Two-Dimensional Conformal Geometry and Vertex Operator Algebras

Notational conventions.- 1. Spheres with tubes.- 1.1. Definitions.- 1.2. The sewing operation.- 1.3. The moduli spaces of spheres with tubes.- 1.4. The sewing equation.- 1.5. Meromorphic functions on the moduli spaces and meromorphic tangent spaces.- 2. Algebraic study of the sewing operation.- 2.1. Formal power series and exponentials of derivations.- 2.2. The formal sewing equation and the sewing identities.- 3. Geometric study of the sewing operation.- 3.1. Moduli spaces, meromorphic functions and meromorphic tangent spaces revisited.- 3.2. The sewing operation and spheres with tubes of type (1,0), (1,1) and (1,2).- 3.3. Generalized spheres with tubes.- 3.4. The sewing formulas and the convergence of the associated series via the Fischer-Grauert Theorem.- 3.5. A Virasoro algebra structure of central charge 0 on the meromorphic tangent space of K(1) at its identity.- 4. Realizations of the sewing identities.- 4.1. The Virasoro algebra and modules.- 4.2. Realizations of the sewing identities for general representations of the Virasoro algebra.- 4.3. Realizations of the sewing identities for positive energy representations of the Virasoro algebra.- 5. Geometric vertex operator algebras.- 5.1. Linear algebra of graded vector spaces with finite-dimensional homogeneous subspaces.- 5.2. The notion of geometric vertex operator algebra.- 5.3. Vertex operator algebras.- 5.4. The isomorphism between the category of geometric vertex operator algebras and the category of vertex operator algebras.- 6. Vertex partial operads.- 6.1. The ?x -rescalable partial operad structure on the sequence K of moduli spaces.- 6.2. The topological and analytic structures on K.- 6.3. The associativity of the sphere partial operad K.- 6.4. Suboperads and partial suboperads of K.- 6.5. Thedeterminant line bundles over K and the partial operad structure.- 6.6. Meromorphic tangent spaces of determinant line bundles and a module for the Virasoro algebra.- 6.7. Proof of the convergence of projective factors in the sewing axiom.- 6.8. Complex powers of the determinant line bundles.- 6.9. ?-extensions of K.- 7. The isomorphism theorem and applications.- 7.1. Vertex associative algebras.- 7.2. The isomorphism theorem.- 7.3. Geometric construction of some Virasoro vertex operator algebras.- 7.4. Isomorphic vertex operator algebras induced from conformal maps.- Appendix A. Answers to selected exercises.- A.1. Exercise 1.3.5: The proof of Proposition 1.3.4.- A.2. Exercise 2.1.8: Another proof of Proposition 2.1.7.- A.3. Exercise 2.1.12: The proof of Proposition 2.1.11.- A.4. Exercise 2.1.17: The proof of Proposition 2.1.16.- A.5. Exercise 2.1.20: The proof of Proposition 2.1.19.- A.6. Exercise 3.4.2: The sewing formulas.- A.7. Exercise 3.5.1: The definition of the Virasoro bracket.- A.8. Exercise 3.5.3: The calculation of the Virasoro bracket.- A.10. Exercise 5.4.3: The proof of the formula (5.4.10).- A.11. Exercise 6.6.3: The proof of the formula (6.6.20).- A.12. Exercise 6.7.2: The proof of Lemma 6.7.1.- Appendix B. (LB)-spaces and complex (LB)-manifolds.- Appendix C. Operads and partial operads.- C.1. Operads, partial operads and associated algebraic structures.- C.2. Rescaling groups for partial operads, rescalable partial operads and associated algebraic structures.- C.3. Another definition of (partial) operad.- Appendix D. Determinant lines and determinant line bundles.- D.1. Some classes of bounded linear operators.- D.2. Determinant lines.- D.3. Determinant lines over Riemann surfaces with parametrized boundaries.- D.4. Canonical isomorphisms associatedto sewing and determinant line bundles over moduli spaces.- D.6. One-dimensional genus-zero modular functors and the Mumford-Segal theorem.