In this section we explain how conditions (3.7)–(3.9) on the existence of a good chamber arise.

Note that the factors in (3.4) corresponding to vertices of degree \(\deg (I)=2\) are trivial and therefore can be ignored.

Factors in (3.4) corresponding to vertices of degree \(\deg (I)=2+K>2\) can be expanded in two different ways:

$$\begin & \left( \frac\right) ^K\nonumber \\ & \quad =\left\ (z_I-1)^K(1-y_I^)^K\sum _^\infty \frac\,z_I^ y_I^, & |z_I|<|y_I|, \\ \;\;(1-y_I)^K(1-z_I^)^K\sum _^\infty \frac\,y_I^ z_I^, & |z_I|>|y_I|. \end \right. \nonumber \\ \end$$

(A.1)

Factors in (3.4) corresponding to vertices of degree \(\deg (I)=1\) can be expanded in four different ways:

$$\begin \frac= & \frac-\frac \nonumber \\= & \left\ (a)\;\;\sum _^\infty y_I^ -\sum _^\infty z_I^, & |y_I|<1,\,|z_I|<1, \\ (b)\;\;\;\sum _^\infty z_I^ -\sum _^\infty y_I^, & |y_I|>1,\,|z_I|>1, \\ (c)\;\;\;\sum _^\infty y_I^ +\sum _^\infty z_I^, & |y_I|<1,\,|z_I|>1, \\ (d)\;\;\;-\sum _^\infty y_I^ -\sum _^\infty z_I^, & |y_I|>1,\,|z_I|<1. \end \right. \nonumber \\ \end$$

(A.2)

First, we argue that the first two options are not allowed in a good chamber for a generic plumbing, in the terminology of Sect. 3. Let \(U\subset V|_\) be the set of vertices of degree one for which we have expansion (a) or (b) and assume that it is non-empty set in a given chamber. Let \(W\subset V|_\) be the subset of vertices of degree one for which expansion (c) or (d) holds, together with all vertices of degree greater than two. Then \(|_=U\sqcup W\) and if the plumbing is generic, by definition, there exist \(J\in U\) and \(K\in W\) such that \(B^_\ne 0\). For such a pair (J, K) consider a contribution of terms from expansions (A.1)–(A.2) into (3.11) corresponding to fixed summation variables \(\ell _I\). In the limit \(\ell _J \gg \ell _K \gg \ell _S,\;\forall S\ne J,K\) it behaves as

$$\begin \sim q^^\ell _J\ell _K} \end$$

(A.3)

in the leading order. Moreover, the signs in the exponent are opposite for the terms coming from two different sums in the expansions (a) or (b). Therefore (3.11) in this chamber has necessarily arbitrary large positive and large negative powers of q. Therefore the set U must actually be empty.

Taking this into account, a good chamber must be necessarily of form (3.10), specified by some vector \(\alpha \in \^}\). To argue inequalities (3.7)–(3.9) we will consider again the contributions of terms from expansions (A.1)–(A.2) into (3.11) corresponding to fixed summation variables \(\ell _I\).

In the regime when \(\ell _J\) is very large for at least one \(J\in V|_\), compared to \(\ell _K,\;\;\forall K \in V|_\), the contribution behaves as

$$\begin \sim q^\sum \limits _} \hspace \alpha _I\alpha _JB_^\ell _J\ell _K} \end$$

(A.4)

and the positivity of the exponent is equivalent to the copositivity of the matrix \(X_=-B^\alpha _I \alpha _J\).

Fix \(J\in V|_\), \(K\in V|_\) and consider the regime \(\ell _J \gg \ell _K \gg \ell _S,\;\forall S\ne J,K\). The contribution then behaves as

$$\begin \sim q^^\alpha _J\alpha _K\ell _J\ell _K} \end$$

(A.5)

in the leading order. The non-negativity of the exponent is equivalent to condition (3.8). In case \(B_^=0\), the subleading term in the exponent will dominate. It is positive due to copositivity of X.

Next fix a pair \(J,K\in V|_\), \(J\ne K\) and consider the regime \(\ell _J\sim \ell _K \gg \ell _S,\;\forall S\ne J,K\). The contribution then behaves as

$$\begin \sim q^^\alpha _J\alpha _K\ell _J\ell _K} \end$$

(A.6)

in the leading order. The positivity of the exponent is equivalent to condition (3.9).

