Chiral Deformation Theory Of 2d Conformal Field Theory

In this thesis,we study the mathematical theory of chiral deformations of conformal field theories.We define the partition function of chiral deformations of conformal field theory and prove Dijkgraaf's master equation.In chiral deformation theory of free-(7(8 system,we construct the n-point correlation function using Feynman diagrams.We establish a correspondence between quantum master equations and chiral homology groups.We also propose a general framework to study the perturbative aspects of nonlinear-models.We carry out this program in details by the example of topological quantum mechanics.We establish a rigorous connection between topological quantum mechanics and algebraic index theorem.As another application,we study the curved -system over a complex manifold viewed as chiral deformation of free -system.For a complex manifold with a trivialization of the second Chern cahracter,we construct a two dimensional vertex algebra analogue of Fedosov connection.We prove that this connection corresponds to a solution of the quantum master equation.Furthermore,we construct the partition function of general quantum observables in curved -system and establish a coupled equation relating to chiral homology groups of the chiral differential operators.