Squashing, mass, and holography for 3d sphere free energy

We consider the sphere free energy F(b; m_I) in N = 6 ABJ(M) theory deformed by both three real masses m_I and the squashing parameter b, which has been computed in terms of an N dimensional matrix model integral using supersymmetric localization. We show that setting m3=ib−b−12 relates F(b; m_I) to the round sphere free energy, which implies infinite relations between m_I and b derivatives of F(b; m_I) evaluated at m_I = 0 and b = 1. For N = 8 ABJ(M) theory, these relations fix all fourth order and some fifth order derivatives in terms of derivatives of m₁, m₂, which were previously computed to all orders in 1/N using the Fermi gas method. This allows us to compute ∂b4F|b=1 and ∂b5F|b=1 to all orders in 1/N, which we precisely match to a recent prediction to sub-leading order in 1/N from the holographically dual AdS₄ bulk theory.