We compute the \(\mathcal{N}=2\) supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kähler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on \(\mathbb {C}^2\). The evaluation of these residues is greatly simplified by using an “abstruse duality” that relates the residues at the poles of the one-loop and instanton parts of the \(\mathbb {C}^2\) partition function. As particular cases, our formulae compute the SU(2) and SU(3) equivariant Donaldson invariants of \(\mathbb {P}^2\) and \(\mathbb {F}_n\) and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the SU(2) case. Finally, we show that the U(1) self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a \(\mathcal {N}=2\) analog of the \(\mathcal {N}=4\) holomorphic anomaly equations.

我们通过等变定位计算四维紧凑复曲面流形上规范理论的\(\mathcal{N}=2\)超对称配分函数。结果由 Kähler 形式的分段常数函数给出，沿壁跳跃，规范对称性得到增强。这种流形上的配分函数写成\(\mathbb {C}^2\)上配分函数乘积的余数的和。通过使用“深奥二元性”，可以极大地简化对这些残基的评估，该对偶性将\(\mathbb {C}^2\)分配函数的单循环和瞬时子部分的极点处的残基联系起来。作为特殊情况，我们的公式计算SU (2) 和SU (3)\(\mathbb {P}^2\)和\(\mathbb {F}_n\) 的等变唐纳森不变量以及在非等变极限中再现了通过墙交叉和爆破方法在SU 中获得的结果(2 ） 案件。最后，我们表明U (1) 自对偶连接引起对规范耦合的异常依赖，结果证明满足\(\mathcal {N}=2\)模拟的\(\mathcal {N} =4\)全纯异常方程。