Topological Orbifold Models and Quantum Cohomology Rings

TL;DR

This paper explores topological sigma models on orbifold target spaces, detailing the moduli space of classical minima, computing quantum cohomology rings for dihedral orbifolds, and discussing applications to nonabelian orbifolds.

Contribution

It provides a detailed method for computing quantum cohomology rings of orbifolds, including explicit calculations for dihedral ${f CP}^1$ and ${f CP}^2/D_4$ orbifolds, and relates these to known algebraic structures.

Findings

Computed quantum cohomology rings for dihedral ${f CP}^1$ orbifolds.

Established a similarity between orbifold rings and $D$-series superpotentials.

Suggested techniques for twist field correlation functions in nonabelian orbifolds.

Abstract

We discuss the toplogical sigma model on an orbifold target space. We describe the moduli space of classical minima for computing correlation functions involving twisted operators, and show, through a detailed computation of an orbifold of ${\bf CP}^1$ by the dihedral group $D_{4},$ how to compute the complete ring of observables. Through this procedure, we compute all the rings from dihedral ${\bf CP}^1$ orbifolds; we note a similarity with rings derived from perturbed $D-$series superpotentials of the $A-D-E$ classification of $N = 2$ minimal models. We then consider ${\bf CP}^2/D_4,$ and show how the techniques of topological-anti-topological fusion might be used to compute twist field correlation functions for nonabelian orbifolds.