$$ \mathcal{N}=2 $$ super Yang-Mills in AdS4 and FAdS-maximization

A bstract We investigate the dynamics of four-dimensional $$ \mathcal{N}=2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> SU(2) super Yang-Mills theory on an AdS background. We propose that the boundary conditions that preserve the AdS super-isometries are determined by maximizing the real part of the AdS partition function F AdS = −log Z AdS . At weak coupling Λ L ≪ 1 the maximization singles out the Dirichlet boundary condition with an SU(2) boundary global symmetry, corresponding to the classical vacuum at the origin of the Coulomb branch with fully un-higgsed gauge group. We find that for Λ L ~ 𝒪(1) new boundary conditions are favored, with gauge-group higgsed down to U(1), matching the expectation from the flat space limit. We use supersymmetric localization to compute Z AdS nonperturbatively. We further provide evidence for a relation between F AdS and the $$ \mathcal{N}=2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> prepotential in AdS background.