Exact probes of orientifolds

We compute the exact vacuum expectation value of circular Wilson loops for Euclidean N $$ \mathcal{N} $$ = 4 super Yang-Mills with G = SO( N ) , Sp( N ), in the fundamental and spinor representations. These field theories are dual to type IIB string theory compactified on AdS 5 × ℝ ℙ 5 $$ \mathbb{R}{\mathrm{\mathbb{P}}}^5 $$ plus certain choices of discrete torsion, and we use our results to probe this holographic duality. We first revisit the LLM-type geometries having AdS 5 × ℝ ℙ 5 $$ \mathbb{R}{\mathrm{\mathbb{P}}}^5 $$ as ground state. Our results clarify and refine the identification of these LLM-type geometries as bubbling geometries arising from fermions on a half harmonic oscillator. We furthermore identify the presence of discrete torsion with the one-fermion Wigner distribution becoming negative at the origin of phase space. We then turn to the string world-sheet interpretation of our results and argue that for the quantities considered they imply two features: first, the contribution coming from world-sheets with a single crosscap is closely related to the contribution coming from orientable world-sheets, and second, world-sheets with two crosscaps don’t contribute to these quantities