Complex Anti-Self-Dual Connections on a Product of Calabi–Yau Surfaces and Triholomorphic Curves

Abstract:We study the adiabatic limit of a sequence of Ω-anti-self-dual connections on unitary bundles over a product of two compact Calabi–Yau surfaces M×N by scaling metrics to shrink N to a point. We show that after fixing gauge transformations, a subsequence of the N-components of these connections converges to a triholomorphic curve from M away from a Cayley cycle in M×N to the moduli space ${cal M}_N$ of instantons on M×N modulo gauge equivalence in the Hausdorff topology, and converges on the blow-up locus to a family, which is parameterized by the Cayley cycle, of triholomorphic curves from C2 to ${cal M}_N$.

摘要：我们通过缩放度量将\(N\)收缩到一个点，来研究在两个紧卡拉比 - 丘流形\(M\times N\)的乘积上的酉丛上的一列\(\Omega\)-反自对偶联络的绝热极限。我们表明，在固定规范变换之后，这些联络的\(N\)分量的一个子列在豪斯多夫拓扑下，从\(M\times N\)中远离一个凯莱圈的\(M\)出发收敛到一个三全纯曲线，该曲线进入到\(M\times N\)上的瞬子的模空间\({\cal M}_N\)（模规范等价），并且在爆破轨迹上收敛到一族由凯莱圈参数化的从\(C^2\)到\({\cal M}_N\)的三全纯曲线。