Squashing, mass, and holography for 3d sphere free energy
Shai M. Chester, Rohit R. Kalloor, Adar Sharon
Abstract
A bstract We consider the sphere free energy F ( b ; m I ) in $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 $$ {m}_3=i\frac{b-{b}^{-1}}{2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>m</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>i</mml:mi> <mml:mfrac> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>b</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> 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 $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 $$ {\partial}_b^4F\left|{}_{b=1}\right. $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>∂</mml:mi> <mml:mi>b</mml:mi> <mml:mn>4</mml:mn> </mml:msubsup> <mml:mi>F</mml:mi> <mml:mfenced> <mml:msub> <mml:mrow/> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mfenced> </mml:math> and $$ {\partial}_b^5F\left|{}_{b=1}\right. $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>∂</mml:mi> <mml:mi>b</mml:mi> <mml:mn>5</mml:mn> </mml:msubsup> <mml:mi>F</mml:mi> <mml:mfenced> <mml:msub> <mml:mrow/> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mfenced> </mml:math> 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.