Towards a q-series for $osp(2|2n)$

John Chae Department of Physics and QMAP, UC Davis, 1 Shields Ave, Davis, CA, 95616, USA
[email protected]

Abstract

A series invariant for a certain class of closed 3-manifolds associated with a type I Lie superalgebra $sl(m|n)$ was introduced recently. We find a q-series for the other Lie superalgebra of the same type of the minimum rank.

1 Introduction

Motivated by a prediction for a categorification of the Witten-Reshitikhin-Turaev invariant of a closed oriented 3-manifold [19, 17, 18] in [11, 10], a topological invariant q-series for graph 3-manifolds associated with a type I Lie superalgebra $sl(m|n)$ was introduced recently in [4]. This q-series denoted as $\hat{Z}_{ab}$ is labeled by a pair of $Spin^{c}$ structures of the manifolds as opposed to one label for a q-series ( $\hat{Z}_{b}$ ) corresponding to a classical Lie algebra [16]. Another core difference is that the q-series invariant $\hat{Z}_{ab}$ decomposes an extension of the WRT invariant, namely, the CGP invariant based on a Lie superalgebra [2]. In the most general topological setting, the CGP invariant is a topological invariant of a triple consisting of a closed oriented 3-manifold, a link and a certain cohomology class ¹¹1It is denoted $\omega\in H^{1}(Y;\mathbb{Z}/2\mathbb{Z})$ , which induces coloring on a surgery link; this link is not part of the triplet [2].. The construction of the CGP invariant associated to a Lie (super)algebra involves a new facet: the modified quantum dimension, which was first introduced in [8] for the quantum groups of Lie superalgebra of type I and then for the quantum groups of $sl(2)$ at roots of unity [9]. In general, for a complex Lie superalgebra, the standard quantum dimension vanishes, which makes the WRT invariant of links and 3-manifolds trivial. The modified quantum dimension overcomes the obstacle. In the case of a complex simple Lie algebra, the modified quantum dimension enables to extend the WRT invariant to the above mentioned triplet. Furthermore, for a fixed Lie (super)algebra, the modified dimension can be defined for semisimple and nonsemisimple ribbon categories. Specific examples of the former type were given in [9] and [5]. The latter type, which is relevant to this paper, was first dealt with in [6]. And then it was expanded into a Lie superalgebra in [1] and [12] in which finite dimensional irreducible representations of the (unrolled) quantum groups of $sl(m|n)\,(m\neq n)$ at roots of unity was analyzed (the latter paper focuses on $sl(2|1)$ ). The CGP invariant contains a variety of information such as the multivariable Alexander polynomial and the ADO polynomial of a link. The relation between the CGP invariant and the WRT invariant for $sl(2;\mathbb{C})$ was conjectured in [3] and was proved for certain classes of 3-manifolds.

In this paper we analyze $\hat{Z}_{ab}$ associated with $osp(2|2)$ for plumbed 3-manifolds and observe that it is either even or odd power series in the examples. The rest of the paper is organized as follows. In Section 2 we review the ingredients involved in the CGP invariant and in $\hat{Z}_{ab}$ for $sl(m|n)$ . In Section 3 we present the formula for homological blocks $\hat{Z}_{ab}$ for $osp(2|2)$ . And then in Section 4, we apply the formula to a few examples.

Acknowledgments. I am grateful to Bertrand Patureau-Mirand and Pavel Putrov for helpful explanations, as well as to Sergei Gukov for reading a draft of this paper.

2 Background

The twist $\theta$ , the S-matrix and the modified quantum dimension $d$ for the quantum group $\mathcal{U}_{h}(g)$ of $g=$ type I $=sl(m|n),osp(2|2n)$ are given in [8], where h is a formal variable related to $q=e^{h/2}$ :

\theta_{V}(\lambda)=q^{\left\langle\lambda,\lambda+2\rho\right\rangle}1_{V},

where $\lambda$ is the highest weight of a highest weight $\mathcal{U}_{h}(g)$ -module $V$ coloring the a link component and $\rho=\rho_{0}-\rho_{1}$ and $\left\langle-,-\right\rangle$ is the bilinear form (see the appendix A for the details of representation theoretic concepts).

