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.

中文翻译：

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对偶和模拟模块化

我们针对规范组$ SO为扭曲的四维$ \ mathcal {N} = 4 $ super Yang-Mills理论在$ \ mathbb {CP} ^ {2} $上推导了Vafa-Witten分区函数的全纯异常方程（3）来自库仑分支上有效理论的路径积分。该方程的全纯内核仅从实例中获得贡献，它不是模块化的，而是“模拟模块化”的。仅在包括反实例的异常非全态边界贡献之后，分区函数才具有$ S $ -duality期望的正确模块化属性。使用M-理论对偶性，我们将此现象与二维非紧致sigma模型的椭圆属的全同型异常联系起来，并在二维中独立地对其进行计算。六维（2,0）理论对张量分支的有效作用可以将四个维度和两个维度上的异常追溯到一个拓扑项。我们考虑了对其他歧管和其他量规组的概括，以表明模拟模量是通用的，并且在相关领域空间不紧凑时表现出对偶性是必不可少的。