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

TL;DR

This paper explores the relationship between sphere free energy, mass deformations, and squashing in 3d supersymmetric theories, deriving exact derivative relations and matching them with holographic predictions.

Contribution

It establishes infinite relations between mass and squashing derivatives of free energy, enabling exact calculations and holographic comparisons in ABJ(M) theories.

Findings

Derived relations between mass and squashing derivatives of free energy.

Computed higher-order derivatives of free energy to all orders in 1/N.

Matched field theory results with holographic predictions at sub-leading order.

Abstract

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.