Witt invariants from q-series ${\hat{Z}}$

Abstract

We present a relation between the Witt invariants of 3-manifolds and the ${\hat{Z}}$-invariants. It provides an alternative approach to compute the Witt invariants of 3-manifolds, which were originally defined geometrically in four dimensions. We analyze various homology spheres including a hyperbolic manifold using this method.

Appendices

Appendix

A Witt invariants for the Lens spaces

We summarize the Witt invariants for L(p, 1), where p is even.

$-L(p ,1 )$	$w(Y) \in \mathbb {Z}/4\mathbb {Z}$	$d(Y) \in \mathbb {Z}$	$d(Y_1) \in \mathbb {Z}$	$\text {def}_{3}(1) \in \mathbb {Z}/4\mathbb {Z}$	$2(1^3) \in \mathbb {Z}/4\mathbb {Z}$
$-L(2,1)$	2	0	0	0	2
$-L(4,1)$	0	0	0	3	0
$-L(6,1)$	3	1	1	3	2
$-L(8,1)$	2	0	0	3	0
$-L(10,1)$	0	0	0	2	2
$-L(12,1)$	3	1	1	2	0
$-L(14,1)$	2	0	0	2	2
$-L(16,1)$	0	0	0	1	0
$-L(18,1)$	3	1	1	3	0
$-L(20,1)$	2	0	0	1	0
$-L(22,1)$	0	0	0	0	2
$-L(24,1)$	3	1	1	2	2

As written in Sect. 5.2, $d(Y_0) = 2d(Y)$ and $\text {def}_{3}(0) = 0$ modulo 4.

B ${\hat{Z}}$-series for Brieskorn spheres and modular forms

We record ${\hat{Z}}$ for the manifolds in (12) and the formulas for the weight 1/2 and 3/2 modular forms at k-th root of unity.

$$\begin{aligned}{} & {} {\hat{Z}} [S^{3}_{-\frac{1}{r}}(T(2,5)) ;q] = q^{\frac{71}{40}-\frac{r}{4}-\frac{5}{2+20 r}} \\{} & {} \quad \left( \Psi ^{(30r -7)}_{100r+10} (q) - \Psi ^{(30r +13 )}_{100r+10}(q) - \Psi ^{(70r -3)}_{100r+10}(q) + \Psi ^{(70r +17)}_{100r+10}(q) \right) \\{} & {} {\hat{Z}} [S^{3}_{-\frac{1}{r}}(T(2,7)) ;q] = q^{\frac{143}{56}-\frac{r}{4}-\frac{7}{2+28 r}} \\{} & {} \quad \left( \Psi ^{(70r -9)}_{196r+14} (q) - \Psi ^{(70r +19)}_{196r+14}(q) - \Psi ^{(126r -5)}_{196r+14}(q) + \Psi ^{(126r +23)}_{196r+14}(q) \right) \\{} & {} {\hat{Z}} [S^{3}_{-\frac{1}{r}}(T(3,5)) ;q] = q^{\frac{191}{60}-\frac{r}{4}-\frac{15}{4+60 r}} \\{} & {} \quad \left( \Psi ^{(105r -8)}_{225r+15} (q) - \Psi ^{(105r +22)}_{225r+15}(q) - \Psi ^{(195 r-2 )}_{225r+15}(q) + \Psi ^{(195 r + 28)}_{225r+15}(q) \right) \\{} & {} {\hat{Z}} [S^{3}_{-\frac{1}{r}}(T(3,7)) ;q] = q^{\frac{383}{84}-\frac{r}{4}-\frac{21}{4+84 r}} \\{} & {} \quad \left( \Psi ^{(231 r -10)}_{441r+21} (q) - \Psi ^{(231 r +32)}_{441r+21}(q) - \Psi ^{(357 r-4 )}_{441r+21}(q) + \Psi ^{(357 r + 38)}_{441r+21}(q) \right) \\{} & {} {\hat{Z}} [S^{3}_{-\frac{1}{r}}(T(2,-5)) ;q] = q^{-\frac{71}{40}+\frac{5}{2-20 r}-\frac{r}{4}}\\{} & {} \quad \left( \Psi ^{(30r -13)}_{100r-10} (q) - \Psi ^{(30r +7 )}_{100r-10}(q) - \Psi ^{(70r -17)}_{100r-10}(q) + \Psi ^{(70r +3)}_{100r-10}(q) \right) \\{} & {} {\hat{Z}} [S^{3}_{-\frac{1}{r}}(T(2,-7)) ;q] = q^{-\frac{143}{56}+\frac{7}{2-28 r}-\frac{r}{4}} \\{} & {} \quad \left( \Psi ^{(70r -19)}_{196r-14} (q) - \Psi ^{(70r + 9)}_{196r-14}(q) - \Psi ^{(126r -23)}_{196r-14}(q) + \Psi ^{(126r +5)}_{196r-14}(q) \right) \\{} & {} {\hat{Z}} [S^{3}_{-\frac{1}{r}}(T(3,-5)) ;q] = q^{-\frac{191}{60}+\frac{15}{4-60 r}-\frac{r}{4}}\\{} & {} \quad \left( \Psi ^{(105r -22)}_{225r-15} (q) - \Psi ^{(105r +8)}_{225r-15}(q) - \Psi ^{(195 r-28 )}_{225r-15}(q) + \Psi ^{(195 r + 2)}_{225r-15}(q) \right) \\{} & {} {\hat{Z}} [S^{3}_{-\frac{1}{r}}(T(3,-7)) ;q] = q^{-\frac{383}{84}+\frac{21}{4-84 r}-\frac{r}{4}} \\{} & {} \quad \left( \Psi ^{(231 r -32)}_{441r-21} (q) - \Psi ^{(231 r +10)}_{441r-21}(q) - \Psi ^{(357 r- 38 )}_{441r-21}(q) + \Psi ^{(357 r + 4)}_{441r-21}(q) \right) \end{aligned}$$

