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

Abstract

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.

28 pages

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Cited by in corpus (2)

• Universal Dynamics in Non-Orientable CFT${}_2$
• Flavor deformations and supersymmetry enhancement in $4d$ $\mathcal{N}=2$ theories