| Lecturers |
| Seminar Speakers |
| Participants |
| Preface (French) |
| Preface (English) |
| Mathematics / Part I.： |
| Fields, Strings and Duality / R. DijkgraafCourse 1.： |
| Introduction / 1.： |
| What is a quantum field theory? / 2.： |
| Axioms vs. path-integrals / 2.1.： |
| Duality / 2.2.： |
| Quantum mechanics / 3.： |
| Supersymmetric quantum mechanics / 3.1.： |
| Quantum mechanics and perturbative field theory / 3.2.： |
| Two-dimensional topological field theory / 4.： |
| Axioms of topological field theory / 4.1.： |
| Topological field theory in two dimensions / 4.2.： |
| Example - quantum cohomology / 4.3.： |
| Riemann surfaces and moduli / 5.： |
| The moduli space of curves / 5.1.： |
| Example - genus one / 5.2.： |
| Surfaces with punctures / 5.3.： |
| The stable compactification / 5.4.： |
| Conformal field theory / 6.： |
| Algebraic approach / 6.1.： |
| Functorial approach / 6.2.： |
| Free bosons / 6.3.： |
| Free fermions / 6.4.： |
| Sigma models and T-duality / 7.： |
| Two-dimensional sigma models / 7.1.： |
| Toroidal models / 7.2.： |
| Intermezzo - lattices / 7.3.： |
| Spectrum and moduli of toroidal models / 7.4.： |
| The two-torus / 7.5.： |
| Path-integral computation of the partition function / 7.6.： |
| Supersymmetric sigma models and Calabi-Yau spaces / 7.7.： |
| Calabi-Yau moduli space and special geometry / 7.8.： |
| Perturbative string theory / 8.： |
| Axioms for string vacua / 8.1.： |
| Intermezzo - twisting and supersymmetry / 8.2.： |
| Example - The critical bosonic string / 8.3.： |
| Example - Twisted N = 2 SCFT / 8.4.： |
| Example - twisted minimal model / 8.5.： |
| Example - topological string / 8.6.： |
| Functorial definition / 8.7.： |
| Tree-level amplitudes / 8.8.： |
| Families of string vacua / 8.9.： |
| The Gauss-Manin connection / 8.10.： |
| Anti-holomorphic dependence and special geometry / 8.11.： |
| Local special geometry / 8.12.： |
| Gauge theories and S-duality / 9.： |
| Introduction to four-dimensional geometry / 9.1.： |
| The Lorentz group / 9.2.： |
| Duality in Maxwell theory / 9.3.： |
| The partition function / 9.4.： |
| Higher rank groups / 9.5.： |
| Dehn twists and monodromy / 9.6.： |
| Moduli spaces / 10.： |
| Supersymmetric or BPS configurations / 10.1.： |
| Localization in topological field theories / 10.2.： |
| Quantization / 10.3.： |
| Families of QFTs / 10.4.： |
| Moduli spaces of vacua / 10.5.： |
| Supersymmetric gauge theories / 11.： |
| Twisting and Donaldson theory / 11.1.： |
| Observables / 11.3.： |
| Abelian models / 11.4.： |
| Rigid special geometry / 11.5.： |
| Families of abelian varieties / 11.6.： |
| BPS states / 11.7.： |
| Non-abelian N = 2 gauge theory / 11.8.： |
| The Seiberg-Witten solution / 11.9.： |
| Physical interpretation of the singularities / 11.10.： |
| Implications for four-manifold invariants / 11.11.： |
| String vacua / 12.： |
| Perturbative string theories / 12.1.： |
| IIA or IIB / 12.2.： |
| D-branes / 12.3.： |
| Compactification / 12.4.： |
| Singularities revisited / 12.5.： |
| String moduli spaces / 12.6.： |
| Example - Type II on T[superscript 6] / 12.7.： |
| BPS states and D-branes / 13.： |
| Perturbative string states / 13.1.： |
| Perturbative BPS states / 13.2.： |
| D-brane states / 13.3.： |
| Example - Type IIA on K3 = Heterotic on T[superscript 4] / 13.4.： |
| Example - Type II on T[superscript 4] / 13.5.： |
| Example - Type II on K3 [times] S[superscript 1] = Heterotic on T[superscript 5] / 13.6.： |
| Example - Type IIA on X = Type IIB on Y / 13.7.： |
| References |
| How the Algebraic Bethe Ansatz Works for Integrable Models / L.D. FaddeevCourse 2.： |
| General outline of the course |
| XXX[subscript 1/2] model. Description |
| XXX[subscript 1/2] model. Bethe Ansatz equations |
| XXX[subscript 1/2] model. Physical spectrum in the ferromagnetic thermodynamic limit |
