Holographic and Localization Calculations of Boundary F for {cal N} = 4 SUSY Yang-Mills Theory

${\cal N} = 4$ Supersymmetric Yang-Mills (SYM) theory can be defined on a half-space with a variety of boundary conditions preserving scale invariance and half of the original supersymmetry; more general theories with the same symmetry can be obtained by coupling to a 3D SCFT at the boundary. Each of these theories is characterized by a quantity called "boundary $F$", conjectured to decrease under boundary renormalization group flows. In this paper, we calculate boundary $F$ for $U(N)$ ${\cal N} = 4$ SYM theory with the most general half-supersymmetric boundary conditions arising from string theory constructions with D3-branes ending on collections of D5-branes and/or NS5-branes. We first perform the calculation holographically by evaluating the entanglement entropy for a half-ball centered on the boundary using the Ryu-Takayanagi formula in the dual type IIB supergravity solutions. For boundary conditions associated with D3-branes ending on D5 branes only or NS5-branes only, we also calculate boundary $F$ exactly by evaluating the hemisphere partition function using supersymmetric localization. The leading term at large $N$ in the supergravity and localization results agree exactly as a function of the t' Hooft coupling $\lambda$.