Three Point Functions on the Sphere of Calabi-Yau d-Folds

TL;DR

This paper explores three-point functions in Calabi-Yau d-folds using mirror symmetry, analyzing their structure and counting maps in one- and two-parameter families, revealing a subring of the quantum cohomology ring.

Contribution

It provides a detailed study of three-point functions on Calabi-Yau d-folds, expanding them in terms of Kähler parameters and identifying their algebraic structure as a subring of quantum cohomology.

Findings

Three-point functions are expanded in terms of exponential parameters.

Fusion structure of operators forms a subring of quantum cohomology.

Charge conservation is classical due to perturbation operator switching.

Abstract

Using mirror symmetry in Calabi-Yau manifolds M, three point functions of A(M)-model operators on the genus $0$ Riemann surface in cases of one-parameter families of $d$-folds realized as Fermat type hypersurfaces embedded in weighted projective spaces and a two-parameter family of $d$-fold embedded in a weighted projective space ${\amb}$ are studied. These three point functions ${\corr{\,{{\cal O}^{(1)}_{a}}\, {{\cal O}^{(l-1)}_{b}}\, {{\cal O}^{(d-l)}_{c}}\, }}$ are expanded by indeterminates ${q_l}$=${e^{2\pi i {t_l}}}$ associated with a set of {\kae} coordinates $\{{t_l}\}$ and their expansion coefficients count the number of maps. From these analyses, we can read fusion structure of Calabi-Yau A(M)-model operators. In our cases they constitute a subring of a total quantum cohomology ring of the A(M)-model operators. In fact we switch off all perturbation operators on the topological theories except for marginal ones associated with {\kae} forms of M. For that reason, the charge conservation of operators turns out to be a classical one.