$\mathcal{N}=2$ Super Yang-Mills in AdS$_4$ and $F_{\text{AdS}}$-maximization

TL;DR

This paper explores boundary conditions in 4D $ ext{N}=2$ super Yang-Mills theory on AdS, proposing a maximization principle for the partition function that reveals phase transitions and links to the prepotential.

Contribution

It introduces an $F_{ ext{AdS}}$-maximization principle for boundary conditions in AdS, connecting boundary dynamics to the prepotential in $ ext{N}=2$ SYM.

Findings

At weak coupling, Dirichlet boundary conditions with unbroken gauge symmetry are favored.

At intermediate coupling, boundary conditions favor gauge symmetry breaking to U(1).

Supersymmetric localization computes the partition function nonperturbatively.

Abstract

We investigate the dynamics of four-dimensional $\mathcal{N}=2$ $SU(2)$ 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 $F_{\text{AdS}}=-\log Z_{\text{AdS}}$. At weak coupling $\Lambda L \ll 1$ 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 $\Lambda L \sim \mathcal{O}(1)$ 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 $Z_{\text{AdS}}$ nonperturbatively. We further provide evidence for a relation between $F_{\text{AdS}}$ and the $\mathcal{N}=2$ prepotential in AdS background.