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

We study the four-dimensional $\mathcal{N}=4$ super-Yang-Mills (SYM) theory on the unorientable spacetime manifold $\mathbb{RP}^4$. Using supersymmetric localization, we find that for a large class of local and extended SYM observables preserving a common supercharge $\mathcal{Q}$, their expectation values 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 the SYM partition function on $\mathbb{RP}^4$ which, when decomposed into discrete holonomy sectors, contains subtle phase factors due to the nontrivial $\eta$-invariant of the Dirac operator on $\mathbb{RP}^4$. We also comment on potential applications of our setup for AGT correspondence, integrability and bulk-reconstruction in AdS/CFT that involve cross-cap states on the boundary.