S(V,V^{\prime})=\varphi_{\mu+\rho}(sch(V(\lambda))),

where $\lambda$ and $\mu$ are weights of (irreducible) $\mathcal{U}_{h}(g)$ -modules $V$ and $V^{\prime}$ , respectively, coloring each link component. Moreover, $sch(V(\lambda))$ is a supercharacter of $V$ and $\varphi_{\beta}$ is a map from a group ring $\mathbb{Z}[\Lambda]$ to $\mathbb{C}[[h]]$ ²²2 $e^{\alpha}\mapsto q^{2\left\langle\alpha,\beta\right\rangle}\qquad\text{for any weight}\quad\beta$ . The modified quantum dimension $d$ is

d(\mu)S(\lambda,\mu)=d(\lambda)S(\mu,\lambda).

In the case of $sl(m|n)$ , for the unrolled quantum group at roots of unity $\mathcal{U}^{H}_{l}(sl(m|n))\,(m\neq n)$ , the above formulas modify [1]:

\theta_{V}(\lambda;l)=\xi^{\left\langle\lambda,\lambda+\pi\right\rangle}1_{V}\qquad\xi=e^{\frac{i2\pi}{l}}\quad l\geq m+n-1

\begin{split}S(V,V^{\prime};l)&=\varphi_{\lambda+\frac{\pi}{2}}(sch(V(\lambda))),\qquad\pi=2\rho-2l\rho_{\bar{0}}\in\Lambda_{R}\\ &=\xi^{2\left\langle\lambda+\frac{\pi}{2},\mu+\frac{\pi}{2}\right\rangle}\prod_{\alpha\in\Delta^{+}_{\bar{0}}}\frac{\left\{l\left\langle\lambda+\frac{\pi}{2},\alpha\right\rangle\right\}_{\xi}}{\left\{\left\langle\lambda+\frac{\pi}{2},\alpha\right\rangle\right\}_{\xi}}\prod_{\alpha\in\Delta^{+}_{\bar{1}}}\left\{\left\langle\lambda+\frac{\pi}{2},\alpha\right\rangle\right\}_{\xi}\end{split}

d(\mu;l)S(\lambda,\mu;l)=d(\lambda;l)S(\mu,\lambda;l).

The second equality in the S-matrix element assumes V is a simple $\mathcal{U}^{H}_{l}(sl(m|n))$ -module that is typical having dimension D. The changes of the formula are due to the different pivotal structure of $\mathcal{U}^{H}_{l}(sl(m|n))$ . Since the modifications are on the representation theory level and don’t seem to involve unique features of $sl(m|n)$ , we assume that the same modifications occur for $osp(2|2n)$ . We next summarize the concepts involved in the homological blocks $\hat{Z}_{ab}$ associated with a Lie superalgebra [4].

Generic graph In type I Lie superalgebra case, plumbing graph used in practice needs to satisfy certain conditions due to the functional form of the edge factor in the integrand of $\hat{Z}$ ; a plumbing graph is called generic, if

1.

at least one vertex of a graph has degree $>2$
2.

$V|_{deg>2}\neq U\sqcup W$ such that $B^{-1}_{IJ}=0$ for some I $\in V$ , J $\in W$ ,

where V is a set of vertices of a graph.

Good Chamber For $\hat{Z}$ associated with type I Lie superalgebra, there is a notion of a good chamber introduced in [4]. Existence of a good chamber guarantees that $\hat{Z}$ is a power series with integer coefficients. Specifically, an adjacency matrix $B$ of a generic plumbing graph needs to admit a good chamber. It turns out that $osp(2|2)$ requires positive definite (generic) plumbing graphs, its criteria for good chamber existence shown below are opposite of $sl(2|1)$ criteria in [4]. Good chamber exists, if there exists a vector $\alpha$ whose components are

\alpha_{I}=\pm 1,\quad I\in V|_{deg\neq 2}

such that

B^{-1}_{IJ}\alpha_{I}\alpha_{J}\geq 0\quad\forall I\in V|_{deg=1},\quad J\in V|_{deg\neq 2}

B^{-1}_{IJ}\alpha_{I}\alpha_{J}>0\quad\forall I,J\in V|_{deg=1},\quad I\neq J

X_{IJ}\quad\text{is copositive}\qquad X_{IJ}=B^{-1}_{IJ}\alpha_{I}\alpha_{J}\quad I,J\in V|_{deg>2}

are satisfied. Furthermore, a good chamber corresponding to such $\alpha$ is

deg(I)=1:\begin{cases}|y_{I}|^{\alpha_{I}}<1\\ |z_{I}|^{\alpha_{I}}>1\end{cases}

deg(I)>2:\bigg{|}\frac{y_{I}}{z_{I}}\bigg{|}^{\alpha_{I}}<1.

