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

摘要: 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 squashing parameter $b$, which has been computed terms of an $N$ dimensional matrix model integral using supersymmetric localization. show that setting $m_3=i\frac{b-b^{-1}}{2}$ relates to round energy, implies infinite relations between $b$ derivatives evaluated at $m_I=0$ $b=1$. For $\mathcal{N}=8$ theory, these fix all fourth order some fifth $m_1,m_2$, were previously orders $1/N$ Fermi gas method. This allows us compute $\partial_b^4 F\vert_{b=1}$ $\partial_b^5 $1/N$, we precisely match a recent prediction sub-leading from holographically dual $AdS_4$ bulk theory.