| XXX[subscript 1/2] model. BAE for an arbitrary configuration |
| XXX[subscript 1/2] model. Physical spectrum in the antiferromagnetic case |
| XXX[subscript s] model |
| XXX[subscript s] spin chain. Applications to the physical systems |
| XXZ model |
| Inhomogeneous chains and discrete time shift |
| Examples of dynamical models in discrete space-time |
| Conclusions and perspectives |
| Comments on the literature on BAE / 14.： |
| Supersymmetric Quantum Theory, Non-Commutative Geometry, and Gravitation / J. Frohlich ; O. Grandjean ; A. RecknagelCourse 3.： |
| The classical theory of gravitation |
| (Non-relativistic) quantum theory |
| Reconciling quantum theory with general relativity: quantum space-time-matter |
| Classical differential topology and -geometry and supersymmetric quantum theory |
| Pauli's electron |
| The special case where M is a Lie group |
| Supersymmetric quantum theory and geometry put into perspective |
| Supersymmetry and non-commutative geometry |
| Spin[superscript c] non-commutative geometry |
| The spectral data of spin[superscript c] NCG / 5.1.1.： |
| Differential forms / 5.1.2.： |
| Integration / 5.1.3.： |
| Vector bundles and Hermitian structures / 5.1.4.： |
| Generalized Hermitian structure on [Omega superscript k](A) / 5.1.5.： |
| Connections / 5.1.6.： |
| Riemannian curvature and torsion / 5.1.7.： |
| Generalized Kahler non-commutative geometry and higher supersymmetry / 5.1.8.： |
| Aspects of the algebraic topology of N = n supersymmetric spectral data / 5.1.9.： |
| Non-commutative Riemannian geometry |
| N = (1, 1) supersymmetry and Riemannian geometry / 5.2.1.： |
| Unitary connections and scalar curvature / 5.2.2.： |
| Remarks on the relation between N = 1 and N = (1, 1) spectral data / 5.2.5.： |
| Riemannian and spin[superscript c] "manifolds" in non-commutative geometry / 5.2.6.： |
| Algebraic topology of N = [characters not reproducible] spectral data / 5.2.7.： |
| Central extensions of supersymmetry, and equivariance / 5.2.8.： |
| N = (n, n) supersymmetry, and supersymmetry breaking / 5.2.9.： |
| Reparametrization invariance, BRST cohomology, and target space supersymmetry |
| The non-commutative torus |
| Spin geometry (N = 1) / 6.1： |
| Integration and Hermitian structure over [Omega superscript 1 subscript D](A[alpha]) / 6.1.1.： |
| Connections on [Omega superscript 1 subscript D](A[alpha]) / 6.1.3.： |
| Riemannian geometry (N = [characters not reproducible] |
| Kahler geometry (N = [characters not reproducible] |
| Applications of non-commutative geometry to quantum theories of gravitation |
| From point-particles to strings |
| A Schwinger-Dyson equation for string Green functions from reparametrization invariance and world-sheet supersymmetry |
| Some remarks on M(atrix) models |
| Two-dimensional conformal field theories |
| Recap of two-dimensional, local quantum field theory / 7.4.1.： |
| A dictionary between conformal field theory and Lie group theory / 7.4.2.： |
| Reconstruction of (non-commutative) target spaces from conformal field theory |
| Superconformal field theories, and the topology of target spaces |
| The N = 1 super-Virasoro algebra / 7.6.1.： |
| N = 2 and N = 4 supersymmetry; mirror symmetry / 7.6.2.： |
| Conclusions |
| Lectures on the Quantum Geometry of String Theory / B.R. GreeneCourse 4.： |
| What is quantum geometry? / 1.1.： |
| The ingredients / 1.2.： |
| The N = 2 superconformal algebra |
| The algebra |
| Representation theory of the N = 2 superconformal algebra |
| Chiral primary fields / 2.3.： |
| Spectral flow and the U(1) projection / 2.4.： |
| Four examples / 2.5.： |
| Example one: free field theory / 2.5.1.： |
| Example two: nonlinear sigma models / 2.5.2.： |
| Example three: Landau-Ginzburg models / 2.5.3.： |
| Example four: minimal models / 2.5.4.： |
| Families of N = 2 theories |
| Marginal operators |