Consider now a general regime of large \(\ell \in }^L\) (i.e., at least one \(\ell _I\) is large). The contribution behaves as

$$\begin \sim q^\sum \limits _} \hspace \alpha _J\alpha _KB_^\ell _J\ell _K \; - \hspace\sum \limits _ J\in V|_\\ K\in V|_ \end} \hspace \alpha _J\alpha _KB_^\ell _J\ell _K \; - \hspace\sum \limits _} \hspace \alpha _J\alpha _KB_^\ell _J\ell _K } \end$$

(A.7)

The exponent is then generically positive due to conditions (3.7)–(3.9). The only issue is the special direction when \(\ell _I=0\) for all I except some \(I_0\in V|_\). The exponent of q is then identically zero for all such contributions from expansions (A.1)–(A.2). This can give an infinite number of non-trivial contributions to the constant term in \(}^(2|1)}_\). However, one can make the sum of all such contributions to the constant term finite using the standard \(\zeta \)-function regularization, as described in Sect. 3 (which is essentially equivalent to the regularization by \(\epsilon \) described in Appendix D).

Consider the following q-series:

$$\begin F(q;\alpha ,\beta ,A,B):=\sum _ \frac}} \end$$

(B.1)

Assume \(A,\beta \ge 0\), \(\alpha ,B>0\). The asymptotic expansion of \(F(q;\alpha ,\beta ,A,B)\) can be obtained via Euler-Maclaurin summation formula:

$$\begin F(q;\alpha ,\beta ,A,B) \approx \int _0^\infty f(x)\textrmx -\sum _\frac\,f^(0) \end$$

(B.2)

where

$$\begin f(x):=\frac}} \end$$

(B.3)

and we use the convention \(B^\pm _1=\pm \frac\) for the two types of Bernoulli numbersFootnote 25. The expansion of f(x) at \(x=0\) reads

$$\begin f(x)= & \sum _\frac B^+_\hbar ^} \nonumber \\= & \sum _\sum _\frac\alpha ^\beta ^A^B^x^\,C_\hbar ^}\nonumber \\ \end$$

(B.4)

where

$$\begin C_:=\left\ \frac}, & M_1,M_2\ge 0, \\ (-1)^, & M_2=-M_1-1,\,M_1\ge 0, \\ 0, & \text . \\ \end \right. \end$$

(B.5)

The second part of (B.2) then reads:

$$\begin & -\sum _\frac\,f^(0) \nonumber \\ & \qquad -\sum _\sum _\frac\alpha ^\beta ^A^B^\,B^-_\,C_\hbar ^}\nonumber \\ & \quad = \sum _ }_L}\hbar ^L \end$$

(B.6)

where

$$\begin \tilde}_L=\sum _ N_1+N_2+M_1+M_2=L \\ N_\ge 0 \end}\frac\alpha ^\beta ^A^B^\,B^+_\,C_}.\nonumber \\ \end$$

(B.7)

The contribution to the power low behavior in the asymptotics of \(c_L\) when \(L\rightarrow \infty \) will be given by the terms in the sum at the boundary of the summation region in (B.7) where \(M_1,N_2,M_2\ll L\). Using the formula

$$\begin B^\pm _=\frac2(2n)!}}\,\zeta (2n), \end$$

(B.8)

the fact the \(B_r^+=0\) for odd \(r\ge 1\), Stirling approximation

$$\begin n!=\sqrt\left( \frac\right) ^n(1+O(1/n)), \end$$

(B.9)

and

$$\begin \zeta (2n)=\sum _ \frac} \end$$

(B.10)

we obtain that in \(L\rightarrow \infty \) limit:

$$\begin \tilde}_L= & \sum _\frac\alpha ^\beta ^A^B^\,C_B^-_} \nonumber \\\approx & \sum _,M_1,K\ge 0} \frac}\,\left( \frac\right) ^L \nonumber \\ & \times \frac\left( \frac\right) ^ \left( \frac\right) ^ \left( \frac\right) ^ C_\left( 1+O\left( \frac\right) \right) \nonumber \\= & \sum _\frac}\,\left( \frac\right) ^L\, \textrm\,\frac}}}}\,\left( 1+O\left( \frac\right) \right) . \end$$

(B.11)

We assumed that

$$\begin K A/\alpha < 2 \end$$

(B.12)

so that the sum with respect to \(M_1\) above is convergent. This asymptotic behavior corresponds to the singularities in the Borel plane at \(\xi =\frac\). From general formula (4.9) it follows that the corresponding Stokes jumps are of the form

$$\begin \Delta Z = }} \, \textrm\,\frac}}}} \,e^}\,(1+O(\hbar )). \end$$

(B.13)

