发表机构
University of Kansas; Ohio State University（堪萨斯大学; 俄亥俄州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明稳定Higgs丛模空间上除子的等变量量子积由Steinberg对应给出，并证明Fourier-Mukai变换与量子积相容，建立了量子上同调与Hitchin系统的联系。
AI 中文摘要
设$X$为光滑射影曲线$C$上具有固定行列式$\Lambda\in\Pic(C)$且$\gcd(r,\deg\Lambda)=1$的稳定无迹秩$r$ Higgs丛的模空间。我们证明了$X$上除子的$\mathbb{C}^*$-等变量量子积由$X\times X$中Lagrangian Steinberg环的Steinberg对应给出。该Steinberg环是$2$点Kontsevich稳定映射空间到$X$的约化虚拟基本环在求值映射下的像。对于$\Gamma=\Pic^0(C)[r]$，我们给出了$\Gamma$-不变曲线类或有限轨道和的特征分解。在秩二和亏格二的情形，我们显式确定了其十五条内窥线和约化谱判别式。对于$\widehat X=[X/\Gamma]$，即$C$上稳定$\PGL_r$-Higgs丛的模空间，我们从gerbe扭曲理论出发考虑轨道量子上同调理论。由万有射影丛的提升$\mu_r$-gerbe $\alpha$给出的复K-理论$\KU^*(X)$与扭曲K-理论$\KU^*([X/\Gamma],\alpha)$之间存在同构，且该同构由Fourier-Mukai变换给出。我们证明了Fourier-Mukai变换与除子量子积相容。
英文摘要
Let $X$ be the moduli space of stable trace-free rank $r$ Higgs bundles over a smooth projective curve $C$ with fixed determinant $Λ\in\Pic(C)$ and $\gcd(r,\degΛ)=1$. We prove that the $\cc^*$-equivariant quantum product of a divisor on $X$ is given by the Steinberg correspondence of the Lagrangian Steinberg cycle in $X\times X$. The Steinberg cycle is the image of the reduced virtual fundamental cycle of the $2$-pointed Kontsevich stable map space to $X$ under evaluation map. For $Γ=\Pic^0(C)[r]$, we give a character decomposition for $Γ$-invariant curve classes or finite orbit sums. In rank two and genus two we explicitly determine its fifteen endoscopic lines and the reduced spectral discriminant. %The resulting endoscopic support statement is cohomological, not a determination of the entire Chow cycle. For $\widehat X=[X/Γ]$, which is the moduli space of stable $\PGL_r$-Higgs bundles over $C$. We consider the orbifold quantum cohomology theory from a gerbe-twisted theory. The complex K-theory $\KU^*(X)$ and twisted K-theory $\KU^*([X/Γ],α)$ by the lifting $μ_r$-gerbe $α$ of the universal projective bundle is an isomorphism, and is given by the Fourier-Mukai transformation. We prove that the Fourier-Mukai transformation is compatible with the quantum product by the divisors.
Comments56 pages, comments are welcome
