论文标题
扭曲的滑轮和$ \ mathrm {su}（r） / \ mathbb {z} _r $ vafa-witten理论
Twisted sheaves and $\mathrm{SU}(r) / \mathbb{Z}_r$ Vafa-Witten theory
论文作者
论文摘要
$ \ mathrm {su}（r）$ vafa-witten分区函数实际上是在投射表面$ s $上计数higgs对的，它是由田中 - 托马斯（Tanaka-Thomas）数学定义的。在Langlands双方，名称为姓名的作者最近在$μ_r$ - gerbes上引入了希格斯对的虚拟计数。在本文中，我们使用Yoshioka的扭曲滑轮的模量空间。使用Chern字符扭曲有理$ b $ - field，我们给出了$ \ mathrm {su}（r） / \ mathbb {z} _r $ vafa-witten分区函数的新数学定义。我们的定义使用de jong的时期索引定理。 $ s $ - 偶性是物理学的概念，预测$ \ mathrm {su}（r）$和$ \ mathrm {su}（r） / \ mathbb {z} _r $ partitions函数与模块化转换相关。我们将其转变为数学猜想，我们证明了所有$ K3 $表面和质数$ r $。
The $\mathrm{SU}(r)$ Vafa-Witten partition function, which virtually counts Higgs pairs on a projective surface $S$, was mathematically defined by Tanaka-Thomas. On the Langlands dual side, the first-named author recently introduced virtual counts of Higgs pairs on $μ_r$-gerbes. In this paper, we instead use Yoshioka's moduli spaces of twisted sheaves. Using Chern character twisted by rational $B$-field, we give a new mathematical definition of the $\mathrm{SU}(r) / \mathbb{Z}_r$ Vafa-Witten partition function when $r$ is prime. Our definition uses the period-index theorem of de Jong. $S$-duality, a concept from physics, predicts that the $\mathrm{SU}(r)$ and $\mathrm{SU}(r) / \mathbb{Z}_r$ partitions functions are related by a modular transformation. We turn this into a mathematical conjecture, which we prove for all $K3$ surfaces and prime numbers $r$.
