Quantum vortices, M2-branes and black holes

A bstract 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 S 2 × 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 N 2 degrees of freedom except N^3{2} N 3 2 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 AdS 4 × S 7. Our Cardy free energy also suggests a finite N version of the N^3{2} N 3 2 degrees of freedom.