Two-Dimensional Conformal Geometry and Vertex Operator Algebras

The focus of this volume is to formulate and prove one main theorem, the equivalance between the algebraic and geometric formulations of the notion of vertex operator algebra. The author introduces a geomatric notion of vertex operator algebra in terms of complex powers of the determinant line bundles over certain moduli spaces (parameter spaces) of spheres (genus-zero Riemann surfaces) with punctures and local analytic co-ordinates, and seeks to prove that this notion is precisely equivalent to the algebraic notion of vertex operator algebra. In particular, a detailed algebraic and analytic study of the sewing operation in the moduli space is presented.

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.