Title: Constructive Wall-Crossing and Seiberg-Witten Show Chinese title

Title: 构造性壁交叉和Seiberg-Witten

Abstract: We outline a comprehensive and first-principle solution to the wall-crossing problem in D=4 N=2 Seiberg-Witten theories. We start with a brief review of the multi-centered nature of the typical BPS states and recall how the wall-crossing problem thus becomes really a bound state formation/dissociation problem. Low energy dynamics for arbitrary collections of dyons is derived, from Seiberg-Witten theory, with the proximity to the so-called marginal stability wall playing the role of the small expansion parameter. We find that, surprisingly, the $\mathbb{R}^{3n}$ low energy dynamics of n+1 BPS dyons cannot be consistently reduced to the classical moduli space, $\CM$, yet the index can be phrased in terms of $\CM$. We also explain how an equivariant version of this index computes the protected spin character of the underlying field theory, where $SO(3)_\CJ$ isometry of $\CM$ turns out to be the diagonal subgroup of $SU(2)_L$ spatial rotation and $SU(2)_R$ R-symmetry. The so-called rational invariants, previously seen in the Kontsevich-Soibelman formalism of wall-crossing, are shown to emerge naturally from the orbifolding projection due to Bose/Fermi statistics. Show Chinese abstract

Abstract: 我们概述了 D=4 N=2 Seiberg-Witten 理论中穿越壁问题的全面第一性原理解决方案。 我们首先简要回顾典型 BPS 态的多中心性质，并回顾穿越壁问题如何真正转化为束缚态的形成/解离问题。 基于 Seiberg-Witten 理论，推导出任意双子集合的低能动力学，其中与所谓边缘稳定壁的接近度充当了小扩展参数的作用。 令人惊讶的是，我们发现，n+1 个 BPS 双子的 $\mathbb{R}^{3n}$ 低能动力学不能一致地简化为经典模空间 $\CM$，但该指标可以用 $\CM$ 来表示。 我们还解释了该指标的等变版本如何计算底层场论的受保护自旋特征，其中 $\CM$ 的 $SO(3)_\CJ$ 等距同构实际上是 $SU(2)_L$ 空间旋转和 $SU(2)_R$ R 对称性的对角子群。 所谓的有理不变量，先前在 Kontsevich-Soibelman 的跨壁形式中出现，被证明是由于 Bose/Fermi 统计而从 orbifolding 投影中自然产生的。