Quantum vortices, M2-branes and black holes

We study the partition functions of BPS vortices and magnetic monopole operators, in gauge theories describing N M2-branes. In particular, we explore two closely related methods to study the Cardy limit of the index on S2 × ℝ. The first method uses the factorization of this index to vortex partition functions, while the second one uses a continuum approximation for the monopole charge sums. Monopole condensation confines most of the N2 degrees of freedom except N32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {N}^{\frac{3}{2}} $$\end{document} of them, even in the high temperature deconfined phase. The resulting large N free energy statistically accounts for the Bekenstein-Hawking entropy of large BPS black holes in AdS4 × S7. Our Cardy free energy also suggests a finite N version of the N32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {N}^{\frac{3}{2}} $$\end{document} degrees of freedom.