Abstract
We study the four-dimensional \( \mathcal{N} \) = 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 \( \mathcal{Q} \), 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 \( \mathcal{N} \) = 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.
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Wang, Y. From \( \mathcal{N} \) = 4 Super-Yang-Mills on ℝℙ4 to bosonic Yang-Mills on ℝℙ2. J. High Energ. Phys. 2021, 203 (2021). https://doi.org/10.1007/JHEP03(2021)203
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DOI: https://doi.org/10.1007/JHEP03(2021)203