Supergroup Invariants and the Brane/Negative Brane Expansion

We propose a Molien–Weyl-type formula computing generating functions of invariants of supergroups $U(N\mathrm{\mid }M)$, i.e. polynomials in supertraces, which arise as gauge groups of brane/negative brane systems in string theory. We either prove or numerically verify the formula in various examples. The formula further leads to a new expansion relating finite-$N$ and infinite-$N$ indices of $U(N)$ gauge theories. We comment on its relation to Murthy’s Giant Graviton expansion, for which we suggest a physical interpretation in terms of ``Koszul dual" branes and negative branes.