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

Abstract

Abstract We consider the sphere free energy F(b; m I ) in N $$ \mathcal{N} $$ = 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 b − b − 1 2 $$ {m}_3=i\frac{b-{b}^{-1}}{2} $$ 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 $$ \mathcal{N} $$ = 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 ∂ b 4 F b = 1 $$ {\partial}_b^4F\left|{}_{b=1}\right. $$ and ∂ b 5 F b = 1 $$ {\partial}_b^5F\left|{}_{b=1}\right. $$ 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.

Keywords

• 1/N Expansion
• AdS-CFT Correspondence
• M-Theory
• Matrix Models