Chamber Expansion The series expansions for the good chambers for $osp(2|2)$ turn out to be the same as that of $sl(2|1)$ in [4]. We record here the expansions.

•

$\text{degree(I)}=2+K>2$

$\left(\frac{(1-z_{I})(1-y_{I})}{y_{I}-z_{I}}\right)^{K}=\begin{cases}(z_{I}-1)^{K}(1-y_{I}^{-1})^{K}\sum_{r_{I}=0}^{\infty}\frac{(r_{I}+1)(r_{I}+2)\cdots(r_{I}+K-1)}{(K-1)!}\left(z_{I}/y_{I}\right)^{r_{I}},&|z_{I}|<|y_{I}|\\ (1-y_{I})^{K}(1-z_{I}^{-1})^{K}\sum_{r_{I}=0}^{\infty}\frac{(r_{I}+1)(r_{I}+2)\cdots(r_{I}+K-1)}{(K-1)!}\left(y_{I}/z_{I}\right)^{r_{I}},&|z_{I}|>|y_{I}|\\ \end{cases}$
•

$\text{degree(I)}=1$

$\frac{y_{I}-z_{I}}{(1-z_{I})(1-y_{I})}=\begin{cases}\sum_{r_{I}=1}^{\infty}y_{I}^{r_{I}}+\sum_{r_{I}=0}^{\infty}z_{I}^{-r_{I}},&|y_{I}|<1,|z_{I}|>1\\ -\sum_{r_{I}=0}^{\infty}y_{I}^{-r_{I}}-\sum_{r_{I}=1}^{\infty}z_{I}^{r_{I}},&|y_{I}|>1,|z_{I}|<1\\ \end{cases}$

3 Result

For closed oriented plumbed 3-manifolds $Y(\Gamma)$ having $b_{1}(Y)=0$ and $a,b\in Spin^{c}(Y)\cong H_{1}(Y)$ , $\hat{Z}_{ab}[Y;q]$ associated with $osp(2|2)$ is

	$\displaystyle\hat{Z}^{osp(2\|2)}_{ab}[Y;q]$	$\displaystyle=(-1)^{\pi}\int_{\Omega}\prod_{I\in V}\frac{dz_{I}}{i2\pi z_{I}}\frac{dy_{I}}{i2\pi y_{I}}\left(\frac{y_{I}-z_{I}}{(1-z_{I})(1-y_{I})}\right)^{2-deg(I)}\sum_{\begin{subarray}{c}n\in BZ^{L}+a\\ m\in BZ^{L}+b\end{subarray}}q^{-2nB^{-1}m}\prod_{J\in V}z_{J}^{m_{J}}y_{J}^{n_{J}}$
		$\displaystyle\in\mathbb{Q}+q^{\Delta_{ab}}\mathbb{Z}[[q]],\qquad\Delta_{ab}\in\mathbb{Q}\qquad H_{1}(Y)=\mathbb{Z}^{L}/B\mathbb{Z}^{L}\qquad\|q\|<1$

where $B=B(\Gamma)$ is an adjacency matrix of a generic plumbing graph $\Gamma$ . Moreover, the convergent q-series in a complex unit disc requires $\Gamma$ to be positive definite plumbing graphs inside a complex unit disc. This is opposite of the $sl(m|n)$ case (and for $su(n)$ ). The edge factor in the integrand turns out to be the same as that of $sl(2|1)$ ³³3This seems to be no longer true in higher rank cases. The integration prescription states that one chooses a chamber for an expansion of the integrand using the chamber expansion in the previous section and then picks constant terms in $y_{I}$ and $z_{I}$ .

4 Examples

We apply the above formula to $\mathbb{Z}HS^{3}$ and $\mathbb{Q}HS^{3}$ . Each $\hat{Z}$ is either even or odd power series and the regularized constants are given by the zeta function $\zeta(s)$ or the Hurwitz zeta function $\zeta(s,x)$ (see [4] for details).

•

$Y=S^{3}$ The adjacency matrix of a generic plumbing graph of $S^{3}$ (Figure 1 in appendix B) is

B(\Gamma)=\begin{pmatrix}4&1&1&1\\ 1&1&0&0\\ 1&0&1&0\\ 1&0&0&1\\ \end{pmatrix}\qquad DetB=1\qquad|H_{1}(Y)|=|DetB|

The two chambers are

\alpha=\pm(1,-1,-1,-1).

