Gromov-Witten theory and Donaldson-Thomas theory, I

TL;DR

The paper develops a precise conjectural framework linking Gromov-Witten and Donaldson-Thomas theories for Calabi-Yau 3-folds, formalized by the variable change $e^{iu}=-q$ between their reduced partition functions. It establishes a degree-0 correspondence and extends to local Calabi-Yau geometries, where the conjectures are proven via virtual localization and the topological vertex, connecting GW counts to DT counts through a melting-crystal/3D-partition formalism. The main technical achievement is identifying the DT equivariant vertex with the GW topological-vertex expansion in toric local CY settings, validating the GW/DT equivalence in these cases and illustrating a broader gauge/string duality perspective. These results pave the way for a full GW/DT correspondence on general Calabi-Yau 3-folds and illuminate the role of integrality, locality, and crystal combinatorics in enumerative geometry.

Abstract

We conjecture an equivalence between the Gromov-Witten theory of 3-folds and the holomorphic Chern-Simons theory of Donaldson-Thomas. For Calabi-Yau 3-folds, the equivalence is defined by the change of variables, exp(iu)=-q, where u is the genus parameter of GW theory and q is charge parameter of DT theory. The conjecture is proven for local Calabi-Yau toric surfaces.

Theorems & Definitions (13)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Theorem 1
• Lemma 9