Donaldson-Thomas Invariants of 2-Dimensional sheaves inside threefolds and modular forms

TL;DR

This paper investigates Donaldson-Thomas invariants of 2D sheaves on threefolds, relating them to modular forms and confirming predictions from S-duality in string theory.

Contribution

It expresses these invariants for K3 fibered threefolds in terms of Hilbert scheme Euler characteristics and Noether-Lefschetz numbers, proving their generating function is a vector modular form.

Findings

Invariants are expressed via Hilbert scheme Euler characteristics and Noether-Lefschetz numbers.

Generated functions of these invariants are proven to be vector modular forms of weight -3/2.

Results support the S-duality conjecture predictions.

Abstract

Motivated by the S-duality conjecture, we study the Donaldson-Thomas invariants of the 2 dimensional Gieseker stable sheaves on a threefold. These sheaves are supported on the fibers of a nonsingular threefold X fibered over a nonsingular curve. In the case where X is a K3 fibration, we express these invariants in terms of the Euler characteristic of the Hilbert scheme of points on the K3 fiber and the Noether-Lefschetz numbers of the fibration. We prove that a certain generating function of these invariants is a vector modular form of weight -3/2 as predicted in S-duality.