\hat{Z}^{osp(2|2)}_{00}[Y;q]=1+2\zeta(-1)+2\zeta(0)-2q^{2}-4q^{4}-4q^{6}-6q^{8}-4q^{10}-8q^{12}-4q^{14}-8q^{16}-6q^{18}+\cdots

•

$Y=\Sigma(2,3,7)$ The adjacency matrix of a (generic) plumbing graph (Figure 2 in appendix B) is

B(\Gamma)=\begin{pmatrix}1&1&1&1\\ 1&2&0&0\\ 1&0&3&0\\ 1&0&0&7\\ \end{pmatrix}\qquad DetB=1

The two chambers are

\alpha=\pm(1,-1,-1,-1).

	$\displaystyle\hat{Z}^{osp(2\|2)}_{00}$	$\displaystyle=1+2\zeta(-1)+2\zeta(0)-2q^{4}-2q^{6}-4q^{8}-2q^{10}-6q^{12}-4q^{14}-6q^{16}-6q^{18}-8q^{20}$
		$\displaystyle-4q^{22}-10q^{24}-6q^{26}-8q^{28}+\cdots$

•

$Y=M(-1|\frac{1}{2},\frac{1}{3},\frac{1}{8})$ The adjacency matrix of a (generic) plumbing graph (Figure 3 in appendix B) is

B(\Gamma)=\begin{pmatrix}1&1&1&1\\ 1&2&0&0\\ 1&0&3&0\\ 1&0&0&8\\ \end{pmatrix}\qquad DetB=2

The two chambers are

\alpha=\pm(1,-1,-1,-1).

The independent $\hat{Z}_{ab}$ are

	$\displaystyle\hat{Z}_{00}$	$\displaystyle=1+2\zeta(-1)+2\zeta(0)-2q^{4}-4q^{8}-2q^{10}-4q^{12}-2q^{14}-6q^{16}-2q^{18}-6q^{20}-2q^{22}-8q^{24}$
		$\displaystyle-2q^{26}-4q^{28}+\cdots$
	$\displaystyle\hat{Z}_{11}$	$\displaystyle=-2q^{3}-2q^{5}-2q^{7}-4q^{9}-4q^{11}-2q^{13}-6q^{15}-4q^{17}-4q^{19}-6q^{21}-4q^{23}-6q^{27}+\cdots$
	$\displaystyle\hat{Z}_{01}$	$\displaystyle=2\zeta(-1,1/2)+\zeta(0,1/2)-q^{2}-q^{4}-3q^{6}-2q^{8}-3q^{10}-4q^{12}-4q^{14}-3q^{16}-5q^{18}-5q^{20}$
		$\displaystyle-4q^{22}-5q^{24}-4q^{26}-5q^{28}+\cdots$

where the labels denote the last components of elements of $H_{1}(Y)$ .

•

$Y=M(-1|\frac{1}{2},\frac{1}{3},\frac{1}{9})$ The adjacency matrix of a (generic) plumbing graph (Figure 4 in appendix B) is

B(\Gamma)=\begin{pmatrix}1&1&1&1\\ 1&2&0&0\\ 1&0&3&0\\ 1&0&0&9\\ \end{pmatrix}\qquad DetB=3

