Squashing, Mass, and Holography for 3d Sphere Free Energy

We consider the sphere free energy $F(b;m_I)$ in $\mathcal{N}=6$ 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 $m_3=i\frac{b-b^{-1}}{2}$ 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 $\mathcal{N}=8$ ABJ(M) theory, these relations fix all fourth order and some fifth order derivatives in terms of derivatives of $m_1,m_2$, which were previously computed to all orders in $1/N$ using the Fermi gas method. This allows us to compute $\partial_b^4 F\vert_{b=1}$ and $\partial_b^5 F\vert_{b=1}$ 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 $AdS_4$ bulk theory.