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In physics, it is believed that the consistency of two dimensional conformalfield theory follows from the bootstrap equation. In this paper, we introducethe notion of a full vertex algebra by analyzing the bootstrap equation, whichis a "real analytic" generalization of a $\mathbb{Z}$-graded vertex algebra. Wealso give a mathematical formulation of the consistency of four pointcorrelation functions in two dimensional conformal field theory and prove itfor a full vertex algebra with additional assumptions on the conformalsymmetry. In particular, we show that the bootstrap equation together with theconformal symmetry implies the consistency of four point correlation functions.As an application, a deformable family of full vertex algebras parametrized bythe Grassmanian is constructed, which appears in the toroidal compactificationof string theory. This give us examples satisfying the above assumptions.

Full vertex algebra and bootstrap -- consistency of four point functions in 2d CFT