Uniform convergence of the ABJM index in transverse fugacities

Establish a uniform summable bound on the signed ABJM index coefficients in the double-scale limit that justifies exchanging the limit with the full infinite transverse-fugacity sum.

Background

The paper proves the ABJM–BMN index relation coefficientwise: for every fixed monopole charge and fixed transverse monomial, the relevant coefficients stabilize as N and k tend to infinity with N/k^2 fixed. This deliberately avoids interchanging the limiting operation with the full infinite fugacity expansion.

The authors note that coefficientwise convergence does not establish convergence of the index as a function of the fugacities. A uniform bound controlling the growth of the signed coefficients with transverse degree and with N and k would be needed to justify such an interchange, and the paper leaves that problem unresolved.