Calabi-Yau 3-folds from 2-folds

TL;DR

This paper explores the geometric and topological structures of Calabi-Yau 3-folds derived from 2-folds via wrapped D6-branes in type IIA string theory, providing explicit conditions, constructions, and examples.

Contribution

It introduces a complete set of supergravity conditions for wrapped D6-branes on 2-cycles, and constructs the resulting Calabi-Yau 3-folds with detailed topological and geometric features.

Findings

Derived conditions for supergravity solutions with wrapped branes

Constructed explicit Calabi-Yau 3-fold examples with new 2-cycles

Connected 3-fold topology to underlying 2-fold and brane configurations

Abstract

We consider type IIA string theory on a Calabi-Yau 2-fold with D6-branes wrapping 2-cycles in the 2-fold. We find a complete set of conditions on the supergravity solution for any given wrapped brane configuration in terms of SU(2) structures. We reduce the problem of finding a supergravity solution for the wrapped branes to finding a harmonic function on R$\times$CY$_2$. We then lift this solution to 11-dimensions as a product of R$^{(4.1)}$ and a Calabi-Yau 3-fold. We show how the metric on the 3-fold is determined in terms of the wrapped brane solution. We write down the distinguished (3,0) form and the K{\"a}hler form of the 3-fold in terms of structures defined on the base 2-d complex manifold. We discuss the topology of the 3-fold in terms of the D6-branes and the underlying 2-fold. We show that in addition to the non-trivial cycles inherited from the underlying 2-fold there are $N-1$ new 2-cycles. We construct closed (1,1) forms corresponding to these new cycles. We also display some explicit examples. One of our examples is that of D6-branes wrapping the 2-cycle in an A$_1$ ALE space, the resulting 3-fold has $h^{(1,1)}=N$, where $N$ is the number of D6-branes.