数理解析研究所講究録
第 1866 巻 2013 年 45-67

45The colored HOMFLY homology and super-A-polynomial
Hiroyuki Fuji1’2
1Interdisciplinary Center for Mathematical Sciences,
Graduate School of Mathematical Sciences,
University of Tokyo,
3-8-1 Komaba, Tokyo 153-8914, Japan
2
Institute for Biology and Mathematics of Dynamical Cell Processes(iBMath), University of Tokyo,
3-8-1 Komaba, Tokyo 153-8914, Japan
Abstract
In this article we will discuss the generalizations of the A-polynomial on basis of thecolored superpolynomial. The colored superpolynomial is the Poincar\’e polynomial of thecolored HOMFLY homology that categorifies the colored HOMFLY polynomial. In particular for the colored superpolynomial with the symmetric representation, the analogues of
the generalized/quantum volume conjecture can be considered. As a result of the study ofthe colored HOMFLY homology for the $(2, 2p+1)$ -torus knots and -twist knots, we findthe 2-parameter deformation of the A-polynomial that is named as super- -polynomial.
$n$

$A$

1

IntroductionLetbe the character variety of the knot $K$ . This variety is defined to be the moduli
space of theholonomy representation of the knot complement:$\mathcal{C}_{K}$$SL_{2}(\mathbb{C})$$\mathbb{S}^{3}\backslash K$$\rho$$Hom(\pi_{1}(K);SL_{2}(\mathbb{C}))/$

conjugation.

(1)is the algebraic curve in the variables
which are given by the eigenvaluesof the holonomy matrices for the meridian and longitude elements:$C_{K}$$(x, y)\in \mathbb{C}^{*}\cross \mathbb{C}^{*}$$\mu,$$\rho(\mu)=(x0x^{-1}*) , \rho(\lambda)=(y0y^{-1}*)$.

$\lambda\in\pi_{1}(K)$

(2)

46
The defining equation of$C_{K}$

is given by the$\mathcal{A}$-polynomial $A_{K}(x, y)[8]$ :As for example, the A-polynomials for
relations of:$3_{1}$

and$4_{1}$(3).

$C_{K}=\{(x, y)\in \mathbb{C}^{*}\cross \mathbb{C}^{*}|A_{K}(x, y)=0\}$

are found manifestly from the generator$\pi_{1}(K)$(4)$A_{3_{1}}(x, y)=(y-1)(y+x^{3})$ ,$A_{4_{1}}(x, y)=(y-1)(y^{2}+(-x^{4}+x^{3}+2x^{2}+x-1)y+1)$

In the genemlized volume conjecture, the colored Jones polynomial andare related in the asymptotic limit.
Conjecture 1.1 $([27, 33, 23J)$ Let $J_{n}(K;q)$ be an
asymptotic limit:$q=e^{\hslash}arrow 1,$$narrow\infty,$

$n$

.

(5)
$A$

-polynomial-colored Jones polynomial. In the(6)(fixed),$x:=q^{n}$there exist a saddle point such that the colored Jones polynomial grows exponentially:$J_{n}(K;q=e^{\hslash}) \simeq\exp(\frac{1}{\hslash}S_{0}(x)+\cdots)$

The leading coefficient$S_{0}(x)$.

(7)

satisfies the relation:$x \frac{\partial S_{0}(x)}{\partial x}=\log y(x) , A_{K}(x, y(x))=0$.

(8)

The quantum version of the volume conjecture (which is also known as $AJ$ conjecture)also relates the colored Jones polynomial and A-polynomial via the -difference equation:$q$

Conjecture 1.2 $([23, 20J)$ The colored Jones polynomial obeys the$A_{K}(\hat{x},\hat{y};q)J_{n}(K;q)=0$$q$-difference equation:(9),$\hat{x}J_{n}(K;q)=q^{n}J_{n}(K;q) , \hat{y}J_{n}(K;q)=J_{n+1}(K;q) , \hat{y}\hat{x}=q\hat{x}\hat{y}$such that the$q$-difference operator$A_{K}(\hat{x},\hat{y};q)$

.

(10)

yields in the limit $qarrow 1$ of (6):$A_{K}(\hat{x},\hat{y};q)arrow^{q\vec {}1}A_{K}(x, y)$

.

(11)

This conjecture is verified directly for some knots via computer talks [20], and the qcan be found explicitly.
difference operator
In recent developments of the string theory, some extensions of the volume conjecture
have been studied. In [1], the analogue of the generalized/quantum volume conjecture
are also proposed for the colored HOMFLY polynomial:$A_{K}(\hat{x},\hat{y};q)$47
Conjecture 1.3 $([1J)$ Let $H_{n}(K;a, q)$ be a colored HOMFLY polynomial of the r-th completely symmetric representation $S^{r}(n=r+1)$ . In the asymptotic limit:$q=e^{\hslash}arrow 1,$$narrow\infty,$$a,$$x:=q^{n}$(fixed),(12)

there exist a saddle point such that the colored Jones polynomial grows exponentially:$H_{n}(K;a, q=e^{\hslash}) \simeq\exp(\frac{1}{\hslash}S_{0}(x;a)+\cdots)$,$x \frac{\partial S_{0}(x;a)}{\partial x}=\log y(x) , A_{K}^{Q-def}(x, y(x);a)=0$

(13)

.

(14)

The -deformed -polynomial $A_{K}^{Q-def}(x, y;a)$ coincides with the augmentation polynomial[$35J$ for the differential graded algebm of the knot contact homology $[12J.$
$A$

$Q$

Conjecture 1.4 ([ ) One finds a
the colored HOMFLY polynomial:
$1J$

$q$

-difference opemtor$A_{K}(\hat{x},\hat{y};q)H_{n}(K;a, q)=0$

that annihilates$A_{K}^{Q-def}(\hat{x},\hat{y};a, q)$

,

(15)

where and acts on $H_{n}(K;a, q)$ as (10), and the -difference opemtor
reduces to the -deformed -polynomial $A_{K}^{Q-def}(x, y;a)$ in the limit (12):$\hat{x}$$A_{K}^{Q-def}(\hat{x},\hat{y};a, q)$$\hat{y}$

$q$

$Q$

$A$

$A_{K}^{Q-def}(\hat{x},\hat{y};a, q)arrow^{q\vec {}1}A_{K}^{Q-def}(x, y;a)$

.

