All-order continuation of the topologically twisted index sectors

Determine whether the finite-depth quadratic-in-t sector decomposition of the selected-orbit ABJM topologically twisted index continues to all instanton orders at k=1, 2, and 4, including the coefficient sequences R_m^(k), A_m^(k), and B_m^(k).

Background

For the selected Bethe-vacuum orbit, the index exponential sectors are reconstructed through orders 17, 15, and 15 at k=1, 2, and 4, respectively. Each resolved sector is quadratic in the instanton action t and exhibits a specific t+1/m structure.

The paper proposes an all-order expression based on the observed finite data, but explicitly distinguishes the reconstructed orders from the conjectural continuation. Establishing the continuation is necessary for a complete nonperturbative finite-rank description of the index.