The Hodge-Elliptic Genus, Spinning BPS States, and Black Holes

We perform a refined count of BPS states in the compactification of M-theory on K3×T2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${K3 \times T^2}$$\end{document}, keeping track of the information provided by both the SU(2)L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${SU(2)_L}$$\end{document} and SU(2)R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${SU(2)_R}$$\end{document} 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×T2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${K3 \times T^2}$$\end{document}, simultaneously refining Katz, Klemm, and Pandharipande’s motivic stable pairs counts on K3 and Oberdieck–Pandharipande’s Gromov–Witten counts on K3×T2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${K3 \times T^2}$$\end{document}. 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.