Orientifold and Integrability: $\mathcal{N}=4$ Super-Yang-Mills on $\mathbb{RP}^4$

Abstract

We study planar $\mathcal{N}=4$ super Yang-Mills theory (SYM) on the real projective space $\mathbb{RP}^4$ using integrability, holography and supersymmetric localization, and discuss its relation to orientifolds in string theory. We show that the integrability of $\mathcal{N}=4$ SYM is preserved when combining the $\mathbb{Z}_2$ involution defining $\mathbb{RP}^4$ with the action of the charge conjugation of the SU$(N)$ gauge group. We demonstrate this by calculating one-point functions of non-BPS single-trace operators in various subsectors at weak coupling, which can be mapped to overlaps between energy eigenstates and crosscap states in integrable spin chains. The results exhibit a parity selection rule, a hallmark of integrability, and are given in terms of a ratio of determinants. Building on these results, we conjecture a closed formula for general single-trace operators. We further provide evidence that this construction is holographically dual to a probe orientifold O1-plane in $\text{AdS}_5\times \text{S}^5$ by constructing the corresponding worldsheet crosscap states in the pp-wave limit. Finally, we use supersymmetric localization to compute one-point functions of half-BPS operators exactly at finite $g_{\rm YM}$ and $N$. The computation reduces to two-dimensional Yang-Mills theory on $\mathbb{RP}^2$ twisted by charge conjugation and leads to a matrix integral of a Wishart-Laguerre type. In the planar limit, the exact result interpolates between the weak-coupling gauge-theory result, the strong-coupling supergravity prediction for the orientifold, and the worldsheet crosscap in the pp-wave limit.