The independent $\hat{Z}^{osp(2|2)}_{ab}[Y]$ are

	$\displaystyle\hat{Z}_{00}$	$\displaystyle=1+2\zeta(-1)+2\zeta(0)-2q^{6}-2q^{8}-4q^{12}-2q^{14}-2q^{16}-4q^{18}-2q^{20}-2q^{22}-6q^{24}-2q^{26}$
		$\displaystyle-2q^{28}+\cdots$
	$\displaystyle\hat{Z}_{11}$	$\displaystyle=q^{\frac{1}{3}}\left(-q-q^{3}-3q^{5}-2q^{7}-3q^{9}-2q^{11}-5q^{13}-2q^{15}-4q^{17}-2q^{19}-5q^{21}-4q^{23}-2q^{25}\right.$
		$\displaystyle\left.-q^{27}-3q^{29}+\cdots\right)$
	$\displaystyle\hat{Z}_{22}$	$\displaystyle=q^{\frac{1}{3}}\left(-q-q^{3}-3q^{5}-2q^{7}-3q^{9}-2q^{11}-5q^{13}-2q^{15}-4q^{17}-2q^{19}-5q^{21}-4q^{23}-2q^{25}\right.$
		$\displaystyle\left.-q^{27}-3q^{29}+\cdots\right)$
	$\displaystyle\hat{Z}_{01}$	$\displaystyle=3\zeta(-1,1/2)+\zeta(0,1/2)-2q^{4}-q^{6}-2q^{8}-3q^{10}-3q^{12}-2q^{14}-4q^{16}-2q^{18}-5q^{20}-3q^{22}$
		$\displaystyle-2q^{24}-2q^{26}-3q^{28}+\cdots$
	$\displaystyle\hat{Z}_{02}$	$\displaystyle=3\zeta(-1,1/2)+\zeta(0,1/2)-q^{2}-q^{4}-q^{6}-3q^{8}-2q^{10}-2q^{12}-3q^{14}-4q^{16}-2q^{18}-4q^{20}$
		$\displaystyle-2q^{22}-3q^{24}-2q^{26}-4q^{28}+\cdots$
	$\displaystyle\hat{Z}_{12}$	$\displaystyle=q^{-\frac{1}{3}}\left(-2q^{3}-2q^{5}-2q^{7}-2q^{9}-4q^{11}-2q^{13}-4q^{15}-2q^{17}-6q^{19}-2q^{21}-4q^{23}-2q^{25}\right.$
		$\displaystyle\left.-4q^{27}+\cdots\right)$

Appendix

Appendix A Type I Lie superalgebra and its quantization

We give a summary of the representation theory of $osp(2|2n)$ and the quantum group $U_{h}(\text{type}\,\textup{I})$ in [8]⁴⁴4For reviews on Lie superalgebras, see [13, 14, 15]. For $osp(2|2n)$ , the set of positive roots is $\Delta^{+}=\Delta^{+}_{\bar{0}}\cup\Delta^{+}_{\bar{1}}$ with

\Delta^{+}_{\bar{0}}=\left\{\delta_{i}\pm\delta_{j}|1\leq i<j\leq n\right\}\cup\left\{2\delta_{i}\right\}\quad\text{and}\quad\Delta^{+}_{\bar{1}}=\left\{\epsilon\pm\delta_{i}\right\},\qquad n\in\mathbb{Z}_{+},

where $\left\{\epsilon,\delta_{1},\cdots,\delta_{n}\right\}$ form the dual basis of the Cartan subalgebra. Their inner products are

\left\langle\epsilon,\epsilon\right\rangle=1\qquad\left\langle\delta_{i},\delta_{j}\right\rangle=-\delta_{ij}\qquad\left\langle\epsilon,\delta_{j}\right\rangle=0

The half sums of positive roots are

\rho_{0}=\sum_{i}(n+1-i)\delta_{i},\quad\rho_{1}=n\epsilon\quad\text{and}\quad\rho=\rho_{0}-\rho_{1}.

The fundamental weights are

w_{1}=\epsilon\qquad w_{k+1}=\epsilon+\sum_{i=1}^{k}\delta_{i}\quad\text{and}\quad k=1,\cdots,n.

The weight $\lambda^{c}_{a}$ decomposes as

\lambda^{c}_{a}=aw_{1}+c_{1}w_{2}+\cdots+c_{n}w_{n+1},

where $c\in\mathbb{N}^{r-1},a\in\mathbb{C}$

Let $g$ be a Lie superalgebra of type I, $sl(m|n)$ or $osp(2|2n)\,(m\neq n)$ . $U_{h}(g)$ is the $\mathbb{C}[[h]]$ -Hopf superalgebra generated by

E_{i},F_{i},h_{i},\quad i=1,\cdots,r=m+n-1\quad\text{or}\quad n+1

satisfying the relations

[h_{i},h_{j}]=0,\quad[h_{i},E_{j}]=A_{ij}E_{j},\quad[h_{i},F_{j}]=A_{ij}F_{j},

[E_{i},F_{j}]=\delta_{i,j}\frac{q^{h_{i}}-q^{-h_{i}}}{q-q^{-1}},\quad E_{s}^{2}=F_{s}^{2}=0

and the quantum Serre-type relations; they are quadratic, cubic and quartic relations among $E_{i}$ or $F_{i}$ (see [20] Definition 4.2.1). $A_{ij}$ is $r\times r$ Cartan matrix and $\left\{s\right\}=\tau\subset\left\{1,\cdots,r\right\}$ determining the parity of the generators. $E_{s},F_{s}$ are the only odd generators. Moreover, the (anti)commutator is $[x,y]:=xy-(-1)^{\bar{x}\bar{y}}yx$ . The Hopf algebra structure is

