The Hodge-elliptic genus, spinning BPS states, and black holes

TL;DR

This paper introduces a refined counting function for BPS states in M-theory compactified on K3×T^2, connecting motivic Donaldson-Thomas and Gromov-Witten counts, and develops a new Hodge-elliptic genus for Calabi-Yau manifolds.

Contribution

It provides the first full motivic curve counts for a compact Calabi-Yau threefold and introduces a new Hodge-elliptic genus interpolating between known invariants.

Findings

First full motivic curve counts for K3×T^2

Development of a Hodge-elliptic genus for Calabi-Yau manifolds

Refined BPS state counting incorporating angular momenta

Abstract

We perform a refined count of BPS states in the compactification of M-theory on $K3 \times T^2$, keeping track of the information provided by both the $SU(2)_L$ and $SU(2)_R$ angular momenta in the $SO(4)$ little group. Mathematically, this four variable counting function may be expressed via the motivic Donaldson-Thomas counts of $K3 \times T^2$, simultaneously refining Katz, Klemm, and Pandharipande's motivic Donaldson-Thomas counts on $K3$ and Oberdieck-Pandharipande's Gromov-Witten counts on $K3 \times T^2$. This provides the first full answer for motivic curve counts of a compact Calabi-Yau threefold. Along the way, we develop a Hodge-elliptic genus for Calabi-Yau manifolds -- a new counting function for BPS states that interpolates between the Hodge polynomial and the elliptic genus of a Calabi-Yau.