The theory of vertex operator algebras and their representations has been showing its power in the solution of concrete mathematical problems and in the understanding of conceptual but subtle mathematical and physical struc tures of conformal field theories. Much of the recent progress has deep connec tions with complex analysis and conformal geometry. Future developments, especially constructions and studies of higher-genus theories, will need a solid geometric theory of vertex operator algebras. Back in 1986, Manin already observed in [Man) that the quantum theory of (super )strings existed (in some sense) in two entirely different mathematical fields. Under canonical quantization this theory appeared to a mathematician as the representation theories of the Heisenberg, Vir as oro and affine Kac Moody algebras and their superextensions. Quantization with the help of the Polyakov path integral led on the other hand to the analytic theory of algebraic (super ) curves and their moduli spaces, to invariants of the type of the analytic curvature, and so on. He pointed out further that establishing direct mathematical connections between these two forms of a single theory was a "big and important problem. " On the one hand, the theory of vertex operator algebras and their repre sentations unifies (and considerably extends) the representation theories of the Heisenberg, Virasoro and Kac-Moody algebras and their superextensions.

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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.