Spherical Branes and the BMN Matrix Quantum Mechanics

We study the maximally supersymmetric Yang-Mills theory on $S^d$ using supersymmetric localisation and holography. We argue that the analytic continuation in dimension to $d=1$ yields a Euclidean version of the BMN matrix quantum mechanics. This system can be analysed at large $N$ using supersymmetric localisation and leads to explicit results for the free energy on $S^d$ and the expectation value of supersymmetric Wilson loops. We show how these results can be reproduced at strong gauge coupling using holography by employing spherical D-brane solutions. We construct these solutions for any value of $d$ using an effective supergravity description and pay particular attention to the subtleties arising in the $d\to1$ limit. Our results have implications for the BMN matrix quantum mechanics and the physics of circular D0-branes.