(16)

In the context of the topological string, this -difference equation reveals the $D$ -module$q$structure of the open topological string, and such aspect has been studied in $aarrow 1$ limit
via the topological recursion on the character variety [9, 10, 25, 5].
On basis of these observations for the $(2, 2p+1)$ -torus knots and -twist knots, we
expect the further extension of the volume conjecture to the colored superpolynomial.
The colored superpolynomial. is the Poincar\’e polynomial of the colored
HOMFLY homology:$n$$\mathcal{P}^{R}(K;a, q, t)$$\mathcal{H}_{ijk}^{R}(K)[24]$$\mathcal{P}^{R}(K;a, q, t)=\sum_{i,j,k}a^{i}q^{j}t^{k}\dim \mathcal{H}_{ijk}^{R}(K)$

.

(17)

The triply-graded homology
categorifies the the colored HOMFLY polynomial$H^{R}(K;a, q)$ , therefore these polynomial invariants are
related by$\mathcal{H}_{ijk}^{R}(K)$$\mathcal{P}^{R}(K;a, q, t=-1)=H^{R}(K;a, q)$

.

(18)

In particular for the completely symmetric representation $R=S^{r}$ , the colored super$:=\mathcal{P}^{S^{n-1}}(K;a, q, t)$polynomial, we can study the natural extension of thequantum volume conjecture.
$\mathcal{P}_{n}(K;a, q, t)$

48
be the colored superpolynomial
Conjecture 1.5 $([18J)$ Let
the completely symmetnc representation $R=S^{r}(r=n-1)$ . One finds aequation:$A_{K}^{\sup er}(\hat{x},\hat{y};a, q, t)\mathcal{P}_{n}(K;a, q, t)=0$

such that the$A_{K}^{\sup er}(x, y;a, t)$

$q$

for
-difference$\mathcal{P}^{R}(K;a, q, t)$$\mathcal{P}_{n}(K;a, q, t)$

$q$

(19)

,

-difference opemtor reduces to the 2-pammeter extended$A$-polynomialin the limit:$q=e^{\hslash}arrow 1,$$narrow\infty,$$a,$$t,$ $x:=q^{n}$(20)(fixed).

reduces to the -deformed -polynomial in the specialization $t=-1$ , andfurther reduces to the -polynomial in the specialization $a=1,$ $t=-1.$$Q$$A_{K}^{\sup er}(x, y;a, t)$

$A$

$A$

defined above is named as the
The 2-parameter extended A-polynomial
super- -polynomial, and this polynomial is also found via asymptotic expansion of thecolored superpolynomial:$A_{K}^{\sup er}(x, y;a, t)$$A$$\mathcal{P}_{n}(K;a, q=e^{\hslash}, t)\simeq\exp(\frac{1}{\hslash}S_{0}(x;a, t)+\cdots)$

(21)

,

.

$x \frac{\partial S_{0}(x;a,t)}{\partial x}=\log y(x) , A_{K}^{\sup er}(x, y(x);a, t)=0$(22)

Conjecture 1.5 is verified for $K=3_{1},4_{1},5_{2},6_{1}$ in [18, 34] via the computer talks on basis.
of the explicit formula for the colored superpolynomial
The organization of this article is as follows. In section 2, we will survey briefly on
the colored superpolynomial [11, 24, 22] and show some computational results obtainedin [19]. In section 3, we will discuss about the super- -polynomial from the explicit
formulae of the colored superpolynomial obtained in section 2, and see their properties.
In the appendix, we discuss the string-theoretical perturbative invariant associated to thesuper- -polynomial which is found via the topological recursion as is discussed in [10, 5]
for the A-polynomial.
$\mathcal{P}_{n}(K;a, q, t)$

$A$

$A$

2

Colored HOMFLY homology and colored superpolynomial

2.1

Categorification of the HOMFLY polymonial

The categorifications of the Jones polynomial and HOMFLY polynomial are established in the celebrated works [28, 30]. In [30] the doubly-graded homology
invariant which is
is constructed via the matrix factorization, and categorifies thegiven by the specialization of the HOMFLY polynomial $H(K;a=q^{N}, q)$ . This homol-Khovanov-Rozansky’s homology. As the decategorification ofogy is known as the$\mathcal{H}_{kp}^{sl_{N}}(K)$$sl_{N}$$sl_{N}$49
the Khovanov-Rozansky’s homology, one finds thecharacteristic:$sl_{N}$invariant via the -graded Euler
$q$

$*$

$H(K;a=q^{N}, q)= \sum_{k,p}(-1)^{k}q^{p}\dim \mathcal{H}_{kp}^{sl_{N}}(K)$

.

(23)

In order to describe thehomologies in the unified manner, the triply-graded homology
is proposed in [11, 29]. This homology categorifies the HOMFLY polynomial:$sl_{N}$$\mathcal{H}_{ijk}(K)$$H(K;a, q)= \sum_{i,j,k}a^{i}q^{j}(-1)^{k}\mathcal{H}_{ijk}(K)$

,

(24)

and is named as the HOMFLY homology. In particular, the Poincarepolynomial of the
HOMFLY homology is known as superpolynomial:$\mathcal{P}(K;a, q, t)=\sum_{i,j,k}a^{i}q^{j}t^{k}\dim \mathcal{H}_{ijk}(K)$

.

(25)

The properties of the HOMFLY homology are given as its definition in [11], and they
are listed as follows:
Properties of the HOMFLY homology
Hl.

H2.

$\mathcal{H}_{*}$

categorifies the HOMFLY polynomial.

$\mathcal{H}_{*}$

has finite support:

H3. The differentials$\dim \mathcal{H}_{*}<\infty$$d_{N}(N\in \mathbb{Z})$acts on$\mathcal{H}_{*}$, and the grading $(ijk)$ is shifted as:\dagger$d_{N}$:$\mathcal{H}_{ijk}arrow \mathcal{H}_{i-1,j+N,k-1}(N>0)$(26)$d_{N}$:$\mathcal{H}_{ijk}arrow \mathcal{H}_{i-1,j+N,k-3}(N\leq 0)$(27)

Thus the differential
are triply-graded of degree: $\deg d_{N}=(-1, N, -1)(N>0)$ ,and $\deg d_{N}=(-1, N, -3)(N\leq 0)$ .
$d_{N}$

H4. The differentials are anticommuting each other:$(N, M\in \mathbb{Z})$ .
Hence for $N=M,$ $d_{N}^{2}=0.$
H5. There exist an involution

H6.

