EFT and the SUSY index on the 2nd sheet

The counting of BPS states in four-dimensional ${\cal N}=1$ theories has attracted a lot attention recent years. For superconformal theories, these are one-to-one correspondence with local operators various short representations. generating function for this problem branch cuts and hence several Cardy-like limits, which analogous to high-temperature limits. Particularly interesting is the second sheet, been shown capture microstates phases supersymmetric black holes AdS$_5$. Here we present 3d Effective Field Theory (EFT) approach limit on sheet. We use EFT derive behavior index at orders $\beta^{-2},\beta^{-1},\beta^0$. also make conjecture $O(\beta)$, where argue that expansion truncates up exponentially small corrections. An important point existence vector multiplet zero modes, unaccompanied by massless matter fields. runaway Affleck-Harvey-Witten however avoided non-perturbative confinement mechanism. This mechanism guarantees our results robust.