论文标题
双重性和模块化
Duality and Mock Modularity
论文作者
论文摘要
我们为Vafa-Witten分区函数得出了一个全体形态异常方程，用于扭曲的四维$ \ Mathcal {n} = 4 $ super Yang-Mills Theory on $ \ Mathbb {cp}^{2} $ for Garuge组$ so（3）$ so（3）$从COULOMB分支上的Path Ote（3）$。该方程式的全态核仅从激体顿获得贡献，不是模块化的，而是“模块化模块”。该分区函数只有在包括反对抗Instantons的异常非核形态边界贡献之后，才具有正确的模块化属性。使用M理论二元性，我们将这种现象与二维非竞争Sigma模型的椭圆形属的全态异常联系起来，并在二维中独立计算。在六维（2,0）理论对张量分支的有效作用中，可以将四个和二维中的异常态度归于拓扑术语。我们考虑对其他歧管和其他量规组进行概括，以表明当相关的野外空间不相同时，模拟模块对于表现出二元性至关重要。
We derive a holomorphic anomaly equation for the Vafa-Witten partition function for twisted four-dimensional $\mathcal{N} =4$ super Yang-Mills theory on $\mathbb{CP}^{2}$ for the gauge group $SO(3)$ from the path integral of the effective theory on the Coulomb branch. The holomorphic kernel of this equation, which receives contributions only from the instantons, is not modular but `mock modular'. The partition function has correct modular properties expected from $S$-duality only after including the anomalous nonholomorphic boundary contributions from anti-instantons. Using M-theory duality, we relate this phenomenon to the holomorphic anomaly of the elliptic genus of a two-dimensional noncompact sigma model and compute it independently in two dimensions. The anomaly both in four and in two dimensions can be traced to a topological term in the effective action of six-dimensional (2,0) theory on the tensor branch. We consider generalizations to other manifolds and other gauge groups to show that mock modularity is generic and essential for exhibiting duality when the relevant field space is noncompact.
