paper

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

arXiv:2102.05643 · doi:10.1007/JHEP04(2021)244

Abstract

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 $m_{3} = i \frac{b - b ^{- 1}}{2}$ 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 $N = 8$ 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 $\partial_{b}^{4} F ∣_{b = 1}$ and $\partial_{b}^{5} F ∣_{b = 1}$ 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 $A d S_{4}$ bulk theory.

23 pages