Comments on the Giant-Graviton Expansion of the Superconformal Index

Work by Gaiotto and Lee, and by Imamura and collaborators suggests that the superconformal index of $U(N)$ gauge theory should be expressible as a convergent series whose terms are indices of associated $U(k)$ gauge theories realized as the worldvolume theories of stacks of $k$ giant-graviton branes. A different expansion for the index provided by Murthy was shown to hold very generally, but the connection to the first expansion was not immediately clear. We study the relation between the two expansions and propose a prescription for extracting the terms of the first series from those of the second. We follow this prescription in the case of the $1/2$-BPS index and show that the contribution of the $m$th term of the first expansion is fully encoded in the first $m$ terms of the second. In addition, we identify the $m$th term of the second expansion with the expectation value of the $N$th power of the superdeterminant in a $U(m|m)$ superunitary matrix integral, which hints at a brane/anti-brane origin for the term.