$\widehat{Z}$在大型$N$：从曲线计数到量子模块化
$\widehat{Z}$ at large $N$: from curve counts to quantum modularity
摘要：
在3-流形$Y$上约化一个6维五膜理论，得到一个$q$ -级数3-流形不变量$\widehat{Z}(Y)$ .我们分析了大$N$行为的$F_K=\widehat{Z}(M_K)$，其中$M_K$是一个结$K$在3-球面的补充，并探讨了一个$a$变形（$a=q^N$）版本的$F_{K}$和HOMFLY-PT多项式之间的关系。一方面，结合纽结补上的全纯环的计数，给出了开全纯曲线的计数对$F_K$的计数解释。另一方面，它导致封闭形式的表达式$a$ -变形$F_K$ $(2,2p+1)$ -环面结。他们建议进一步$t$ -变形的基础上超多项式，这可以用来获得$t$ -变形的ADO多项式，预计与分类。此外，研究$F_K$如何在$K$上的自然几何操作下变换，表明在新的设置中与量子模块化的关系。
英文摘要：
Reducing a 6d fivebrane theory on a 3-manifold $Y$ gives a $q$-series 3-manifold invariant $\widehat{Z}(Y)$. We analyse the large-$N$ behaviour of $F_K=\widehat{Z}(M_K)$, where $M_K$ is the complement of a knot $K$ in the 3-sphere, and explore the relationship between an $a$-deformed ($a=q^N$) version of $F_{K}$ and HOMFLY-PT polynomials. On the one hand, in combination with counts of holomorphic annuli on knot complements, this gives an enumerative interpretation of $F_K$ in terms of counts of open holomorphic curves. On the other, it leads to closed form expressions for $a$-deformed $F_K$ for $(2,2p+1)$-torus knots. They suggest a further $t$-deformation based on superpolynomials, which can be used to obtain a $t$-deformation of ADO polynomials, expected to be related to categorification. Moreover, studying how $F_K$ transforms under natural geometric operations on $K$ indicates relations to quantum modularity in a new setting.
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