Two-Dimensional Conformal Geometry and Vertex Operator Algebras

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.