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关于“分支”Seiberg-Witten方程及其不变量的注记

Notes on the "Ramified" Seiberg-Witten Equations and Invariants

Meng-Chwan Tan

备注：
21 pages. Published version
发表信息：
JHEP 1201:067,2012
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摘要：

在这些注记中，我们仔细分析了定向闭四维流形X上的Seiberg-Witten阿贝尔规范理论的超对称组态所对应的“分歧”Seiberg-Witten方程的性质。我们发现，为了有合理的解决方案，这些方程，只有表面运营商与某些参数和嵌入在X，是允许的。此外，X上相应的具有正数量曲率且b^+_2 > 1的“分歧”塞伯格-威滕不变量为零，而如果X具有b^+_2 = 1，则可以存在不变量跳跃的壁交叉。一般来说，对于对应于具有零虚维数的解的模空间的有限数量的基本类中的每一个，卡勒流形上的扰动“分歧”塞伯格-威滕不变量将取决于-除了与表面算子相关的其他参数之外--参数“l”和完整参数“alpha”。尽管如此，（扰动）的“分歧”和普通的不变量被发现是一致的，虽然到一个符号，在一些例子中。

英文摘要：

In these notes, we carefully analyze the properties of the "ramified" Seiberg-Witten equations associated with supersymmetric configurations of the Seiberg-Witten abelian gauge theory with surface operators on an oriented closed four-manifold X. We find that in order to have sensible solutions to these equations, only surface operators with certain parameters and embeddings in X, are admissible. In addition, the corresponding "ramified" Seiberg-Witten invariants on X with positive scalar curvature and b^+_2 > 1, vanish, while if X has b^+_2 = 1, there can be wall-crossings whence the invariants will jump. In general, for each of the finite number of basic classes that corresponds to a moduli space of solutions with zero virtual dimension, the perturbed "ramified" Seiberg-Witten invariants on Kahler manifolds will depend - among other parameters associated with the surface operator - on the monopole number "l" and the holonomy parameter "alpha". Nonetheless, the (perturbed) "ramified" and ordinary invariants are found to coincide, albeit up to a sign, in some examples.

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