内容

目次

to Group Actions in Symplectic and Complex Geometry.- I. Finite-dimensional manifolds.- 1. Vector space structures.- 2. Local theory.- 3. Global differentiable objects.- 4. A sketch of integration theory.- 5. Smooth submanifolds.- 6. Induced orientation and Stokes’ theorem.- 7. Functionals on de Rham cohomology.- II. Elements of Lie groups and their actions.- 1. Introduction to actions and quotients.- 2. Examples of Lie groups.- 3. Smooth actions of Lie groups.- 4. Fiber bundles.- III. Manifolds with additional structure.- 1. Geometric structures on vector spaces.- 2. The elements of function theory.- 3. A brief introduction to complex analysis in higher dimensions.- 4. Complex manifolds.- 5. Symplectic manifolds.- 6. Kähler manifolds.- IV. Symplectic manifolds with symmetry.- 1. Introduction to the moment map.- 2. Central extensions.- 3. Existence and uniqueness of the moment map.- 4. Basic examples of the moment map.- 5. The Poisson structure on (Lie G)* and on coadjoint orbits.- 6. The basic formula and some consequences.- 7. Moment maps associated to representations.- V. Kählerian structures on coadjoint orbits of compact groups and associated representations.- 1. Generalities on compact groups.- 2. Root decomposition for $${\mathfrak{k}^\mathbb{C}}$$.- 3. Complexification of compact groups.- 4. Algebraicity properties of complexifications of compact groups.- 5. Compact complex homogeneous spaces.- 6. The root groups SL2(?) and H2(G/P, ?).- 7. Representations of complex semisimple groups.- Literature.- Infinite-dimensional Groups and their Representations.- I. Calculus in locally convex spaces.- 1. Differentiable functions.- 2. Differentiable functions on Banach spaces.- 3. Holomorphic functions.- 4. Differentiable manifolds.- 5 Infinite-dimensional Lie groups.- II. Dual spaces of locally convex spaces.- 1. Metrizability.- 2. Semireflexivity.- 3. Completeness properties of the dual space.- III. Topologies on function spaces.- 1. The space C? (M, V).- 2. Smooth mappings between function spaces.- 3. Applications to groups of continuous mappings.- 4. Spaces of holomorphic functions.- IV. Representations of infinite-dimensional groups.- V. Generalized coherent state representations.- 1. The line bundle over the projective space of a topological vector space.- 2. Applications to representation theory.- References.- Borel-Weil Theory for Loop Groups.- I. Compact groups.- II. Loop groups and their central extensions.- 1. Groups of smooth maps.- 2. Central extensions of loop groups.- 3. Appendix IIa: Central extensions and semidirect products.- 4. Appendix IIb: Smoothness of group actions.- 5. Appendix IIc: Lifting automorphisms to central extensions.- 6. Appendix IId: Lifting automorphic group actions to central extensions.- III. Root decompositions.- 1. The Weyl group.- 2. Root decomposition of the central extension.- IV. Representations of loop groups.- 1. Lowest weight vectors and antidominant weights.- 2. The Casimir operator.- V. Representations of involutive semigroups.- VI. Borel-Weil theory.- VII. Consequences for general representations.- References.- Coadjoint Representation of Virasoro-type Lie Algebras and Differential Operators on Tensor-densities.- I. Coadjoint representation of Virasoro group and Sturm-Liouville operators; Schwarzian derivative as a 1-cocycle.- 1. Virasoro group and Virasoro algebra.- 2. Regularized dual space.- 3. Coadjoint representation of the Virasoro algebra.- 4. The coadjoint action of Virasoro group and Schwarzian derivative.- 5. Space of Sturm-Liouville equations as a Diff+(S1)-module.- 6. The isomorphism.- 7. Vect(S1)-action on the space of Sturm-Liouville operators.- II. Projectively invariant version of the Gelfand-Fuchs cocycle and of the Schwarzian derivative.- 1. Modified Gelfand-Fuchs cocycle.- 2. Modified Schwarzian derivative.- 3. Energy shift.- 4. Projective structures.- III. Kirillov’s method of Lie superalgebras.- 1. Lie superalgebras.- 2. Ramond and Neveu-Schwarz superalgebras.- 3. Coadjoint representation.- 4. Projective equivariance and Lie superalgebra osp(1|2).- IV. Invariants of coadjoint representation of the Virasoro group.- 1. Monodromy operator as a conjugation class of $$\widetilde {SL}(2,R)$$.- 2. Classification theorem.- V. Extension of the Lie algebra of first order linear differential operators on S1 and matrix analogue of the Sturm-Liouville operator.- 1. Lie algebra of first order differential operators on S1 and its central extensions.- 2. Matrix Sturm-Liouville operators.- 3. Action of Lie algebra of differential operators.- 4. Generalized Neveu-Schwarz superalgebra.- VI. Geometrical definition of the Gelfand-Dickey bracket and the relation to the Moyal-Weil star-product.- 1. Moyal-Weyl star-product.- 2. Moyal-Weyl star-product on tensor-densities, the transvectants.- 3. Space of third order linear differential operators as a Diff+(S1)-module.- 4. Second order Lie derivative.- 5. Adler-Gelfand-Dickey Poisson structure.- References.- From Group Actions to Determinant Bundles Using (Heat-kernel) Renormalization Techniques.- I. Renormalization techniques.- 1. Renormalized limits.- 2. Renormalization procedures.- 3. Heat-kernel renormalization procedures.- 4. Renormalized determinants.- II. The first Chern form on a class of hermitian vector bundles.- 1. Renormalization procedures on vector bundles.- 2. Weighted first Chern forms on infinite dimensional vector bundles.- III. The geometry of gauge orbits.- 1. The finite dimensional setting.- 2. The infinite dimensional setting.- IV. The geometry of determinant bundles.- 1. Determinant bundles.- 2. A metric on the determinant bundle.- 3. A connection on the determinant bundle.- 4. Curvature on the determinant bundle.- V. An example: the action of diffeomorphisms on complex structures.- 1. The orbit picture.- 2. Riemannian structures.- 3. A super vector bundle arising from the group action.- 4. The determinant bundle picture.- 5. First Chern form on the vector bundle.- References.- Fermionic Second Quantization and the Geometry of the Restricted Grassmannian.- I. Fermionic second quantization.- 1. The Dirac equation and the negative energy problem.- 2. Fermionic multiparticle formalism: Fock space and the CAR-algebra.- II. Bogoliubov transformations and the Schwinger term.- 1. Implementation of operators on the Fock space.- 2. The Schwinger term.- 3. The central extensions Ures~ and GLres~.- III. The restricted Grassmannian of a polarized Hilbert space.- 1. The restricted Grassmannian as a homogeneous complex manifold.- 2. The basic differential geometry of the restricted Grassmannian.- IV. The non-equivariant moment map of the restricted Grassmannian.- 1. Differential k-forms in infinite dimensions.- 2. Symplectic manifolds, group actions and the co-moment map.- 3. Co-momentum and momentum maps (in infinite dimensions).- 4. Examples of symplectic actions and (co-)momementum maps.- 5. The Ures-moment map on Gres and the Schwinger term.- V. The determinant line bundle on the restricted Grassmannian.- 1. The C*-algebraic construction of the determinant bundle DET.- 2. Comparison to other approaches to the determinant bundle.- 3. Holomorphic sections of the dual of DET.- References.