$\phi$:$\mathcal{H}_{ij*}arrow \mathcal{H}_{i,-j,*}$$d_{N}od_{M}=-d_{M}\circ d_{N}$. The action of$\phi$

on$d_{N}$Taking homology with respect to $d_{N}(N\geq 2)$ , one finds theRozansky’s homology
after the amalgamation:

for all

is $\phi d_{N}=d_{-N}\phi.$$sl_{N}$

Khovanov-$\mathcal{H}^{sl_{N}}$$\bigoplus_{iN+j=p}(\mathcal{H}_{*}, d_{N})_{ijk}=\mathcal{H}_{kp}^{sl_{N}}$

.

(28)

’In this article, we discuss the reduced homology which is the trivial for the unknot.
\dag er Here
we obey the convention of grading in [24]. In [11] another degree is assigned for $d_{N}(N<0)$ such that $\deg d_{N}=$$(-1, N, N-1)$ These two gradings are combined into the quadruply-graded homology [22].

50
H7. Taking homology with respect to[31]:$d_{\pm 1}$, one finds the one dimensional$(\mathcal{H}_{*}, d_{1})=\mathcal{H}_{S,-S,0}, (\mathcal{H}_{*}, d_{-1})=\mathcal{H}_{S,S,2S}$$sl_{1}$-homology(29).

The grading of the surviving generator is specified by the Rasmussen’s invariant[37] (e.g. $S=(p-1)(q-1)/2$ for $(p, q)$ -torus knot).
H8. Taking homology with respect toafter a regrading:$d_{0}$$S$, one finds the knot Floer homology $HFK[36]$$\bigoplus_{-2i+k=k’}(\mathcal{H}_{*}, d_{0})_{ijk}=HFK_{k’j}$

.

(30)

KhovanovThese properties implies that the HOMFLY homology unifies not only theRozansky’s homology, but also their deformations [21, 26] and knot Floer homology.
Conversely speaking, one can determine the structure of the triply-graded homology
which satisfies the above properties consistently, and the following conjecture is proposed:$sl_{N}$$\mathcal{H}_{ijk}$

Conjecture 2.1 $([11J)$ There exists a triply-graded homology
properties $HI$ -H8.

$\mathcal{H}_{ijk}$

which satisfies theThis conjecture is verified for various knots up to 8 crossings, and the superpolynomials(i.e. homological structure of the triply-graded homology) are specified uniquely.

2.2

Colored HOMFLY homology

The categorification of the colored HOMFLY polynomial $H^{R}(K;a, q)$ is proposed inwhich categorifies $H^{R}(K;a, q)$ is named as the[24]. The triply-graded homology
colored HOMFLY homology:$\mathcal{H}_{ijk}^{R}$$H^{R}(K;a, q)= \sum_{ijk}a^{i}\phi(-1)^{k}\dim \mathcal{H}_{ijk}^{R}(K)$

.

(31)

Here we focus on the colored HOMFLY homology in the completely symmetric repreare proposed in [24], and they consists of thesentation $R=S^{r}$ . The properties ofnatural generalization of the properties $HI$ -H7 and new aspects are introduced via thecolored differentials. Such properties listed as follows:
Properties of the colored HOMFLY homology$\mathcal{H}_{ijk}^{s^{r}}$Cl.

$\mathcal{H}_{*}^{s^{r}}$C2.

$\mathcal{H}_{*}$

categorifies the colored HOMFLY polynomial $H^{S^{r}}(K;a, q)$ .

has finite support:$\dim \mathcal{H}_{*}<\infty$

51
C3. The differentials$d_{N}^{S^{r}}(N\in \mathbb{Z})$acts on$\mathcal{H}_{*}$

with degrees:$\deg d_{N}^{S^{r}}=(-1, N, -1)(N\geq 1-r) , \deg d_{N}^{S^{r}}=(-1, N, -3)(N\leq-r)$

C4. The differentials are anticommuting each other:. Hence for $N=M,$$(N, M\in \mathbb{Z})$.

$d_{N}^{S^{r}}\circ d_{M}^{s^{r}}=-d_{M}^{S^{r}}\circ d_{N}^{S^{r}}$(32)

for all$(d_{N}^{S^{r}})^{2}=0.$

C5. There exist an involution:
symmetric representation such that$\phi$$\mathcal{H}_{ij*}^{s^{r}}arrow \mathcal{H}_{i,-j,*}^{\Lambda^{r}}$$(S^{r})^{t}=\Lambda^{r}$where
denotes the r-th anti. The involution acts onas:$\Lambda^{r}$$d_{N}^{S^{r}}$$\phi d_{N}^{S^{r}}=d_{-N}^{\Lambda^{r}}\phi.$

C6. Taking homology with respect to$d_{N}^{S^{r}}(N\geq 2)$, one finds the$S^{r}$-colored$sl_{N}$homology$\mathcal{H}^{s}\iota_{N},s^{r}:(\mathcal{H}_{*}^{S^{r}}, d_{N}^{S^{r}})_{ijk}\simeq \mathcal{H}^{sl_{N},S^{r}}$C7. Taking homology with respect to$d_{1}^{S^{r}}$and$d_{r}^{\underline{S}^{r}}$, one finds the one dimensional ho-mology:$(\mathcal{H}_{*}^{S^{r}}, d_{1}^{S^{r}})=\mathcal{H}_{rS,-rS,0}^{s^{r}}, (\mathcal{H}_{*}^{S^{r}}, d_{-r}^{S^{r}})=\mathcal{H}_{rS,r^{2}S,2rS}^{s^{r}}$C8. Taking homology with respect tohomology of the smaller representation $R=S^{k}$ :$d_{1-k}^{s^{r}}(1\leq k\leq r)$.

, one finds the colored HOMFLY(34)$(\mathcal{H}_{*}^{S^{r}}, d_{1-k}^{S^{r}})\simeq \mathcal{H}^{S^{k}}$This class of differentials$d_{1-k}^{S^{r}}$are named as the vertical colored differentials.

C9. There exists yet another class of the differential
tation $S^{r}arrow S^{m}(r>m)$ :$d_{rarrow m}$which reduces the represen(35)$(\mathcal{H}_{*}^{S^{r}}, d_{rarrow m})\simeq \mathcal{H}^{s^{m}}$For example, the degree of(33)$d_{rarrow r-1}$

is $(0,1,0)$ .

