ABJM to BMN in a Double Scale Limit: Indices and Bubbling Geometries

Abstract

We study a double scale limit of $U(N)_k\times U(N)_{-k}$ ABJM theory, with $N,k\to\infty$ and $N/k^2\toν\in(0,\infty)$, at fixed positive total monopole charge $q$. In this limit, the Penrose geometry retains a compact null circle of finite radius, and the sector of monopole charge $q$ carries $q$ units of longitudinal momentum. It is therefore naturally associated with the rank-$q$ BMN matrix model, while $ν$ fixes its dimensionless coupling. We establish this relation from both gravity and the supersymmetric index. On the gravity side, we take the double scale limit of Hopf quotients of the Donos--Simón half-BPS geometries. The disk carrying the growing background flux becomes an infinite conducting plane, while the remaining finite disks reproduce the Lin--Maldacena electrostatic problem. Their quantized fluxes map directly to the partition data labeling BMN vacua. On the field-theory side, starting from the finite-$N$, finite-$k$ ABJM localization formula, we prove that, at fixed positive monopole charge $q$, the ABJM superconformal index factorizes coefficientwise in the transverse fugacities into a universal neutral contribution and the refined Witten index of the rank-$q$ BMN matrix model, together with the longitudinal momentum weight. The neutral contribution is precisely the limiting zero-monopole-charge index. Dividing by this universal factor therefore isolates the BMN index summed over all its supersymmetric vacua.