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Equivariant K-Theory and Refined Vafa–Witten Invariants
Communications in Mathematical Physics ( IF 2.2 ) Pub Date : 2020-07-20 , DOI: 10.1007/s00220-020-03821-1
Richard P. Thomas

In [MT2] the Vafa-Witten theory of complex projective surfaces is lifted to oriented $\mathbb C^*$-equivariant cohomology theories. Here we study the K-theoretic refinement. It gives rational functions in $t^{1/2}$ invariant under $t^{1/2}\leftrightarrow t^{-1/2}$ which specialise to numerical Vafa-Witten invariants at $t=1$. On the "instanton branch" the invariants give the virtual $\chi_{-t}^{}$-genus refinement of Gottsche-Kool. Applying modularity to their calculations gives predictions for the contribution of the "monopole branch". We calculate some cases and find perfect agreement. We also do calculations on K3 surfaces, finding Jacobi forms refining the usual modular forms, proving a conjecture of Gottsche-Kool. We determine the K-theoretic virtual classes of degeneracy loci using Eagon-Northcott complexes, and show they calculate refined Vafa-Witten invariants. Using this Laarakker [Laa] proves universality results for the invariants.

中文翻译:

等变 K 理论和改进的 Vafa-Witten 不变量

在 [MT2] 中,复杂射影曲面的 Vafa-Witten 理论被提升为有向的 $\mathbb C^*$-等变上同调理论。在这里,我们研究 K 理论的细化。它在 $t^{1/2}\leftrightarrow t^{-1/2}$ 下的 $t^{1/2}$ 不变式中给出有理函数,专门用于 $t=1$ 处的数值 Vafa-Witten 不变式。在“instanton 分支”上,不变量给出 Gottsche-Kool 的虚拟 $\chi_{-t}^{}$-genus 细化。将模块化应用于他们的计算可以预测“单极子分支”的贡献。我们计算了一些案例并找到了完美的一致性。我们还在 K3 曲面上进行计算,发现 Jacobi 形式改进了通常的模形式,证明了 Gottsche-Kool 的猜想。我们使用 Eagon-Northcott 复合物确定简并位点的 K 理论虚拟类,并展示他们计算精炼的 Vafa-Witten 不变量。使用这个 Laarakker [Laa] 证明了不变量的普遍性结果。
更新日期:2020-07-20
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