In the statement of the properties C8 and C9, one can identify the generators after a
re-grading $(i,j, k)arrow(i’, j’, k’)$ . Although the re-grading rules are not simply describedin the triply-graded homology [24], such rules become manifest when we discuss in theframework of the quadruply-graded homology [22], and larger hidden symmetry of thehomology can be found.
Assuming these properties as the axioms of the colored HOMFLY homology, one canalso determine the triply-graded homology from the consistency of conditions $CI$ -C9.
Actually, the explicit expressions of the colored superpolynomial:$\mathcal{P}^{S^{r}}(K;a, q, t)$$\mathcal{P}^{S^{r}}(K;a, q, t)=\sum_{ijk}a^{i}q^{j}t^{k}\dim \mathcal{H}_{ijk}^{S^{r}}(K)$

(36)

52
are determined for the $(2, 2p+1)$ -torus knots $T^{2,2p+1}[17,19]$ and the twist knots with$n$crossings $TK_{n}[34,19]$ :$\mathcal{P}^{S^{r}}(T^{2,2p+1};a, q, t)$$=$$a^{pr}q^{-pr} \sum_{0\leq k_{p}\leq\cdots\leq k_{2}\leq k_{1}\leq k_{0}=r}\{\begin{array}{l}rk_{1}\end{array}\}\{\begin{array}{l}k_{1}k_{2}\end{array}\}\cdots\{\begin{array}{l}k_{p-1}k_{p}\end{array}\}$$\cross q^{(2r+1)\Sigma_{=1}^{p}k_{i}-\Sigma_{=1}^{\dot{p}}k_{:-1}k_{i}}t^{2\Sigma_{i=1}^{\rho}k_{i}}\prod_{i=1}^{k_{1}}(1+aq^{i-2}t)$$\mathcal{P}^{s^{r}}(TK_{2n+2};a, q, t)$$=$$\{\begin{array}{l}nk\end{array}\}$(37)$\sum_{0\leq k_{n}\leq\cdots\leq k_{2}\leq k_{1}\leq k_{0}=r}\{\begin{array}{l}rk_{1}\end{array}\}\{\begin{array}{l}k_{1}k_{2}\end{array}\}\cdots\{\begin{array}{l}k_{n-1}k_{n}\end{array}\}a^{\Sigma_{=1}^{\dot{n}}k_{i}}t^{2\Sigma_{i=1}^{n}k}\dot{\cdot}$$\cross q^{\Sigma_{=1}^{n}(k_{i}^{2}-k.)}\prod_{i=1}^{k_{1}}(1+a^{-1}q^{2-i}t^{-1})(1+a^{-1}q^{1-r-i}t^{-3})$

where

,

(38)

,

denotes the -binomial coefficient:$q$$\{\begin{array}{l}nk\end{array}\}=\frac{(q;q)_{n}}{(q;q)_{k}(q;q)_{n-k}},$$(q;q)_{n}= \prod_{i=1}^{n}(1-q^{i})$

.

(39)

The colored superpolynomial for the twist knots with odd number of crossings can be
expressed in the similar manner.

3

Super- $A$ -polynomial

Now we shall discuss about the super- -polynomial and Conjecture 1.5 based on (37)and (38). $Rom$ the asymptotic expansion of these expressions for the colored superpolyknots,
omials, one finds the super- -polynomialfrom (7). Forandthe super- -polynomials are
$A$

$A$

$A_{K}^{\sup er}(x, y;a, t)$$3_{1}$$4_{1}$$A$$A_{3_{1}}^{\sup er}(x, y;a, t)=a^{2}t^{4}(x-1)x^{3}+(1+at^{3}x)y^{2}$$-a(1-t^{2}x+2t^{2}(1+at)x^{2}+at^{5}x^{3}+a^{2}t^{6}x^{4})y$,(40)$A_{4_{1}}^{\sup er}(x, y;a, t)=a^{2}t^{5}(x-1)^{2}x^{2}+at^{2}x^{2}(1+at^{3}x)^{2}y^{3}$$+at(x-1)(1+t(1-t)x+2at^{3}(t+1)x^{2}-2at^{4}(t+1)x^{3}+a^{2}t^{6}(1-t)x^{4}-a^{2}t^{8}x^{5})y$$-(1+at^{3}x)(1+at(1-t)x+2at^{2}(t+1)x^{2}+2a^{2}t^{4}(t+1)x^{3}+a^{2}t^{5}(t-1)x^{4}+a^{3}t^{7}x^{5})y^{2}.$(41)Specializing to $a=1$ and $t=-1$ , one finds that these expressions reduce to the Apolynomials (4) and (5).

53
Now our conjecture 1.5 can be verified explicitly. For
and , one finds the quantum
super- -polynomials
manifestly via the computer talks [20]:$4_{1}$$3_{1}$$A$$A_{K}^{\sup er}(\hat{x},\hat{y};a, q, t)$$A_{3_{1}}^{\sup er}( \hat{x},\hat{y};a, q, t)=\sum_{k=0}^{2}\hat{a}_{3_{1}}^{(k)}(\hat{x};a, q, t)\hat{y}^{k}$

,

(42)

$\hat{a}_{3_{1}}^{(0)}=\frac{a^{2}t^{4}(\hat{x}-1)\hat{x}^{3}(1+aqt^{3}\hat{x}^{2})}{q(1+at^{3}\hat{x})(1+at^{3}q^{-1}\hat{x}^{2})}, \hat{a}_{3_{1}}^{(2)}=1,$$\hat{a}_{3_{1}}^{(1)}=-\frac{a(1+at^{3}\hat{x}^{2})(q-q^{2}t^{2}\hat{x}+t^{2}(q^{2}+q^{3}+at+aq^{2}t)\hat{x}^{2}+aq^{2}t^{5}\hat{x}^{3}+a^{2}qt^{6}\hat{x}^{4})}{q^{2}(1+at^{3}\hat{x})(1+at^{3}q^{-1}\hat{x}^{2})}.$$A_{4_{1}}^{\sup er}( \hat{x},\hat{y};a, q, t)=\sum_{k=0}^{3}\hat{a}_{4_{1}}^{(k)}(\hat{x};a, q, t)\hat{y}^{k}$

,

(43)