\Delta(E_{i})=\Delta(E_{i})\otimes 1+q^{-h_{i}}\otimes\Delta(E_{i}),\quad\epsilon(\Delta(E_{i}))=0,\quad S(\Delta(E_{i}))=-q^{h_{i}}\Delta(E_{i})

\Delta(F_{i})=\Delta(F_{i})\otimes 1+q^{-h_{i}}\otimes\Delta(F_{i}),\quad\epsilon(\Delta(F_{i}))=0,\quad S(\Delta(E_{i}))=-\Delta(F_{i})q^{h_{i}}

\Delta(h_{i})=\Delta(F_{i})\otimes 1+1\otimes\Delta(h_{i}),\quad\epsilon(\Delta(h_{i}))=0,\quad S(\Delta(h_{i}))=-h_{i}

Appendix B Invariance checks

We display the weighted positive definite generic plumbing graphs and their equivalent graphs of the examples in Section 4. Each vertex corresponds to a $S^{1}$ -bundle over $S^{2}$ ; a positive integer is the Euler number of a bundle. The graphs are related by the Kirby-Neumman moves:

The invariance of $\hat{Z}_{ab}$ under the Kirby-Neumman moves for the exhibited graphs and the graphs mentioned in the captions of Figure 3 and 4 were verified.

Figure 1: A generic plumbing graph of

S^{3}

(left most) and its equivalent graphs

Figure 2: A generic plumbing graph of

\Sigma(2,3,7)

(the first graph) and its equivalent graphs

Figure 3: A generic plumbing graph of

M(-1|\frac{1}{2},\frac{1}{3},\frac{1}{8})

and its equivalent graphs can be obtained from Figure 2 by replacing the weight 7 by 8 or 8 by 9.

Figure 4: A generic plumbing graph of

M(-1|\frac{1}{2},\frac{1}{3},\frac{1}{9})

and its equivalent graphs can be obtained from Figure 2 by replacing the weight 7 by 9 or 8 by 10.

Appendix C Derivations

We setup for the derivations of the ingredients in the CGP invariant formula using the appendix A and outline the derivation of $\hat{Z}^{osp(2|2)}$ in Section 3.

The root lattice $\Lambda_{R}$ of $osp(2|2)$ are generated by $2\delta$ and $\epsilon-\delta$ , hence, two dimensional. The pivotal element $\pi$

\begin{split}\pi&=2(\rho_{0}-\rho_{1})-2l\rho_{0}\in\Lambda_{R}\\ &=-2(\epsilon-\delta)-l(2\delta)\end{split}

This implies that

K_{\pi}=K_{1}^{-l}K_{2}^{-2}\in U^{H}_{\xi}(osp(2|2))

The weight $\lambda$ of $V$ is

\begin{split}\lambda&=\mu_{1}w_{1}+\mu_{2}w_{2}\\ &=(\mu_{1}+\mu_{2})\epsilon+\mu_{2}\delta,\end{split}

where $w_{1}=\epsilon$ and $w_{2}=\epsilon+\delta$ . Under the assumption mentioned in Section 2, we arrive at

\theta_{V}(\vec{\mu};l)=\xi^{2l\mu_{2}}\xi^{\mu_{1}^{2}+2\mu_{1}\mu_{2}-4\mu_{2}-2\mu_{1}}\,1_{V}\qquad l=\text{odd and}\geq 3.

\begin{split}S(\vec{\mu},\vec{\mu}^{\prime};l)&=\xi^{2l(\mu_{2}+\mu^{\prime}_{2})}\xi^{2\left(\mu_{1}\mu^{\prime}_{1}+\mu_{1}\mu^{\prime}_{2}+\mu^{\prime}_{1}\mu_{2}\right)-2\left(\mu_{1}+\mu^{\prime}_{1}+2\mu_{2}+2\mu^{\prime}_{2}\right)}\\ &\times\frac{\left\{2l(\mu^{\prime}_{2}+1-l)\right\}}{\left\{2(\mu^{\prime}_{2}+1-l)\right\}}\left\{\mu^{\prime}_{1}-2+l\right\}\left\{\mu^{\prime}_{1}+2\mu^{\prime}_{2}-l\right\}\end{split}

d(\vec{\mu};l)=\frac{\left\{2(\mu_{2}+1-l)\right\}}{\left\{2l(\mu_{2}+1-l)\right\}}\frac{1}{\left\{\mu_{1}-2+l\right\}\left\{\mu_{1}+2\mu_{2}-l\right\}},

\left\{x\right\}_{\xi}:=\xi^{x}-\xi^{-x}\quad\xi=q^{1/2}=e^{i2\pi/l},

where $(\mu_{1},\mu_{2})$ and $(\mu^{\prime}_{1},\mu^{\prime}_{2})$ are the coefficients in the weight decompositions of $\lambda$ and $\mu$ for $V$ and $V^{\prime}$ , respectively (see appendix A). For the S-matrix, notations for $V$ and $V^{\prime}$ are switched.