Now consider the asymptotic expansion of the first term in (B.2) at \(\hbar \rightarrow \infty \):

$$\begin \int _0^\infty f(x)\textrmx= & \frac} \int _^ \textrmt\frac\,t}}}\,t -B\hbar }}\nonumber \\\approx & \frac}} F\left( \frac}\right) + \sum _ L\ge -1 \\ L\in \frac} \end} \tilde_L \Gamma \left( L+1\right) \hbar ^ \end$$

(B.14)

where

$$\begin & F(x):=\frac} (\pi \textrm(x)-\textrm(x)) \nonumber \\ & \quad =\sqrt \sum _\frac} -\frac}\left( \gamma +2\log x +\sum _\frac} \right) \nonumber \\ \end$$

(B.15)

is a function which can be expressed as a series with infinite radius of convergence (therefore its Borel transform does not have singularities away from the origin) and

$$\begin \tilde_L= \frac\sum _+M_2-\frac=L} \frac B^\alpha ^}C_\,\Gamma \left( \frac\right) }.\nonumber \\ \end$$

(B.16)

At \(L\rightarrow \infty \) we have

$$\begin \tilde_L\approx \sum _ \frac \textrm\left[ \left( -\frac\right) ^\, e^\left( 1+O\left( \frac\right) \right) \right] .\nonumber \\ \end$$

(B.17)

It follows that the Borel transform has corresponding singularities at \(\xi =-4\pi ^2K^2\alpha /A^2\). This is in agreement with the observation that the Stokes jumps in the first term of (B.2) originate from the poles in the integrand at \(x=\pm \frac-\frac\), \(K\in }_+\). They are of the form

$$\begin \Delta Z =-\frac\, e^ \, e^} \,(1+O(\hbar )). \end$$

(B.18)

The sign in the exponential depends on the choice of the branch of the Borel sum \(B(\xi )\), which has a branching point at the origin.

To summarize, the asymptotic expansion of (B.1) is given by

$$\begin \sum _ \frac}} & \approx \frac}} F\left( \frac}\right) + \sum _}_+} \tilde_L \Gamma \left( L+1\right) \hbar ^\nonumber \\ & \quad + \sum _}_+} }_L}\hbar ^L \end$$

(B.19)

where the function F is defined by a convergent series in (B.15) and the coefficients \(\tilde}_L\) and \(\tilde_L\) are given by explicit expressions (B.7) and (B.16) with asymptotic behavior at \(L\rightarrow \infty \) as in (B.11) and (B.17).

The aim of this appendix is to provide two proofs of the transformation properties of \(}^(2|1)}_(\tau )\) for the lens space L(2, 1) reported in Sect. 5.2.

First, notice that these homological blocks can be expressed in terms of Lambert series (5.3) as

$$\begin }^(2|1)}_(\tau )&= -\frac+2F(2\tau ), \end$$

(C.1)

$$\begin }^(2|1)}_(\tau )&= \frac + 2F(\tau )-2F(2\tau ), \end$$

(C.2)

$$\begin }^(2|1)}_(\tau )&= 2F(\tau /2)-4F(\tau )+2F(2\tau ). \end$$

(C.3)

Then, equation (5.9) immediately follows from the above relations and the invariance of \(F(\tau )\) under \(\tau \rightarrow \tau +1\). Equation (5.8) is readily derived from the relations above together with (5.4) and (5.6).

Another proof of equation (5.8), along the lines of [67, 76], is reported below. The \(\mathfrak (2|1)\) homological blocks of L(2, 1) can be written via the inverse Mellin transformFootnote 26 as

$$\begin }^(2|1)}_(\tau )&= \text _ + \frac\int _(s)=3/2} \textrms\, e^ (2\pi \tau )^ \Gamma (s) 2^\zeta (s,a/2)\zeta (s,b/2) \end$$

(C.5)

where we assume that \(|\text (-i\tau )|<\pi /2\) and use the fact that the Dirichlet series associated to \((}^(2|1)}_(\tau )-\text _)/2\) is

$$\begin L(d(a,b),s):=2^\sum _\sum _ (k+b/2)^(n+c/2)^= 2^\zeta (s,a/2)\zeta (s,b/2)\nonumber \\ \end$$

(C.7)

with d(a, b) being an abbreviation for the coefficients defined in Eq. (3.20) in the case \(p=2\) and

$$\begin \zeta (s,g):=\sum _\frac \end$$

(C.8)

being the Hurwitz zeta function, which reduces to the Riemann zeta function for \(g=1\), and at \(s=0\) is given by \(\zeta (0,g)=-g+1/2\).