| Moduli spaces: I |
| Interrelations between various N = 2 superconformal theories |
| Landau-Ginzburg theories and minimal models |
| Minimal models and Calabi-Yau manifolds: a conjectured correspondence |
| Arguments establishing minimal-model/Calabi-Yau correspondence |
| Mirror manifolds |
| Strategy of the construction |
| Minimal models and their automorphisms |
| Direct calculation |
| Constructing mirror manifolds |
| Examples / 5.5.： |
| Implications / 5.6.： |
| Spacetime topology change |
| Basic ideas |
| Mild topology change |
| Kahler moduli space / 6.2.1.： |
| Complex structure moduli space / 6.2.3.： |
| Implications of mirror manifolds: revisited / 6.2.4.： |
| Flop transitions / 6.2.5.： |
| An example / 6.2.6.： |
| Drastic topology change |
| Strominger's resolution of the conifold singularity / 6.3.1.： |
| Conifold transitions and topology change / 6.3.3.： |
| Symmetry Approach to the XXZ Model / T. MiwaCourse 5.： |
| The XXZ Hamiltonian for [Delta] [ -1 |
| Transfer matrix |
| Symmetry of U[subscript q](sl[subscript 2]) |
| Corner transfer matrix |
| Level 1 highest weight module |
| Half transfer matrix |
| Intertwiners |
| The vacuum vector |
| Diagonalization of the transfer matrix |
| Local operators and difference equations |
| Superstring Dualities, Dirichlet Branes and the Small Scale Structure of Space / M.R. DouglasSeminar 1.： |
| Duality and solitons in supersymmetric field theory |
| Duality and solitons in superstring theory |
| Dirichlet branes |
| Short distances in superstring theory |
| Further directions |
| Testing the Standard Model and Beyond / J. EllisSeminar 2.： |
| Introduction to the Standard Model and its (non-topological) defects |
| Testing the Standard Model |
| The electroweak vacuum |
| Motivations for supersymmetry |
| Model building |
| Physics with the LHC |
| Quantum Group Approach to Strongly Coupled Two Dimensional Gravity / J.-L. GervaisSeminar 3.： |
| Basic points about Liouville theory |
| The basic relations between 6j symbols |
| The Liouville string |
| Concluding remarks |
| N = 2 Superalgebra and Non-Commutative Geometry / H. Grosse ; C. Klimcik ; P. PresnajderSeminar 4.： |
| Commutative supersphere |
| Non-commutative supersphere |
| Outlook |
| Lecture on N = 2 Supersymmetric Gauge Theory / W. LercheSeminar 5.： |
| Semi-classical N = 2 Yang-Mills theory for G = SU(2) |
| The exact quantum moduli space |
| Solving the monodromy problem |
| Picard-Fuchs equations |
| Generalization to SU(n) |
| Physics / Part II.： |
| Noncommutative Geometry: The Spectral Aspect / A. ConnesCourse 6.： |
| Noncommutative geometry: an introduction |
| Infinitesimal calculus |
| Local index formula and the transverse fundamental class |
| The notion of manifold and the axioms of geometry |
| The spectral geometry of space-time |
| The KZB Equations on Riemann Surfaces / G. FelderCourse 7.： |
| Conformal blocks on Riemann surfaces |
| Kac-Moody groups |
| Principal G-bundles |
| Conformal blocks |
| The connection |
| The energy-momentum tensor |
| Flat structures |
| Connections on bundles of projective spaces / 3.3.： |
| The Friedan-Shenker connection / 3.4.： |
| The Knizhnik-Zamolodchikov-Bernard equations |
| Dynamical r-matrices |
| An explicit form for the connection |
| Transformation properties |
| Moving points |
| Fixing the complex structure |
| Proof of Theorem 5.2 |
| From Diffeomorphism Groups to Loop Spaces via Cyclic Homology / J.-L. LodayCourse 8.： |
| Diffeomorphism group and pseudo-isotopy space |
| Algebraic K-theory via Quillen +-construction |
| The +-construction |
| First definition of Waldhausen's space A(X) |
| The Grothendieck group K[subscript 0] |
| Hochschild and cyclic homology, Lie algebras |
| Hochschild homology |
| Cyclic homology |
| Relationship with the Lie algebra homology of matrices |
