Two-Dimensional Conformal Geometry and Vertex Operator Algebras [Hardcover]

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. The determinant 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 projectivelcä