$\hat{a}_{4_{1}}^{(0)}=\frac{at^{3}(1-\hat{x})(1-q\hat{x})(1+at^{3}q^{2}\hat{x}^{2})(1+at^{3}q^{3}\hat{x}^{2})}{q^{3}(1+at^{3}\hat{x})(1+at^{3}\hat{x}^{2})(1+at^{3}q\hat{x})(1+at^{3-1}q\hat{x}^{2})}, \hat{a}_{4_{1}}^{(3)}=1,$$\hat{a}_{4_{1}}^{(1)}=-\frac{(1-q\hat{x})(1+at^{3}q^{3}\hat{x}^{2})}{tq^{3}\hat{x}^{2}(1+at^{3}\hat{x})(1+at^{3}q\hat{x})(1+at^{3}q^{-1}\hat{x}^{2})}$$\cross(1-t(t-1)q\hat{x}+at^{3}q^{-1}(1+q^{3}+qt+q^{2}t)\hat{x}^{2}$-at4$(q+q^{2}+t+q^{3}t)\hat{x}^{3}-a^{2}(t-1)t^{6}q\hat{x}^{4}-a^{2}t^{8}q^{2}\hat{x}^{5})$,$\hat{a}_{4_{1}}^{(2)}=-\frac{(1+at^{3}q^{2}\hat{x}^{2})}{at^{2}q^{2}\hat{x}^{2}(1+at^{3}\hat{x}^{2})(1+at^{3}q\hat{x})}$$\cross(1-at(t-1)\hat{x}+at^{2}(q+q^{2}+t+q^{3}t)\hat{x}^{2}$$+a^{2}t^{4}(1+q^{3}+qt+q^{2}t)\hat{x}^{3}+a^{2}(t-1)t^{5}q^{3}\hat{x}^{4}+a^{3}t^{7}q^{3}\hat{x}^{5})$.

.

These -difference operators satisfies the following properties:$\bullet$$q$$A_{K}^{\sup er}(x, y;a, q=1, t)$

reduces to the super- -polynomial$A$$A_{K}^{\sup er}(x, y;a, t)$

all factors.

Under the specializations $t=-1$ and $a=q^{2},$colored HOMFLY polynomial and colored Jones polynomial:$A_{K}^{\sup er}(\hat{x},\hat{y};a, q, t)$$A_{K}^{\sup er}(\hat{x},\hat{y};a, q, t=-1)H^{S^{n-1}}(K;a, q)=0$

annihilates the
(44)

,

.

(45)

$A_{K}^{\sup er}(x=1, y;a, t)=y^{k}+y^{k-1}\mathcal{P}(K;a, q=1, t)$ ,(46)$A_{K}^{\sup er}(\hat{x},\hat{y};a=q^{2}, q, t=-1)J_{n}(K;q)=0$$\bullet$up to over-

Under the specialization $x=1$ , one finds thewhere is the maximum -degree of the super- -polynomial, andsuperpolynomial.
$k$

$y$

$A$

$\mathcal{P}(K;a, q, t)$is the54$H1$ :

Hierarchy of -polynomials$A$

The second property implies an existence of the hierarchical structure of the $(a, t)$ -deformationsof the A-polynomial. As a completion of this hierarchy, one also finds the refined Apolynomial [17]:$A_{K}^{ref}(x, y;t):=A_{K}^{\sup er}(x, y;a=1, t)$,$A_{K}^{ref}(\hat{x},\hat{y};t):=A_{K}^{\sup er}(\hat{x},\hat{y};a=q^{2}, t)$

.

(47)

In conclusion, the super- -polynomial reveals the integrable structure behind the colored superpolynomials, and we expect that this polynomial contains a plenty of information for the topology of the knot complement. Via string theoretical interpretation of the
A-polynomial [9, 10, 1], the invariants associated to the symplectic structure of the thecharacter variety can be explored. As the first step, we will study such invariant for thesuper- -polynomial ofknot in Appendix.
Acknowledgment
The auther would like to thank to K. Hikami, T. Ohtsuki, and Y. Terashima for discussions
and comments. The work of H. . is supported by the Grant-in-Aid for Young Scientists(B)
25800137 , and Platform for Dynamic Approaches to Living System from the
Ministry of Education, Culture, Sports, Science and Technology, Japan.
$A$

$A$

$3_{1}$

$s$

$F$

$[\#$

A

$]$

Topological recursion on the character variety

For the character variety whose defining equation is given by the A-polynomial, one
finds a class of geometric invariants associated to the symplectic structure of via string
theoretical interpretations [9, 10, 1] by utilizing the formalism of the topological recursion.
This formalism is originally developed in the study of the matrix models, and one can
$C$

$C$

55
calculate such symplectic invariants systematically. In this appendix, we will compute(2 2)-Baker-Akhierzer kernelknot.\ddagger
for the (super-) character variety of$\psi^{[2|2]}$

$|$

A.l

$3_{1}$

Topological recursion

In the analytic study of the matrix models [3, 2] and topological strings [6, 7], the freeenergy and correlator $W^{(g,h)}(p_{1}, \cdots,p_{h})$ are studied in detail on basis of the symplectic
data of the spectral curve . In [13, 16], a systematic method for such analysis has been
established, and called as the Eynard-Omntin’s topological recursion. Here we focus on
the topological recursion for the genus one curve given by the algebraic equation:$F_{g}$$C$$C$$C_{0}=\{(x, y)\in \mathbb{C}^{2}|y^{2}=M(x)^{2}S(x)\}$

,

(48)

$S(x)=(x-x_{1})(x-x_{2})(x-x_{3})(x-x_{4})$ .

(49)

We denoted a rational function $M(x)$ , and the branch points $q_{i}(i=1, \cdots, 4)$ of whichobey $(x(q_{i}), y(q_{i}))=(x_{i}, 0)$ . In the computation of the topological recursion, we mainly
use the symplectic data of the smooth model :$C$$C_{0}$$C_{0}=\{(x, y)\in \mathbb{C}^{2}|y_{0}^{2}=S(x)\}$

,

(50)

where a compact cycle which encircles aroundis denoted as , and its symplectic
dual cycle is denoted by . These cycles obeys, andFor this genus one curve, the Bergman kernel $B(p_{1},p_{2})$ is the meromorphic (1, 1)-form on
which obeys the following properties:$[q_{1}, q_{2}]$$\mathcal{B}$$\mathcal{A}$$\mathcal{A}\cap \mathcal{A}=0,$$\mathcal{B}\cap \mathcal{B}=0$$\mathcal{A}\cap \mathcal{B}=1.$$C_{0^{\otimes 2}}\ni(x(p_{1}), y_{0}(p_{1});x(p_{2}), y_{0}(p_{2}))$$B(p_{1},p_{2})^{P1}\vec{\sim}^{p_{2}}dp_{1}\otimes dp_{2}$$\overline{(p_{1}-p_{2})^{2}}+$holomorphic,$\oint_{\mathcal{A}}B(p_{1}, \cdot)=0, \oint_{\mathcal{B}}B(p_{1}, \cdot)=2\pi id\omega_{0}$$\oint_{\mathcal{A}}d\omega_{0}=1, \oint_{\mathcal{B}}d\omega_{0}=\tau$

