The free energy of N=2 supersymmetric AdS_4 solutions of M-theory

TL;DR

This work develops a geometric framework for general ${\cal N}=2$ AdS$_4$ solutions of M-theory with nonzero M2 charge by exhibiting a canonical contact structure on the internal space $Y_7$ and a Reeb vector $\xi$ that encodes the $U(1)_R$ symmetry. The authors derive holographic expressions for the free energy on $S^3$ and for the scaling dimensions of BPS M5-brane operators in terms of the contact data, notably ${\cal F} = N^{3/2}\sqrt{\frac{32\pi^6}{9\int_{Y_7} \sigma\wedge (\mathrm{d}\sigma)^3}}$ and $\Delta(\mathcal{O}_{\Sigma_5}) = \pi N \left| \frac{\int_{\Sigma_5} \sigma\wedge (\mathrm{d}\sigma)^2}{\int_{Y_7} \sigma\wedge (\mathrm{d}\sigma)^3} \right|$. The paper also connects these quantities to topological data via a Duistermaat–Heckman formalism and shows a universal IR/UV free-energy ratio for mass deformations of CY$_3\times\mathbb{C}$, highlighting the power of contact geometry in controlling holographic observables.

Abstract

We show that general N=2 supersymmetric AdS_4 solutions of M-theory with non-zero M2-brane charge admit a canonical contact structure. The free energy of the dual superconformal field theory on S^3 and the scaling dimensions of operators dual to supersymmetric wrapped M5-branes are expressed via AdS/CFT in terms of contact volumes. In particular, this leads to topological and localization formulae for the coefficient of N^{3/2} in the free energy of such solutions.