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 $N^{\frac{3}{2}}$ \( {N}^{\frac{3}{2}} \) 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 AdS₄ × S⁷. Our Cardy free energy also suggests a finite N version of the $N^{\frac{3}{2}}$ \( {N}^{\frac{3}{2}} \) degrees of freedom.