String Theory near a Conifold Singularity

TL;DR

This paper reveals a deep connection between type II string theory on a conifold singularity and $c=1$ string theory at the self-dual radius, using mirror symmetry, two-dimensional QCD, and Chern-Simons theory, supporting S-duality conjectures.

Contribution

It establishes a novel correspondence between complex structure deformations of the conifold and $c=1$ string theory, providing new evidence for S-duality between different string theories.

Findings

Complex structure deformations relate to maps from surfaces to $S^2$.

The problem is equivalent to $c=1$ string theory via two-dimensional QCD.

Alternative derivation using Chern-Simons theory on $S^3$.

Abstract

We demonstrate that type II string theory compactified on a singular Calabi-Yau manifold is related to $c=1$ string theory compactified at the self-dual radius. We establish this result in two ways. First we show that complex structure deformations of the conifold correspond, on the mirror manifold, to the problem of maps from two dimensional surfaces to $S^2$. Using two dimensional QCD we show that this problem is identical to $c=1$ string theory. We then give an alternative derivation of this correspondence by mapping the theory of complex structure deformations of the conifold to Chern-Simons theory on $S^3$. These results, in conjunction with similar results obtained for the compactification of the heterotic string on $K_3\times T^2$, provide strong evidence in favour of S-duality between type II strings compactified on a Calabi-Yau manifold and the heterotic string on $K_3\times T^2$.