The u-plane integral is the contribution of the Coulomb branch to correlation functions of \({\mathcal {N}}=2\) gauge theory on a compact four-manifold. We consider the u-plane integral for correlators of point and surface observables of topologically twisted theories with gauge group \(\mathrm{SU}(2)\), for an arbitrary four-manifold with \((b_1,b_2^+)=(0,1)\). The u-plane contribution equals the full correlator in the absence of Seiberg–Witten contributions at strong coupling, and coincides with the mathematically defined Donaldson invariants in such cases. We demonstrate that the u-plane correlators are efficiently determined using mock modular forms for point observables, and Appell–Lerch sums for surface observables. We use these results to discuss the asymptotic behavior of correlators as function of the number of observables. Our findings suggest that the vev of exponentiated point and surface observables is an entire function of the fugacities.

的ü -平面积分是库仑分支的贡献的相关函数\（{\ mathcal {N}} = 2 \）规范理论上的紧凑四歧管。我们考虑具有规范群\(\mathrm{SU}(2)\)的拓扑扭曲理论的点和表面可观测量的相关器的u平面积分，对于具有\((b_1,b_2^+)的任意四流形=(0,1)\)。该ü -平面贡献等于在不存在中的强耦合Seiberg及-威滕的贡献，以及在这种情况下，数学上定义唐纳森不变一致的全相关。我们证明你使用点可观察量的模拟模块化形式和表面可观察量的 Appell-Lerch 和来有效地确定 -平面相关器。我们使用这些结果来讨论相关器作为可观察数量的函数的渐近行为。我们的研究结果表明，指数点和表面可观测值的 vev 是逸度的完整函数。