In order to apply the derivation of the superalgebra $\hat{Z}$ given in [4], we need to adapt the above three expressions into a plumbing graph setup. We first let

\vec{\alpha}=(\alpha_{1}=\mu_{1}-2+l,\,\alpha_{2}=2\mu_{2}+2-2l)\quad\in\mathbb{C}^{2}

Then

\theta_{V}=\xi^{\alpha_{1}^{2}+\alpha_{1}\alpha_{2}}

S^{\prime}=\xi^{2\alpha_{1}\alpha^{\prime}_{1}+\alpha_{1}\alpha^{\prime}_{2}+\alpha_{2}\alpha^{\prime}_{1}}\frac{\left\{l\alpha^{\prime}_{2}\right\}_{\xi}\left\{\alpha^{\prime}_{1}\right\}_{\xi}\left\{\alpha^{\prime}_{1}+\alpha^{\prime}_{2}\right\}_{\xi}}{\left\{\alpha^{\prime}_{2}\right\}_{\xi}}

We next shift $\alpha_{1}$ and $\alpha_{2}$ by s and t, respectively. And then we associate $\theta_{V}$ to each vertex. This leads to

\theta_{V_{\alpha^{I}_{s^{I}t^{I}}}}=\xi^{(\mu_{1}-2+s)^{I}(\mu_{1}+2\mu_{2}+s+t)^{I}}.

Similarly, the S-matrix elements between to vertices I and J of graphs contain

\prod_{(I,J)\in Edges}S^{\prime}(\alpha^{J}_{s^{J}t^{J}},\alpha^{I}_{s^{I}t^{I}})\supset\xi^{\sum_{IJ}B_{IJ}(\mu_{1}-2+s)^{I}(\mu_{1}+2\mu_{2}+s+t)^{J}},

which is the relevant part for the derivation. For the modified quantum dimension $d(\vec{\alpha})$

d(\vec{\alpha})=\frac{\left\{\alpha_{2}\right\}_{\xi}}{\left\{l\alpha_{2}\right\}_{\xi}\left\{\alpha_{1}\right\}_{\xi}\left\{\alpha_{1}+\alpha_{2}\right\}_{\xi}},

after shifting by s and t as above and some manipulations, the roots of unity dependent factor becomes

d(\vec{\alpha})\supset\frac{\xi^{2(\mu_{1}+2\mu_{2}+s+t)}-\xi^{2(\mu_{1}+s-2)}}{\left(1-\xi^{2(\mu_{1}+2\mu_{2}+s+t)}\right)\left(1-\xi^{2(\mu_{1}+s-2)}\right)}

We define coordinates of the 2D Cartan subalgebra of $osp(2|2)$ to be

y=\xi^{2(\mu_{1}+2\mu_{2}+s+t)}\qquad z=\xi^{2(\mu_{1}+s-2)}

Then the modified quantum dimension for a graph contains

d(y,z)\supset\frac{y_{I}-z_{I}}{(1-y_{I})(1-z_{I})}

We next sketch the derivation of $\hat{Z}^{osp(2|2)}$ in Section 3 by applying the procedure in the appendix D of [4]. For a closed oriented graph 3-manifold $Y$ equipped with $\omega\in H^{1}(Y;\mathbb{Q}/\mathbb{Z}\times\mathbb{Q}/\mathbb{Z})$ , the CGP invariant for a pair $(Y,\omega)$ in which Y is presented by Dehn surgery is

N_{l}(Y,\omega)=\sum_{s^{I},t^{I}=0}^{l-1}d(\alpha^{I_{0}}_{s^{I_{0}}t^{I_{0}}})\prod_{I\in Vert}d(\alpha^{I}_{s^{I}t^{I}})\left\langle\theta_{V_{\alpha^{I}_{s^{I}t^{I}}}}\right\rangle^{B_{II}}\prod_{(I,J)\in Edges}S^{\prime}(\alpha^{J}_{s^{J}t^{J}},\alpha^{I}_{s^{I}t^{I}})

