Symétries quantiques : Les Houches, session LXIV, 1 août - 8 septembre 1995 = Quantum symmetries

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Symétries quantiques : Les Houches, session LXIV, 1 août - 8 septembre 1995 = Quantum symmetries / édité par A. Connes, K. Gawedzki, et J. Zinn-Justin

資料種別:

図書

出版情報:

Amsterdam ; Tokyo : Elsevier Science, c1998

形態:

xxxvii, 990 p. ; 23 cm

著者名:

ISBN:

9780444828675 [0444828672]

書誌ID:

BA36052407

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フルテキスト

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書誌詳細

注記:

At head of title: UJFG, NATO ASI
Includes bibliographical references and index

主題:

Geometric quantization--Congresses; Mathematical physics--Congresses; Quantum groups--Congresses; Symmetry (Physics)--Congresses

言語:

英語

目次情報:

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

Preface (French)

Preface (English)

Mathematics / Part I.：

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