The Eichler integrals of the $w=1/2$ and $w=3/2$ false theta functions $\Phi _{m,r}(q)$ and $\Psi _{m,r}(q)$ at k-th primitive root of unity are given by [15, 16].

$$\begin{aligned} \Phi _{m,r}(e^{i 2 \pi /k})&= -(m k) \sum _{n=1}^{2 m k} \left( \left( \frac{n}{2 m k}\right) ^2-\frac{n}{2 m k}+\frac{1}{6}\right) \psi ^{\prime , (r)}_{2m}(n) e^{\frac{i \pi n^2}{2 m k}}\\ \psi ^{\prime , (r)}_{2m}(n)&: = {\left\{ \begin{array}{ll} 1, n \equiv \pm r\quad \text {mod}\quad 2m\\ 0, \text {otherwise} \end{array}\right. } \end{aligned}$$

$$\begin{aligned} \Psi ^{(r)}_{m} \equiv \Psi _{m,r}(e^{i 2 \pi /k})&= \sum _{n=1}^{2 m k} \left( \frac{1}{2} - \frac{n}{2mk} \right) \psi ^{(r)}_{2m}(n) e^{\frac{i \pi n^2}{2 m k}} \\ \psi ^{(r)}_{2m}(n)&: = {\left\{ \begin{array}{ll} \pm 1, n \equiv \pm r\quad \text {mod}\quad 2m\\ 0, \text {otherwise} \end{array}\right. } \end{aligned}$$

C ${\hat{Z}}$-series for plumbed 3-manifolds

We state a formula for ${\hat{Z}}$ of (weakly) negative definite plumbed 3-manifolds $Y(\Gamma )$ having $b_{1} (Y(\Gamma ))=0$ [10, 13]:

$$\begin{aligned} {\hat{Z}}_{b}[Y(\Gamma ) ;q]= & {} (-1)^{\pi } q^{\frac{3\sigma - Tr B}{4}}\, \prod _{v \in Vert} PV \oint _{|z_v|=1} \nonumber \\{} & {} \frac{d z_v}{i2\pi z_v} \left( z_v - \frac{1}{z_v} \right) ^{2-deg(v)} \Theta ^{Y}_{b}(\vec {z};q), \end{aligned}$$

(17)

where

$$\begin{aligned}{} & {} \Theta ^{Y}_{b}(\vec {z};q) = \sum _{\vec {w} \in 2B\mathbb {Z}^{L} + \vec {b}} q^{-\frac{(\vec {w},B^{-1}\vec {w})}{4}} \prod _{v \in Vert} z_{v}^{w_v},\qquad b \in \textrm{Spin}^c(Y)\cong H_1(Y) \\{} & {} \quad B= \text {adjacency matrix of}\, \Gamma ,\qquad \pi = \sharp \text { (positive eigenvalues of B)},\qquad \\{} & {} \quad \sigma = \text {signature}(B), PV= \lim _{\epsilon \rightarrow 0 } \frac{1}{2} \left( \oint _{|z_v|=1 + \epsilon } + \oint _{|z_v|=1 - \epsilon } \right) \end{aligned}$$