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${\cal N} = 4$ Supersymmetric Yang-Mills (SYM) theory can be defined on ahalf-space with a variety of boundary conditions preserving scale invarianceand half of the original supersymmetry; more general theories with the samesymmetry can be obtained by coupling to a 3D SCFT at the boundary. Each ofthese theories is characterized by a quantity called "boundary $F$",conjectured to decrease under boundary renormalization group flows. In thispaper, we calculate boundary $F$ for $U(N)$ ${\cal N} = 4$ SYM theory with themost general half-supersymmetric boundary conditions arising from string theoryconstructions with D3-branes ending on collections of D5-branes and/orNS5-branes. We first perform the calculation holographically by evaluating theentanglement entropy for a half-ball centered on the boundary using theRyu-Takayanagi formula in the dual type IIB supergravity solutions. Forboundary conditions associated with D3-branes ending on D5 branes only orNS5-branes only, we also calculate boundary $F$ exactly by evaluating thehemisphere partition function using supersymmetric localization. The leadingterm at large $N$ in the supergravity and localization results agree exactly asa function of the t' Hooft coupling $\lambda$.

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