From N=4 Super-Yang-Mills on P4 to bosonic Yang-Mills on P2

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

Automated summary

Through the use of supersymmetric localization, it is found that the N = 4 super-Yang-Mills (SYM) theory on unorientable spacetime manifold can be effectively captured by a two-dimensional bosonic Yang-Mills (YM) theory. This provides a foundation for understanding the relationship between the two theories.