A Mirror Theorem for Non-split Toric Bundles: Genus-zero Gromov-Witten Theory

Table of Links

• Abstract and Intro
• Genus-zero Gromov-Witten Theory
• Toric Bundles
• Lagrangian cones of Toric bundles
• Mirror theorem for a product of projectives bundles
• Mirror Theorem for Toric Bundles
• Appendix A. Equivariant Fourier Transformation and References

2. Genus-zero Gromov-Witten theory

In this section, we briefly recall the (torus-equivariant/twisted) genus-zero Gromov-Witten theory. We will introduce Gromov-Witten invariants, Givental Lagrangian cones and the quantum Riemann-Roch theorem.

2.1. Gromov-Witten invariant and its variants. We recall the definition of Gromov-Witten invariant. We also introduce an torus-equivariant version and a twisted version of it.

2.3. Quantum Riemann-Roch theorem and twisted theory. We introduce quantum Riemann-Roch theorem [9, Corollary 4], which relates twisted Givental cones via some transcendental operators. We also explain relationships between the Gromow-Witten theory of a vector bundle (resp. a subvariety) and that of a base space (resp. an ambient space) in terms of twisted theories. Note that we will use the material in this subsection only in Section 5.