\omega\in H^{1}(Y;\mathbb{Q}/\mathbb{Z}\times\mathbb{Q}/\mathbb{Z})\cong B^{-1}\mathbb{Z}^{L}/\mathbb{Z}^{L}\times B^{-1}\mathbb{Z}^{L}/\mathbb{Z}^{L}

\omega([m_{I}])=\mu^{I}=(\mu^{I}_{1},\mu^{I}_{2})\in\mathbb{Q}/\mathbb{Z}\times\mathbb{Q}/\mathbb{Z},\qquad\sum_{J}B_{IJ}\mu^{J}=0\quad\text{mod}\,\mathbb{Z}\times\mathbb{Z},

where $m_{I}$ is a meridian of I-th component $L_{I}$ of a surgery link $L$ and $[m_{I}]$ is its homology class and $B$ is a linking matrix of $L$ . A color of $L_{I}$ is set by $\omega([m_{I}])$ . The origin of $\omega$ can be found in the footnote in Section 1.3 of [2]. After substituting the ingredients, the right hand side becomes

\begin{split}N_{l}(Y,\omega)&=\frac{1}{l^{L+1}}\prod_{I\in V}\left(e^{i2\pi\mu^{I}_{1}}-e^{-i2\pi\mu^{I}_{1}}\right)^{deg(I)-2}\\ &\times\sum_{a^{I},b^{I}\in\mathbb{Z}^{L}/l\mathbb{Z}^{L}}F\left(\left\{\xi^{2(\mu_{1}+2\mu_{2}+a)},\xi^{2(\mu_{1}+b)}\right\}_{I\in V}\right)\xi^{\sum_{IJ}B_{IJ}(\mu_{1}+2\mu_{2}+a)^{I}(\mu_{1}+b)^{J}},\end{split}

where $a^{I}=s^{I}+t^{I},\,b^{I}=s^{I}-2$ and

F(y,z)=\prod_{I\in V}\left(\frac{y_{I}-z_{I}}{(1-y_{I})(1-z_{I})}\right)^{2-deg(I)}.

We next expand $F(y,z)$ , which modifies the summand as

\sum_{a^{I},b^{I}\in\mathbb{Z}^{L}/l\mathbb{Z}^{L}}\xi^{\sum_{IJ}B_{IJ}(\mu_{1}+2\mu_{2}+a)^{I}(\mu_{1}+b)^{J}+2\sum_{I}n_{I}(\mu_{1}+2\mu_{2}+a)^{I}+m_{I}(\mu_{1}+b)^{I}}

And then we recast it in terms of matrices by forming

\mathcal{M}=\frac{1}{2}\begin{pmatrix}0&B\\ B&0\end{pmatrix}\qquad r=\begin{pmatrix}b\\ a\end{pmatrix}\qquad p=\begin{pmatrix}B(\mu_{1}+2\mu_{2})+2m\\ B\mu_{1}+2n\end{pmatrix},

This enables us to apply the appropriate Gauss reciprocity formula

\sum_{r\in\mathbb{Z}^{L}/l\mathbb{Z}^{L}}exp\left(\frac{i2\pi}{l}r^{T}\mathcal{M}r+\frac{i2\pi}{l}p^{T}r\right)=\\ \frac{e^{i\pi\sigma(\mathcal{M})/4}(l/2)^{N/2}}{|Det\mathcal{M}|^{1/2}}\sum_{\delta\in\mathbb{Z}^{L}/2\mathcal{M}\mathbb{Z}^{L}}exp\left(-\frac{i\pi l}{2}\left(\delta+\frac{p}{l}\right)^{T}\mathcal{M}^{-1}\left(\delta+\frac{p}{l}\right)\right),

where $\mathcal{M}$ is a non-degenerate symmetric $2L\times 2L$ matrix over $\mathbb{Z}$ and $\sigma(\mathcal{M})$ is its signature. From the p-quadratic term in the exponential we read off

RHS\supset\xi^{-4m^{T}B^{-1}n},

which becomes the q-term in the summand in Section 3 after we analytically continue from a complex unit circle into an interior of an unit disc coordinatized by q.

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Towards a q-series for o​s​p​(2|2​n)osp(2|2n)

Abstract

1 Introduction

2 Background

3 Result

4 Examples

Appendix

Appendix A Type I Lie superalgebra and its quantization

Appendix B Invariance checks

Appendix C Derivations

References

Towards a q-series for $osp(2|2n)$