论文标题
计算不变曲线：gopakumar-vafa不变的calabi-yau三倍的理论
Counting Invariant Curves: a theory of Gopakumar-Vafa invariants for Calabi-Yau threefolds with an involution
论文作者
论文摘要
我们开发了一种gopakumar-vafa（GV）的理论，用于Calabi-yau三倍（CY3）$ x $，该理论配备有互联级的$ \ imath $保留圆锥形体积形式。我们定义整数$ n_ {g，h}（β）$，该$虚拟地计算了$ g $ curves $ c $ c $ th $ x $的属$ x $，in h_ {2}（x）$中的$β\ in Class $β\，它们是$ \ imath $不变的$ \ imath $，其Quienient $ C/\ IMATH $ imath $ $ imath $ H $ nus $ H $。我们给出了$ n_ {g，h}（β）$的两个定义，我们认为这是等效的：一个在pandharipande-thomas理论的版本中，另一个在maulik-toda theory的版本方面。 我们计算我们的不变性，并在几种情况下为我们的猜想提供证据。特别是，当$ x = s = s \ times \ mathbb {c} $时，我们计算我们的不变性，其中$ s $是带有$ \ imath（a）= - a $或$ k3 $表面具有符号相关性（Nikulin $ k3 $ k3 $）的Abelian表面。对于这些情况，我们就雅各比模块化形式为不变的公式提供了公式。对于Abelian表面案例，我们不变的$ n_ {g，h}（β）$ to $ h = 0 $的专业化恢复了由Bryan-Oberdieck-Pandharipande-Yin首先计算出的Abelian表面上的高纤维曲线计数。
We develop a theory of Gopakumar-Vafa (GV) invariants for a Calabi-Yau threefold (CY3) $X$ which is equipped with an involution $\imath$ preserving the holomorphic volume form. We define integers $n_{g,h}(β) $ which give a virtual count of the number of genus $g$ curves $C$ on $X$, in the class $β\in H_{2}(X)$, which are invariant under $\imath$, and whose quotient $C/\imath$ has genus $h$. We give two definitions of $n_{g,h}(β) $ which we conjecture to be equivalent: one in terms of a version of Pandharipande-Thomas theory and one in terms of a version of Maulik-Toda theory. We compute our invariants and give evidence for our conjecture in several cases. In particular, we compute our invariants when $X=S\times \mathbb{C}$ where $S$ is an Abelian surface with $\imath (a)=-a$ or a $K3$ surface with a symplectic involution (a Nikulin $K3$ surface). For these cases, we give formulas for our invariants in terms of Jacobi modular forms. For the Abelian surface case, the specialization of our invariants $n_{g,h}(β) $ to $h=0$ recovers the count of hyperelliptic curves on an Abelian surface first computed by Bryan-Oberdieck-Pandharipande-Yin.
