标题： $q$-三维流形和纽结-拟箭图对应的系列不变量 显示英文标题

标题： $q$-Series Invariants of Three-Manifolds and Knots-Quivers Correspondence

摘要： Gukov-Pei-Putrov-Vafa（GPPV）猜想是关于两个三维流形不变量之间的一个关系：Witten-Reshetikhin-Turaev（WRT）不变量和\(\widehat{Z}\)（“Z-hat”）不变量。 事实上，WRT 不变量在单位根处定义，即$\mathbbm{q}\left(\exp\left(\frac{2\pi i}{k+2}\right),~k\in\mathbb{Z}_+,~\text{for}~SU(2)\right)$，并且通常是一个复数，而$\widehat{Z}$-不变量是一个具有整数系数的$q$-级数，满足$|q|<1$。 因此，可以通过执行特定的解析延拓从WRT不变量得到$\widehat{Z}$-不变量，即$\mathbbm{q}\rightarrow q$。 本论文中，我们首先研究了这一猜想对超群$SO(3)$和正交辛超群$OSp(1|2)$的情况。 这是通过分别为相应的群建立 WRT 不变量，然后执行特定的解析延拓以提取$\widehat{Z}$实现的。 通过这个过程，我们发现了$\widehat{Z}^{SU(2)}=\widehat{Z}^{SO(3)}$，并确定了$\widehat{Z}^{SU(2)}$和$\widehat{Z}^{OSp(1|2)}$之间的关系。 受$\widehat{Z}$对于$SU(2)$和$SO(3)$群相等的启发，我们在第二篇论文中研究了这一猜想对于$SU(N)/\mathbb{Z}_m$群的情况，其中$\mathbb{Z}_m$是$\mathbb{Z}_N$的一个子群。 我们随后发现$\widehat{Z}^{SU(N)/\mathbb{Z}_m}=\widehat{Z}^{SU(N)}$。 本论文的另一个主题是研究纽结理论与拟向量表示理论之间的猜想。 更精确地说，这个猜想将对称的$r$-着色HOMFLY-PT多项式的生成函数与对称箭图相关的动机生成级数联系起来。 特别是，我们得到了一类称为双扭结$K(p,-m)$的一族结的箭图表示。 主要地，我们利用Melvin-Morton-Rozansky（MMR）形式主义的逆向工程来推导这些箭图的矩阵模式。 显示英文摘要

摘要： The Gukov-Pei-Putrov-Vafa (GPPV) conjecture is a relationship between two three-manifold invariants: the Witten-Reshetikhin-Turaev (WRT) invariant and the \(\widehat{Z}\) (``Z-hat'') invariant. In fact, WRT invariant is defined at roots of unity, $\mathbbm{q}\left(\exp\left(\frac{2\pi i}{k+2}\right),~k\in\mathbb{Z}_+,~\text{for}~SU(2)\right)$, and is generally a complex number, whereas $\widehat{Z}$-invariant is a $q$-series with integer coefficients such that $|q|<1$. Therefore, $\widehat{Z}$-invariant can be obtained from WRT-invariant by performing a particular analytic continuation, $\mathbbm{q}\rightarrow q$. In this thesis, we first examine this conjecture for $SO(3)$ and the ortho-symplectic supergroup $OSp(1|2)$. This is done by setting up the WRT invariant for the respective groups and then performing the particular analytic continuation to extract $\widehat{Z}$. As a result of this exercise, we found that $\widehat{Z}^{SU(2)}=\widehat{Z}^{SO(3)}$ and identified a relation between $\widehat{Z}^{SU(2)}$ and $\widehat{Z}^{OSp(1|2)}$. Motivated by the equality of $\widehat{Z}$ for $SU(2)$ and $SO(3)$ groups, we study this conjecture for $SU(N)/\mathbb{Z}_m$ groups, where $\mathbb{Z}_m$ is a subgroup of $\mathbb{Z}_N$, in our second paper. We subsequently found that $\widehat{Z}^{SU(N)/\mathbb{Z}_m}=\widehat{Z}^{SU(N)}$. Another theme of the thesis is to study a conjecture between knot theory and quiver representation theory. More precisely, this conjecture relates the generating function of the symmetric $r$-colored HOMFLY-PT polynomial with the motivic generating series associated with a symmetric quiver. In particular, we obtain a quiver representation for a family of knots called double twist knots $K(p,-m)$. Primarily, we exploit the reverse engineering of Melvin-Morton-Rozansky (MMR) formalism to deduce the pattern of the matrix for these quivers.