(51)
(52)

,

,

(53)

where in (52) we denoted the integration of the Bergman kernel $B(p_{1},p_{2})$ with respect tofor an abbreviation, and is the period of the elliptic curvethese symplectic data of the spectral curve , the recursion relations for thecorrelators $W^{(g,h)}(p_{1}, \cdots,p_{h})$ are introduced as follows:$p_{2}$$C_{0}.$$\tau$$\mathbb{R}om$$C_{0}$

Definition A.1 $([16J)$ The meromorphic $(1, \cdots, 1)$ -form$\ddagger$

This point is presented in the problem session of the workshop ILDT 2013.

$W^{(g,h)}(p_{1}, \cdots,p_{h})$on$C^{\otimes h}$

sat-

56

$+$

$g$

$\otimes 2$: Diagrammatic representation of the Eynard-Orantin’s topological recursion relation (55)

isfies the following recursion relation:$W^{(0,1)}(p)$,$W^{(0,2)}(p_{1},p_{2})=B(p_{1},p_{2})$

(54)

,

${\rm Res}_{q=q:} \frac{-\frac{1}{2}\int_{\overline{q}}^{q}B(\cdot,p_{0})}{(y(q)-y(\overline{q}))}[W^{(g-1,h+2)}(q,\overline{q},p_{1}, \cdots,p_{h})$$W^{(g,h+1)}(p_{0},p_{1}, \cdots,p_{h})=\sum_{q|:Branch}$pts.
$+ \sum_{\ell=0}^{9}\sum_{J\subset H}W^{(g-\ell,|J|+1)}(q,p_{J})W^{(\ell,|H|-|J|+1)}(\overline{q},p_{H\backslash J})],$(55)

where $H=\{1, \cdots, h\},$ $J=\{i_{1}, \cdots i_{j}\}\subset H$ , and $|H|=h,$ $|J|=j$ . The points
arerelated by $(x(\overline{q}), y(\overline{q}))=(x(q), -y(q))$ . The diagrammatic representation of the recursion(55) is descmbed in Fig.2.
$q,\overline{q}\in C$

The free energy
is defined on basis of the above data of the spectral curve$g=0,1$ , the free energies
are defined by
and$F_{g}$,$\partial_{\epsilon}F_{0}=\oint_{B}dxy_{0}(x) , \epsilon:=\frac{1}{2\pi i}\ointdxy_{0}(x) , \tau=\frac{1}{2\pi i}\partial_{\epsilon}^{2}F_{0}$$F_{1}= \frac{1}{24}\log(\tau_{B}\prod_{i}y’(q_{i}))$$\tau_{B}$${\rm Res}_{q=q_{1}}(B(q,\overline{q})/dx(q))$,$\tau$$\partial\log\tau_{B}/x=$$F_{g}$$F_{g}= \frac{1}{2g-2}{\rm Res}_{q=q_{l}}\Phi(q)W^{(g,1)}(q) , \Phi(x)=l^{x}dxy_{0}(x).]$In this way, the free energies and correlators forthe equation (55) and (58).

A.2

(56)
(57)

-function which is defined via the relation$g\geq2$, the free energy. For
is given byis the Bergmann. For$F_{1}$$F_{0}$

where

$C$

$C$

(58)

are calculated recursively by solving

Non-perturbative partition function and $(n|n)$ -kernel

The above definition of the topological recursion gives the perturbative part of the freeenergy and correlator. When we take into account of the analyticity of these quantities, the57
non-perturbative effects must be treated carefully. To incorporate the non-perturbative
effects systematically, the non-perturbative partition function
is introduced in [14, 15]:$\mathcal{T}_{9s}$Definition A.2 $([14, 15J)$ Let
whose period
defined in (53):$\theta$$C_{0}$$:=\theta[_{\nu}^{\mu}](\zeta|\tau)$be the theta function of the spectral curve$\tau lS$$\theta=\sum_{n\in \mathbb{Z}}e^{i\pi(n+\mu)^{2}\tau+2i\pi(n+\mu)(\zeta+\nu)}$

where$\mu,$$\nu\in \mathbb{C}/\mathbb{Z}$

,

(59)

. The theta function satwfies the heat equation$D\theta=\nabla^{\otimes 2}\theta, \nabla=\partial/\partial\zeta, D=4\pi i\partial/\partial\tau$Using this theta function and the free energy
is defined as follows:$F_{g}$.

(60), the non-perturbative partition function$\mathcal{T}_{g_{s}}[y_{0}dx]$$\mathcal{T}_{g_{s}}[y_{0}dx] = \exp(\sum_{h\geq 0}g_{S}^{2h-2}F_{h})$$\cross\{_{r\geq h_{j}\geq 0,d_{j}\geq 0}j\}\theta$ ,where$\zeta u$

specialized to$\zeta_{g_{s}}$:$\zeta_{g_{s}} = [\frac{1}{2\pi ig_{s}}\oint_{\mathcal{B}}y_{0}(x)dx]+\tau[\frac{1}{2\pi ig_{s}}\oint_{\mathcal{A}}y_{0}(x)dx]$

which is defined modulo

(61)

,

(62)

$\mathbb{Z}.$In the case of the $N\cross N$ Hermitian 1-matrix model, this non-perturbative partition
function is given by summing all possible filling fractions [14], and multi-instanton corrections to the perturbative free energy which is defined by $1/N$ expansion of the partition
function are included. The non-perturbative corrections to the correlator is defined to begiven by the Schlesinger tmnsformation [38] of, and such quantity is named
as the Baker-Akhierzer’s $(n|n)$ -kernel(abbreviated as $(n|n)-BA$
kernel). The $(n|n)-BA$ -kernel is defined as follows:$F_{g}$$\mathcal{T}_{g_{s}}[y_{0}dx]$$\psi_{g_{S}}^{[n|n]}(p_{1},0_{1};\cdots ; p_{n}, 0_{n})$Definition A.3 $([4J)$ Let
be the Abelian differential of the third kind on the genus
one curve
which is given by the Bergman kemel as:$dS_{o,p}$$C_{0}$$dS_{o,p}(q)= \int_{0}^{p}B(\cdot, q)$

Shifting 1-form$(n|n)$ -kernel$y_{0}dx$by$y_{0}dx+ \sum_{i=1}^{n}dS_{o_{i},p_{i}}$$\psi_{g_{s}}^{[n|n]}(p_{1},0_{1};\cdots,p_{n}, 0_{n})$in.

