Bogomolov-Gieseker type inequality and counting invariants

TL;DR

This paper explores a conjectural link between different enumerative invariants on Calabi-Yau 3-folds, connecting string theory conjectures with algebraic geometry through a proven inequality assumption.

Contribution

It proves a conjecture relating Donaldson-Thomas invariants and stable pairs assuming a Bogomolov-Gieseker type inequality.

Findings

Established a proof of the conjecture under the inequality assumption

Connected enumerative invariants with string theory conjectures

Provided a mathematical framework for black hole entropy and topological string relations

Abstract

We study a conjectural relationship among Donaldson-Thomas type invariants on Calabi-Yau 3-folds counting torsion sheaves supported on ample divisors, ideal sheaves of curves and Pandharipande-Thomas's stable pairs. The conjecture is a mathematical formulation of Denef-Moore's formula derived in the study of Ooguri-Strominger-Vafa's conjecture relating black hole entropy and topological string. The main result of this paper is to prove our conjecture assuming a conjectural Bogomolov-Gieseker type inequality proposed by Bayer, Macri and the author.