Moving the path of integration to \(\text (s)=-1/2\), we obtain

$$\begin }^(2|1)}_(\tau )&= \text _+r_(\tau )+\frac\int _(s)=-1/2} \textrms\, e^ (2\pi \tau )^ \Gamma (s) L(d,s)\,. \end$$

(C.9)

The function \(r_(\tau )\) encodes the contributions from the poles of the Hurwitz zeta functions at \(s=1\) and the simple pole at \(s=0\) of the gamma function. Hence, we have

$$\begin r_(\tau )&= \text _\bigl ((-2\pi i\tau )^ 2\Gamma (s) L(d,s)\bigr ) + \text _\bigl ((-2\pi i\tau )^ 2\Gamma (s) L(d,s)\bigr ) \nonumber \\&= \frac(-4\pi i\tau )+\gamma -\gamma _0(a/2)-\gamma _0(b/2)} + \frac(a-1)(b-1). \end$$

(C.10)

Indeed, as s approaches 1,

$$\begin&\zeta (s,g)=\frac+\sum _\frac \gamma _n(g)(s-1)^n, \end$$

(C.11)

$$\begin&\Gamma (s) = 1-\gamma (s-1)+O(|(s-1)|^2), \end$$

(C.12)

$$\begin&x^ = x^ - \frac(x)}(s-1)+O(|(s-1)|^2), \end$$

(C.13)

where \(\gamma _0(g)= -\psi (g)\) is the digamma function and \(\gamma _0(1)=\gamma \) the Euler-Mascheroni constant.

Consider now

$$\begin \frac \bigl (&}^(2|1)}_(-1/\tau ) -_}\bigr ) \nonumber \\&\quad = \frac \int _(s)=3/2} \textrms\, e^ (2\pi )^(-1/\tau )^ \Gamma (s) L(d(a',b'),s) \end$$

(C.14)

$$\begin&\quad = \frac\int _(s)=3/2} \textrms\, e^ (2\pi )^\tau ^ \Gamma (s) 2^ \zeta (s,a'/2)\zeta (s,b'/2) \end$$

(C.15)

$$\begin&\quad = \frac\int _(s)=3/2} \textrms\, e^ (2\pi \tau )^ \Gamma (1-s)2^ \frac\bigl ( \frac\bigr )}\bigl ( \frac\bigr )} \end$$

(C.16)

$$\begin&\qquad \times ((-1)^\zeta (1-s,1/2)+\zeta (1-s))((-1)^\zeta (1-s,1/2)+\zeta (1-s)) \nonumber \\&\quad = \frac\int _(s)=-1/2} \textrms\, e^ (2\pi \tau )^ \Gamma (s) 2^\frac\bigl ( \frac\bigr )}\bigl ( \frac\bigr )}\nonumber \\&\qquad \times 2^((-1)^\zeta (s,1/2)^2+\zeta (s)^2+((-1)^+(-1)^)\zeta (s,1/2)\zeta (s)). \end$$

(C.17)

Above we assumed that \(0<\text (\tau )<\pi \) and \(0<\text (-1/\tau )<\pi \), and thus \(\text (-1/\tau )=\pi -\text (\tau )\). To go from the third to the fourth line we used the functional equation of the Hurwitz zeta function

$$\begin \zeta (s,a/p)=2\Gamma (1-s)(2\pi p)^ \sum _^p \text \biggl (\frac+\frac\biggr ) \zeta (1-s,k/p)\qquad \end$$

(C.18)

together with some identities satisfied by the gamma function. In the last step, we simply changed the integration variable from s to \((1-s)\).

Finally, we obtain

$$\begin&}^(2|1)}_(\tau ) - \frac \sum _(-1)^}^(2|1)}_(-1/\tau ) \end$$

(C.19)

$$\begin&\quad = }_(\tau ) + \frac\int _(s)=-1/2} \textrms\, e^ (2\pi \tau )^ \Gamma (s)L(d(a,b),s) \biggl (1+ i \frac\bigl (\frac\bigr )}\bigl (\frac\bigr )}\biggr ) \end$$

(C.20)

$$\begin&\quad =}_ + \frac\int _(s)=-1/2} \textrms\, (2\pi \tau )^ \frac\bigl (\frac\bigr )}2^\zeta (s,a/2)\zeta (s,b/2) \end$$

(C.21)

where we denote by

$$\begin }_(\tau )&= \text _+ _(\tau )\sum _(-1)^\frac_} \nonumber \\&= \text _+ \frac(a-1)(b-1)+\frac\nonumber \\&\quad \left( \text (-4\pi i\tau )+\gamma -\gamma _0(a/2)-\gamma _0(b/2)2\pi i\sum _(-1)^\text _\right) \,. \end$$

(C.22)

Note that last integral (C.21) converges for all \(\tau \in }\) with \(|\text (\tau )|<\pi \) and thus it provides an extension of the period function \(\psi _(\tau )\) to the slit plane.