$\mathcal{T}_{g_{s}}[y_{0}dx]$(63), one finds the Baker-Akhierzer’s:$\psi_{g_{S}}^{[n|n]}(p_{1},0_{1};\cdots;p_{n}, 0_{n})=\frac{\mathcal{T}_{g_{s}}[y_{0}dxarrow y_{0}dx+\sum_{i=1}^{n}dS_{o_{i},p_{i}}]}{\mathcal{T}_{g_{s}}[y_{0}dx]}$

.

(64)

58
can be described more explicitly in terms of the data
This definition is formal, andof . For details of the above definitions, see [4, 5].
To evaluate the above quantities explicitly, we have to take care about the dependenceof the spectral curve . Since the free energy,
on the choice of the symplectic basis
correlator, and $(n|n)-BA$ kernel are sensitive to the choice of the symplectic basis, one. Therefore, we have to specify
finds a shift of these quantities under the change ofthe choice of symplectic basis when we study the topological recursion. The shift of the
topological recursion can be treated simply as follows. When one changes the symplectic
bybasis$\psi_{g_{s}}^{[n|n]}$$C$$C_{0}$$(\mathcal{A}, \mathcal{B})$$(\mathcal{A}, \mathcal{B})$$(\mathcal{A}, \mathcal{B})$$(\mathcal{A}^{\kappa}, \mathcal{B}^{\kappa})=(\mathcal{A}-\kappa \mathcal{B}^{\kappa}, \mathcal{B}-\tau \mathcal{A})$

(65)

,

the Bergman kemel is shifted as:$B_{\kappa}(p_{1},p_{2})=B(p_{1},p_{2})+2i\pi d\omega_{0}(p_{1})\cdot\kappa\cdotd\omega_{0}(p_{2})$

(66)

,

by adopting this shifted Bergman
are also shifted toand the correlators
kernelto the topological recursion (55) [7, 5].
In [5], the asymptotic expansion of $(2|2)-BA$ kernel in $g_{s}arrow 0$ is found explicitly by
as follows:
using the -shifted correlator$W^{(g,h)}$$W_{\kappa}^{(g,h)}$$B_{\kappa}(p_{1},p_{2})$$W_{\kappa}^{(g,h)}$$\kappa$$\psi_{g_{s}}^{[2|2]}(p_{1},0_{1};p_{2},0_{2})\sim\exp[\sum_{k}g_{s}^{k}G_{k}(p_{1},0_{1};p_{2},0_{2})]$

(67)

,

$G_{1}=G_{1|\kappa}^{1,(0)}+G_{3|\kappa}^{0,(0)}+T_{2},.\kappa G_{1}^{0}+T_{1,.|\kappa}G_{2}^{0}+(T_{3,.|\kappa}-T_{3|\kappa})G_{0|\kappa}^{0,(3)}$,$c_{2}=G_{2|\kappa}^{1,(0)}+G_{4}^{0}+T_{2,.|\kappa}G_{2}^{0}\cdots$ ,where$G_{n|\kappa}^{h,(d)}$and$T_{d|\kappa}$

(68)
(69)

are given by$G_{n|\kappa}^{h,(d)}(p_{1},0_{1};p_{2},0_{2}) = \frac{1}{n!}\frac{1}{(2\pi i)^{d}d!}\int_{0_{1},02}^{p_{1},p_{2}}\cdots\int_{0_{1},02}^{p_{1},p_{2}}\oint_{\mathcal{B}_{\kappa}}\cdots\oint_{\mathcal{B}_{\kappa}}W_{\kappa}^{(n+d,h)}$

,

(70)

$\overline{ntimes}\tilde{dtimes}$.

$T_{d|\kappa} = \sum_{d=0}^{[d/2]}\frac{d!(-1)^{d’}(2i\pi)^{d’}}{2^{d}’ d!}\kappa^{\otimes d’}\otimes\frac{\nabla^{\otimes(d-2d’)}\theta}{\theta}$(71). In (69), we haveformeans the evaluation of
The notation
described only the terms which are relevent in the evaluation on the spectral curve witha reciprocal symmetry:$T_{d|\kappa}$$T_{d,.|\kappa}$$\iota_{*}B_{\kappa}=-B_{\kappa}.$$\zeta=\zeta_{g_{s}}+\int_{0_{1},0_{2}}^{p_{1},p_{2}}d\omega_{0}$

59
A.3

Quantum volume conjecture and $(2|2)-BA$ kernel

Choosing the spectral curve as the character variety
by the A-polynomial $A_{K}(x, y)$ :$C$$C_{K}$of the knot $K$ that is defined$C_{K}=\{(x, y)\in \mathbb{C}^{*}\cross \mathbb{C}^{*}|A_{K}(x, y)=0\}$

(72)

,

one can evaluate the $(2|2)-BA$ kemel as the invariant associated to the symplectic structureof the character variety. In particular for $K=4_{1},$is genus one curve, and we can$(2|2)-BA$
evaluate
kernel explicitly using the formulae (67)-(71). Furthermore, since thecharacter variety has reciprocal symmetry that is invariant under the involution :$C_{4_{1}}$$\iota$(73)$x(\iota(p))=1/x(p) , y(\iota(p))=1/y(p)$ ,we can consider the $(2|2)-BA$ kernel

invariant under the reciprocal
symmetry (73).
On the other hand, in the quantum volume conjecture [23, 20], it is proposed that thecolored Jones polynomial satisfies a -difference equation:, andthe asymptotic solution of the -difference equation can be found iteratively solving thedifferential equation:$\psi_{g_{S}}^{[2|2]}(p, 0;\iota(p), \iota(0))$$A_{K}(\hat{x},\hat{y};q)J_{n}(K_{\rangle}\cdot q)=0$

$q$

$q$

$A_{K}(\hat{x},\hat{y};q=e^{2\hslash})\mathcal{J}_{K}(x;\hslash)=0$

(74)

,

$\hat{x}J_{K}(x;\hslash)=x\mathcal{J}_{K}(x;\hslash) , \hat{y}\mathcal{J}_{K}(x;\hslash)=e^{\gamma_{1\partial_{x}}}J_{K}(x;\hslash)$$\mathcal{J}_{K}(x;\hslash)\sim\hslash^{\delta/2}\exp(\sum_{k\geq-1}\hslash^{k}j_{k}(x))$

.

