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A bstract We study the four-dimensional    $$ \mathcal{N} $$  N = 4 super-Yang-Mills (SYM) theory on the unorientable spacetime manifold ℝℙ 4 . Using supersymmetric localization, we find that for a large class of local and extended SYM observables preserving a common supercharge    $$ \mathcal{Q} $$  Q , their expectation values are captured by an effective two-dimensional bosonic Yang-Mills (YM) theory on an ℝℙ 2  submanifold. This paves the way for understanding    $$ \mathcal{N} $$  N = 4 SYM on ℝℙ 4  using known results of YM on ℝℙ 2 . As an illustration, we derive a matrix integral form of the SYM partition function on ℝℙ 4  which, when decomposed into discrete holonomy sectors, contains subtle phase factors due to the nontrivial  η -invariant of the Dirac operator on 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.

From $$ \mathcal{N} $$ = 4 Super-Yang-Mills on ℝℙ4 to bosonic Yang-Mills on ℝℙ2