LAUSR

Abstract We study the four-dimensional $$ \mathcal{N} $$ N = 4 super-Yang-Mills (SYM) theory on the unorientable spacetime manifold ℝℙ4. 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 ℝℙ2 submanifold. This paves the way for understanding $$ \mathcal{N} $$ N = 4 SYM on ℝℙ4 using known results of YM on ℝℙ2. As an illustration, we derive a matrix integral form of the SYM partition function on ℝℙ4 which, when decomposed into discrete holonomy sectors, contains subtle phase factors due to the nontrivial η-invariant of the Dirac operator on ℝℙ4. 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.