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 |