arXiv 2005.071972020-05-14hep-th

从$\mathbb{RP}^4$上的$\mathcal{N}=4$超级杨米尔斯到$\mathbb{RP}^2$上的玻色子杨米尔斯

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

Yifan Wang

备注：: 28 pages

摘要：

研究了不可定向时空流形$\mathbb{RP}^4$上的四维$\mathcal{N}=4$ super-Yang-Mills（SYM）理论.利用超对称局部化方法，我们发现对于一类保持共同超荷$\mathcal{Q}$的局部和扩展SYM观测量，它们的期望值被有效的二维玻色子Yang-Mills（YM）理论在$\mathbb{RP}^2$子流形上捕获.这为利用YM在$\mathbb{RP}^2$上的已知结果理解$\mathbb{RP}^4$上的$\mathcal{N}=4$ SYM铺平了道路。作为一个例子，我们推导出一个矩阵积分形式的SYM配分函数的$\mathbb{RP}^4$，当分解成离散的holonomy部门，包含微妙的相位因子，由于非平凡的$η$ -不变量的狄拉克运营商的$\mathbb{RP}^4$。我们还评论了我们的设置AGT对应，可积性和批量重建的AdS/CFT，涉及交叉帽状态的边界上的潜在应用。

英文摘要：

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 $η$-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.

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