theories with adjoint matter on S3 x R -- 3. Path integral derivation of the matrix integral and an order parameter -- 3.1. Basic set-up -- 3.2. The integration measure -- 3.3. Evaluation of Seff at one-loop -- 4. Solution of the free Yang-Mills matrix model -- 4.1. Low temperature behavior -- 4.2. Behavior near the transition -- 4.3. High temperature behavior -- 4.4. Summary of thermodynamic behavior -- 4.5. The Polyakov loop as an order parameter at finite volume -- 5. Phase structure at weak coupling -- 5.1. General properties of the effective action -- 5.2. The general form of the effective action in perturbation theory -- 5.3. Possible phase structures at weak coupling -- 5.4. Density of states as a function of energy -- 6. Extrapolation to strong coupling and the dual description -- 6.1. Dual interpretation of the N = 4 SYM thermodynamics at strong coupling -- 6.2. Deconfinement and black holes -- References -- Two-Loop Commuting Charges and the String/Gauge Duality G. Arutyunov and M. Staudacher -- 1. The status quo -- 2. Two-loop stringlgauge matching of infinitely many higher commuting charges -- 2.1. Folded string -- 2.2. Circular string -- Acknowledgments -- References.

Boundary Ground Ring and Disc Correlation Functions in Liouville Quantum Gravity I. Kostov -- 1. Introduction -- 2. Preliminaries: world-sheet CFT for the disc amplitudes -- 2.1. The effective action -- 2.2 . Bulk and boundary vertex operators -- 2.3. Normalization of the bulk and boundary cosmological constants and the screening operators -- 2.4. Degenerate fields -- 3. Bulk ground ring -- 3.1. The ring relations -- 3.2. Functional equations for the bulk correlators -- 4. The boundary ground ring -- 4.1. The ring relations -- 4.2. Deformation of the ring relations by the boundary Liouville term and the screening charges -- 4.3. Functional and difference equations for the correlation functions of boundary primary fields -- 5. Discussion -- Acknowledgments -- References -- Strings, Integrable Systems, Geometry and Statistical Models A. Marshakov -- 1. Introduction -- 2. Matrix models and integrable systems -- 3. Seiberg-Witten geometry, Young diagrams and topological strings -- 4. Discussion -- Acknowledgments -- References -- Impact of Dynamical Tensions in Modified String and Brane Theories E.I. Guendelman, A. Kaganovich, E. Nissimov and S. Pacheva -- 1. Main Motivation -- 2. Dp-Branes with Dynamical Tension -- 3. Conformal Anomaly and Its Impact on the Dynamical String Tension -- 4. Conclusions -- Acknowledgments -- References -- IV. Integrable Systems -- Quantum Group in Roots of Unity and the Restriction of X X Z Model A. Belavin -- 1. Center of the quantum group and the restriction in roots of unity -- 2. Restriction of the XXZ model in roots of unity -- Acknowledgments -- References -- Spaces of Boundary Values Related to a Multipoint Version of the KP-Hierarchy G. F. Helminck -- 1. Introduction -- 2. An algebraic set-up for the linearization -- 3. Spaces of boundary values -- 4. The Grassmannian related to H = H* H+.

5. The Line bundles Det and Det* over Gro(H).