Elliptic non-Abelian Donaldson-Thomas invariants of $\mathbb{C}^3$

TL;DR

This paper computes elliptic genera related to D-brane systems and proposes a conjectural formula connecting these to higher-rank Donaldson-Thomas invariants on ^3, revealing new links between string theory, algebraic geometry, and M-theory.

Contribution

It introduces a novel computation of elliptic genera for D-brane systems and proposes a conjectural formula for higher-rank Donaldson-Thomas invariants on ^3.

Findings

Elliptic genus depends non-trivially on the number of D7 branes.

JK-residues correspond to coloured plane partitions.

A conjectural formula connects to M-theory graviton index.

Abstract

We compute the elliptic genus of the D1/D7 brane system in flat space, finding a non-trivial dependence on the number of D7 branes, and provide an F-theory interpretation of the result. We show that the JK-residues contributing to the elliptic genus are in one-to-one correspondence with coloured plane partitions and that the elliptic genus can be written as a chiral correlator of vertex operators on the torus. We also study the quantum mechanical system describing D0/D6 bound states on a circle, which leads to a plethystic exponential formula that can be connected to the M-theory graviton index on a multi-Taub-NUT background. The formula is a conjectural expression for higher-rank equivariant K-theoretic Donaldson-Thomas invariants on $\mathbb{C}^3$.