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We study the four-dimensional $\mathcal{N}=4$ super-Yang-Mills (SYM) theoryon the unorientable spacetime manifold $\mathbb{RP}^4$. Using supersymmetriclocalization, we find that for a large class of local and extended SYMobservables preserving a common supercharge $\mathcal{Q}$, their expectationvalues are captured by an effective two-dimensional bosonic Yang-Mills (YM)theory on an $\mathbb{RP}^2$ submanifold. This paves the way for understanding$\mathcal{N}=4$ SYM on $\mathbb{RP}^4$ using known results of YM on$\mathbb{RP}^2$. As an illustration, we derive a matrix integral form of theSYM partition function on $\mathbb{RP}^4$ which, when decomposed into discreteholonomy sectors, contains subtle phase factors due to the nontrivial$\eta$-invariant of the Dirac operator on $\mathbb{RP}^4$. We also comment onpotential applications of our setup for AGT correspondence, integrability andbulk-reconstruction in AdS/CFT that involve cross-cap states on the boundary.

From $\mathcal{N}=4$ Super-Yang-Mills on $\mathbb{RP}^4$ to bosonic Yang-Mills on $\mathbb{RP}^2$