Consider the case when \(M^3\) is obtained from a tree plumbing, that is \(M^3\) is a surgery on the collection of linked unknots forming a tree. We will follow the conventions of Sect. 2.2. Let us pick some node in the plumbing tree (corresponding to a particular unknot) and consider this tree as a rooted tree and with edges oriented according to the direction opposite to the root. Denote the index of the root node by \(I_0\). The topological invariant of [26] corresponding to the quantum deformation parameter \(\xi =q^\frac=\exp \frac\) (for odd \(\ell \ge 3\)) of \(}_q(\mathfrak (2|1))\) then reads:

$$\begin & N_\ell (M^3,\omega ) \nonumber \\ & \quad =\sum _^d(\alpha ^_t^})\prod _} d(\alpha ^I_) \langle \theta _}} \rangle ^} \prod _} S'(\alpha ^J_,\alpha ^I_)\nonumber \\ \end$$

(D.1)

where

$$\begin & \:=\xi ^x-\xi ^, \end$$

(D.2)

$$\begin & \alpha _:=(\alpha _1+s,\alpha _2+t)\;\in }\times }, \end$$

(D.3)

$$\begin & (\alpha _1,\alpha _2)\equiv (\mu _1-\ell +1,\mu _2+\ell /2)\;\in }\times }, \end$$

(D.4)

$$\begin & d(\alpha )=\frac}\\}, \end$$

(D.5)

$$\begin & S'(\alpha ',\alpha )=\frac \,\xi ^, \end$$

(D.6)

$$\begin & \langle \theta _ \rangle = -\xi ^. \end$$

(D.7)

The element \(\omega \in H^1(M^3,}/}\times }/})\cong B^}/}\times B^}/}\) is specified by its values on the meridians \(}_I,\; I\in \text \) of the link components. That is

$$\begin \omega (}_I)=\mu ^I=(\mu _1^I,\mu _2^I)\in }/}\times }/},\;\;\;\sum _B_\mu ^J=0\mod }\times }.\nonumber \\ \end$$

(D.8)

After some manipulations and change of summation variables \((a^I,b^I)=(s^I+t^I+1,t^I)\) we have:

$$\begin N_\ell (M^3,\omega )= & \frac}(e^-e^)^(I)-2}}} \nonumber \\ & \quad \times \sum _}_\ell } F\left( \,\xi ^\}_}\right) \nonumber \\ & \quad \xi ^B_(a^I+\mu _1^I+\mu _2^I)(b^J+\mu _2^J)} \end$$

(D.9)

where F is the following rational function of 2L variables:

$$\begin F(y,z)=\prod _}\left( \frac \right) ^(I)}. \end$$

(D.10)

We will need to use the following general Gauss reciprocity formula [94, 95]:

$$\begin & \sum _}^N/\ell }^N} \exp \left( \frac\,r^T}r+\frac\,p^Tr\right) \nonumber \\ & \qquad =\frac})}}\,(\ell /2)^}}|^} \sum _}^N/2}}^N} \exp \left( -\frac\left( \delta +\frac\right) ^T}^\left( \delta +\frac\right) \right) \nonumber \\ \end$$

(D.11)

where \(}\) is a symmetric non-degenerate \(N\times N\) matrix with integer entries and \(\sigma (})\) is its signature.

