Ivancevic_Applied-Diff-Geom

1180 Applied Differential Geometry: A Modern Introduction6.7 Application: Topological String Theory6.7.1 Quantum Geometry FrameworkTo start our review on topological string theory, here we depict a generalquantum geometry framework (see e.g., [Witten (1998a)]).SPECIAL RELATIVITYQUANTUM FIELD THEORYv/c ↑→CLASSICAL DYNAMICSQUANTUM MECHANICSFig. 6.22 The deformation from classical dynamics to quantum field theory (see textfor explanation).The relationship between non–relativistic classical mechanics and quantumfield theory (see [Coleman (1988)]) can be summarized as in Figure6.22. We see that the horizontal axis corresponds to the Planck constant (divided by the typical action of the system being studied), while thevertical axis corresponds to v/c, the ration of motion velocity and lightvelocity.Similarly, in the superstring theory there are also two relevant expansionparameters, as shown in Figure 6.23. Here we see that the horizontal axiscorresponds to the value of the string coupling constant, g s , while the verticalaxis corresponds to the value of the dimensionless sigma model couplingα ′ /R 2 with R being a typical radius of a compactified portion of space). Inthe extreme α ′ = g s = 0 limit, for instance, we recover relativistic particledynamics. For nonzero g s we recover point particle quantum field theory.For g s = 0 and nonzero α ′ we are studying classical string theory. In generalthough, we need to understand the theory for arbitrary values of theseparameters (see [Greene (1996)]).Quantum stringy geometry postulates the existence of 6D Calabi–Yaumanifolds at every point of the space–time (see, e.g., [Candelas et. al.(1985)]). These curled–up local manifolds transform according to the generalorbifolding procedure, as will be described below.