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図書

図書
édité par A. Connes, K. Gawedzki, et J. Zinn-Justin
出版情報: Amsterdam ; Tokyo : Elsevier Science, c1998  xxxvii, 990 p. ; 23 cm
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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
Lecturers
Seminar Speakers
Participants
2.

図書

図書
édité par H. Maître et J. Zinn-Justin
出版情報: Amsterdam ; Tokyo : Elsevier, 1996  xxxii, 349 p. ; 23 cm
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3.

図書

図書
édité par J.P. Hansen, D. Levesque et J. Zinn-Justin
出版情報: Amsterdam ; New York : North Holland , New York, N.Y., USA : Sole distributors for the U.S.A. and Canada, Elsevier Science Pub. Co., 1991  2 v. (xxxiii, 995 p.) ; 23 cm
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4.

図書

図書
édité par E. Brézin et J. Zinn-Justin
出版情報: Amsterdam ; Tokyo : North-Holland, 1990 , New York : Sole distributors for the U.S.A. and Canada, Elsevier Science, 1990  xxix, 640 p. ; 23 cm
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