| Computing A(X) out of the loop space [Lambda]X |
| Algebraic K-theory via Waldhausen S.-construction and Wh(X) |
| Waldhausen S.-construction |
| A(X) and Wh(X) via the S.-construction |
| Relating Wh(X) to pseudo-isotopy |
| Notation and terminology in algebraic topology / Appendix A.： |
| Homotopy theory / A.1.： |
| Classifying spaces / A.2.： |
| Simplicial sets and classifying spaces / Appendix B.： |
| More on classifying spaces of categories / B.1.： |
| Bisimplicial sets / B.2.： |
| References with comments |
| Quantum Groups and Braid Groups / M. RossoCourse 9.： |
| The Yang-Baxter equation, braid groups and Hopf algebras |
| Drinfeld's quantum double |
| The dual double construction |
| The quantum double and its properties |
| Hopf pairings and a generalized double |
| The quantized enveloping algebra U[subscript q]G |
| Construction of U[subscript q]G |
| A Hopf pairing U[subscript +] [times] U[subscript -] [right arrow] C(q) / 4.1.1.： |
| Some results from representation theory |
| The quantum shuffle construction |
| The quantum shuffle Hopf algebra |
| Hopf bimodules |
| Braidings |
| The cotensor Hopf algebra |
| The quantum symmetric algebra |
| The examples from abelian group algebras |
| A classification result |
| Multiplicative bases in the quantum shuffle algebra / 5.3.1.： |
| Consequences of growth conditions / 5.3.2.： |
| From Index Theory to Non-Commutative Geometry / N. TelemanCourse 10.： |
| Differential forms on smooth and Lipschitz manifolds |
| Riemannian metrics and L[subscript 2]-forms on smooth and Lipschitz manifolds |
| Hodge theory on smooth and Lipschitz manifolds |
| Analytical index of Fredholm operators on smooth and Lipschitz manifolds |
| Topological K-theory |
| Symbols of elliptic operators on smooth manifolds and their index |
| Characteristic classes, Chern character |
| Stiefel-Whitney classes of real vector bundles |
| Chern classes of complex vector bundles |
| Pontrjagin classes of real vector bundles |
| Chern-Weyl theory on smooth manifolds |
| Thom isomorphism |
| Thom isomorphism in cohomology |
| Thom isomorphism in K-theory |
| Comparison between the Thom isomorphism in cohomology and K-theory |
| Index theorem for smooth manifolds |
| Index theorem for Lipschitz manifolds |
| Quasi local formulas for Thom-Hirzebruch classes on quasi conformal manifolds |
| Compact Quantum Groups / S.L. WoronowiczCourse 11.： |
| Definitions and results |
| The Haar measure |
| Unitary representations |
| Right regular representation |
| The Hopf algebras |
| Peter-Weyl theory |
| Groups with faithful Haar measure |
| Seiberg-Witten Invariants and Vortex Equations / O. Garcia-PradaSeminar 6.： |
| Preliminaries on spin geometry, almost-complex geometry and self-duality |
| The Seiberg-Witten invariants |
| Kahler complex surfaces |
| Non-Kahler complex surfaces |
| Symplectic four-manifolds |
| Non-Abelian monopole equations |
| Quantization of Poisson Algebraic Groups and Poisson Homogeneous Spaces / P. Eting of ; D. KazhdanSeminar 7.： |
| Quantization of Poisson algebraic and Lie groups |
| Quantization of Poisson homogeneous spaces |
| Eta and Torsion / J. LottSeminar 8.： |
| Eta-invariant |
| Analytic torsion |
| Eta-forms |
| Analytic torsion forms |
| Symplectic Formalism in Conformal Field Theory / A. SchwarzSeminar 9.： |
| Symplectic formalism in classical field theory |
| Superconformal geometry |
| Superconformal field theory |
| Quantization of geometry associated to the quantized Knizhnik-Zamolodchikov equations / A. VarchenkoSeminar 10.： |
| KZ equations |
| Hypergeometric functions |
| Geometry of hypergeometric functions |
| qKZ equations |
| Solutions to the qKZ equations and eigenvectors of commuting Hamiltonians |
| Solutions to the qKZ equations |
| Difference equations of the discrete connection |
| p-Homology theory |
| Conclusion |
図書