Consider then expansion of F(y, z) with respect to \(y_I,z_I\) in some chamber. In (D.9) we will need then to plug in \(y_I=\xi ^,\,z_I=\xi ^\). Because \(|\xi |=1\) this in principle violates the convergence of the series. However, one can cure it by introducing a regularization parameter \(\epsilon >0\) so that the arguments in F are deformed to \(y_I=e^ \,\xi ^,\,z_I=e^ \,\xi ^\), where \(\alpha _I(=\pm 1)\) determines the expansion chamber as in (3.10). This makes the series to be convergent. Below we will consider analytic continuation with respect to \(q=e^}\), away from the unit circle to the region with \(|q|<1\). Taking the radial limit \(q\rightarrow e^}\) and then \(\epsilon \rightarrow 0\), we recover the original \(N_\ell (M^3,\omega )\). On the other hand, one can first take the limit \(\epsilon \rightarrow 0\). As we will see below, it will give a well-defined q-series when the expansion chamber, specified by \(\alpha \), is good, in the terminology of Sect. 3. The result can be expressed using the q-series \(}^(2|1)}_\). The invariant \(N_\ell (M^3,\omega )\) then can be recovered by taking the radial limit \(q\rightarrow e^}\), assuming that it commutes with the limit \(\epsilon \rightarrow 0\). In our work we do not perform a rigorous mathematical analysis of whether the two limits commute, but we will assume it.

The contribution of the monomial \(\prod _y_I^z_I^\) (in the expansion of F described above) to the sum in the second line of (D.9) reads

$$\begin & \sum _}_\ell } \xi ^B_(a^I+\mu _1^I+\mu _2^I)(b^J+\mu _2^J)+2\sum _I \left( n_I(a^I+\mu _1^I+\mu _2^I)+m_I(b^I+\mu _2^I)\right) } \nonumber \\ & \quad =\frac \sum _},}\in }^L/2B}^L} e^}^T B^}+2\pi i(n-B\mu _2)^TB^}+2\pi i(m-M(\mu _1+\mu _2))^TB^}} \cdot \xi ^m} \nonumber \\ & \quad = \frac \sum _}^L/B}^L} e^\beta +4\pi i(n-B\mu _2)^TB^\gamma +2\pi i(m-B(\mu _1+\mu _2))^TB^\beta } \cdot \xi ^m}. \nonumber \\ \end$$

(D.12)

To go from the first line to the second we used Gauss reciprocity formula (D.11) with

$$\begin & N = 2L \end$$

(D.13)

$$\begin & r = \left( \begin a \\ b \end \right) \end$$

(D.14)

$$\begin & } = -\left( \begin 0 & B \\ B & 0 \end \right) , \;\;\sigma (})=\sigma (B)-\sigma (B)=0 \end$$

(D.15)

$$\begin & p = 2\left( \begin n-B\mu _2 \\ m-B(\mu _1+\mu _2) \end \right) \end$$

(D.16)

$$\begin & \delta = \left( \begin } \\ } \end \right) \end$$

(D.17)

From the second line to the third one, we made a change of variables \(}= B\beta '+\beta ,\;\beta '\in }_2^L,\,\beta \in }^L/B}^L\) and performed the sum over \(\beta '\), which gives the \(2^L\) factor times the delta function with the condition \(} =0\mod 2\), that is explicitly solved by \(} =2\gamma \). Combining (D.9) and (D.12) we can write:

$$\begin N_\ell (M^3,\omega )= & \frac}(e^-e^)^(I)-2}}}\nonumber \\ & \times \sum _ \beta ,\gamma \in }^L/B}^L \\ b,c\in B^}^L/}^L \end} e^\beta +4\pi i(b-\mu _2)^T\gamma +2\pi i(c-(\mu _1+\mu _2))^T\beta } \cdot (-1)^\Pi \, }^(2|1)}_|_\nonumber \\ \end$$

(D.18)

where, as in (3.2),

$$\begin }_^(2|1)}= & (-1)^ \textrm_\left\\right) ^ \right| _\alpha } \right. \nonumber \\ & \quad \times \left. \sum _ n=Bb \mod B}^L \\ m=Bc \mod B}^L \end} q^B^_\,n_Im_J}\, \prod _ z_J^y_J^ \right\} \quad \;\; \in q^}}[[q]]. \nonumber \\ \end$$

(D.19)

This relation has the following natural conjectural generalization for rational homology spheres:

$$\begin N_\ell (M^3,\omega ) & = \frac}([2\omega _1])}})|} \sum _ \beta ,\gamma \in H_1(M^3,}) \\ b,c\in H^1(M^3,}/}) \end}\nonumber \\ & \quad \times e^ \cdot }^(2|1)}_|_\nonumber \\ \end$$

(D.20)

where \(}\) is the Reidemeister torsion (equal to the analytic torsion) of the U(1) flat connection \([2\omega _1]:=(2\omega _1\mod H^1(M^3,}))\in H^1(M^3,}/})\), the same one as the one appeared in the context of U(1|1) Chern–Simons theory [23, 24, 82,83,84,85]. The overall sign \(\pm 1\) reflects the sign ambiguity in the definition of the torsion.