AI 中文总结
研究五维\(\mathcal{N}=1\) \(U(N)\)超对称杨 - 米尔斯理论在复Kähler曲面相关空间上的配分函数，扩展前人工作，通过路径积分定域及处理极点，发现非偶数陈类贡献与精细化Vafa-Witten不变量一致，偶数陈类情况待明确。
AI 中文摘要
我们研究了在复Kähler曲面\(S\)乘以半径为\(\boldsymbol{\beta}\)的圆上，具有质量为\(m_{\rm adj}\)的伴随超多重态的五维\(\mathcal{N}=1\) \(U(N)\)超对称杨-米尔斯（SYM）理论的配分函数。扩展了之前在\(S\)上的\(\mathcal{N}=2^*\) SYM理论以及\(S\times \mathbb{S}^1_{\boldsymbol{\beta}}\)上的纯\(\mathcal{N}=1\) SYM理论的工作，我们发现路径积分定域到沿着每个仿射补丁的Nekrasov 5D配分函数乘积的嘉当环面的积分。为简单起见，限制规范群为\(U(2)\)，被积函数有一组度数至多为\(\chi(S)-2\)的无穷多个极点。通过围绕这些极点积分的自然规定，我们发现有贡献的极点与\(S\)上半稳定无挠层的模空间中的环面不动点一一对应。此外，对于非偶数的第一陈类，它们的贡献与等变参数\(\epsilon_1,\epsilon_2\)无关，并且加起来等于该模空间的\(\chi_{y^2}\)亏格，其中\(y^2=e^{-\boldsymbol{\beta} m_{\rm adj}}\)，因此与精细化Vafa-Witten不变量一致。对于偶数陈类，配分函数依赖于等变参数\(\epsilon_1,\epsilon_2\)以及\(y\)，并且它与有理、精细化Vafa-Witten不变量的关系仍不清楚。
英文摘要
We study the partition function of five-dimensional $\mathcal{N}=1$ $U(N)$ supersymmetric Yang-Mills (SYM) theory with an adjoint hypermultiplet of mass $m_{\rm adj}$ on a toric Kähler surface $S$ times a circle of radius $\boldsymbolβ$. Extending earlier work in $\mathcal{N}=2^*$ SYM theory on $S$, and in pure $\mathcal{N}=1$ SYM on $S\times \mathbb{S}^1_{\boldsymbolβ}$, we find that the path integral localizes to an integral along the Cartan torus of the product of Nekrasov 5D partition functions for each affine patch. Restricting to the gauge group $U(2)$ for simplicity, the integrand has an infinite set of poles of degree at most $χ(S)-2$. With a natural prescription for integrating around such poles, we find that the contributing poles are in one-to-one correspondence with the torus-fixed points in the moduli space of semi-stable torsion-free sheaves on $S$. Moreover, for non-even first Chern class, their contributions are independent of the equivariant parameters $ε_1,ε_2$ and add up to the $χ_{y^2}$-genus of that moduli space, where $y^2=e^{-\boldsymbolβ m_{\rm adj}}$, and hence coincide with the refined Vafa-Witten invariants. For even Chern class, the partition function depends on the equivariant parameters $ε_1,ε_2$ as well as $y$, and its relation to rational, refined Vafa-Witten invariants remains unclear.
Comments65 pages + 3 appendices