,

(75)
(76)

On basis of the observation for the figure eight knot and m009, the following conjecture
is proposed in 10,$[$$5]:^{\S}$Conjecture A.l $([5J)$ Choose as
point andfor the theta function$\kappa$$0$$\mu,$$\nu\in \mathbb{C}/\mathbb{Z}$$\kappa(\tau)=-\frac{1}{2i\pi}\frac{E_{2}(\tau)}{3\pi^{2}}$. There exist a choice of the endsuch that$\theta[_{\nu}^{\mu}](\zeta|\tau)$$\mathcal{J}_{K}(x(p), 2\hslash)=(\psi_{\hslash}^{[2|2]}(p, 0;\iota(p), \iota(0)))^{1/2}$

.

(77)

At each level of the asymptotic expansion, (77) implies$2j_{k}(x(p))=G_{k}(p, 0;\iota(p), \iota(0))$

.

(78)

\S Physically this conjecture is proposed in [10]. But in [10] only the perturbative part is discussed, and the nonperturbative part is introduced as an $ad$ hoc regularization. This $ad$ hoc point is resolved in [5] by the completion of
the non-perturbative completion, and here we use the statement of the conjecture proposed in [5].

60$(2|2)-BA$ kernel for the super- $A$ -polynomial of

A.4$3_{1}$

Now we discuss the invariants associated to the super- -polynomial. The superpolynomial forknot is$A$$A$ -$3_{1}$$A_{3_{1}}^{\sup er}(x, y;a, t) = a^{2}t^{4}(x-1)x^{3}+(1+at^{3}x)y^{2}$$-a(1-t^{2}x+2t^{2}(1+at)x^{2}+at^{5}x^{3}+a^{2}t^{6}x^{4})y$and the super-character variety
one curve in this case. Furthermore,

,

(79)

is genus
has the deformed reciprocal symmetry:$C_{K}^{\sup er}=\{(x, y)\in \mathbb{C}^{*}\cross \mathbb{C}^{*}|A_{K}^{\sup er}(x, y;a, t)=0\}$$A_{3_{1}}^{\sup er}$$x( \iota(p))=-\frac{1}{at^{3}}x(p)^{-1}, y(\iota(p))=\frac{1}{t^{2}}y(p)^{-1}$

.

(80)

To discuss the topological recursion for the super-character variety as the genus one
curve, we rewrite the super- -polynomial (79) [32]:$A$$C_{3_{1}}^{\sup er}=\{(x, y)\in \mathbb{C}^{2}|y^{2}=M(X)^{2}S(x)\}$$M(x)= \frac{1}{x\sqrt{S(x)}}\log\frac{A(x)+\sqrt{S(x)}}{A(x)-\sqrt{S(x)}}$

(81)

,

(82)

,

$A(x)= \frac{1-t^{2}x+2t^{2}x^{2}+2at^{3}x^{2}+at^{5}x^{3}+a^{2}t^{6}x^{4}}{at^{3}(1+at^{3}x^{2})}$

(83)

,

$S(x)=a^{-2}t^{-6}(1-2t^{2}x+4t^{2}x^{2}+2at^{3}x^{2}+t^{4}x^{2}+2at^{5}x^{3}+a^{2}t^{6}x^{4})$

.

(84)

In terms of this coordinate, the smooth model is given by$C_{3_{1}0}^{\sup er}=\{(x, y_{0})\in \mathbb{C}^{2}|y_{0}^{2}=S(x)\}$The smooth model$C_{3_{1}0}^{\sup er}$

.

(85)

can be rewritten in the Weierstrass form:$y^{2}=4x^{3}-g_{2}x-g_{3}$

(86)

,

$g_{2}= \frac{16+16at+8t^{2}+16a^{2}t^{2}+16at^{3}+t^{4}}{12a^{4}t^{8}}$

(87)

,

$g_{3}= \frac{(4+8at+t^{2})(-16+8at-8t^{2}+8a^{2}t^{2}-16at^{3}-t^{4})}{216a^{6}t^{12}}$The value of the theta functions of$\theta_{2}^{8}+\theta_{4}^{8}+\theta_{2}^{4}\theta_{4}^{4}=3(2\varpi_{A})^{4}g_{2}/4\pi^{4},$where$2\varpi_{A}$$C_{3_{1}0}^{\sup er}$$E_{6}=-\theta_{2}^{12}-3\theta_{2}^{8}\theta_{4}^{4}/2+3\theta_{2}^{4}\theta_{4}^{8}/2+\theta_{4}^{12}=27(2\varpi_{A})^{6}g_{3}/8\pi^{6}$

with $(x, y)=(\wp(z/2\varpi_{A}|\tau), \wp’(z/2\varpi_{A}|\tau))$ . The coefficientis given by [5]$T_{2d|\kappa(\tau)}=D_{2d+1/2^{O}}\cdots\circ D_{2d-3/2}\circ D_{1/2}\theta_{i}(\tau)/\theta_{i}(\tau)$

The choice ofwith$\theta_{i}(\tau)$$\mu,$

$\nu$

(88)

are found explicitly via the relations: $E_{4}=$$:= \oint_{A}dz$$\kappa(\tau)=-\frac{1}{2i\pi}\frac{E_{2}(\tau)}{3\pi^{2}}$.

,$T_{2d|\kappa(\tau)}$$D_{k}f:=Df+ \frac{2}{3\pi^{2}}kE_{2}$

.

for

(89)

corresponds to the choice of $i=2,3,4$ for the Jacobi’s theta functions
is found for one ofknot, the appropriate choice of. For$\theta_{2}^{4}+\theta_{4}^{4}=\theta_{3}^{4}$$4_{1}$$\mu,$$\nu$

61
the solutions forof the above relations between
andsuper-character variety of , we choose the appropriate values of$\theta_{i}$$E_{a}(\tau)$$\theta_{i}(\tau)[5]$$3_{1}$$T_{2|\kappa(\tau)}/(2 \varpi_{A})^{2}=-\frac{4+8at+t^{2}}{12a^{2}t^{4}}$$T_{2|\kappa(\tau)}$. In the case ofandas:$T_{4|\kappa(\tau)}$

,

(90)

$T_{4|\kappa(\tau)}/(2 \varpi_{A})^{4}=-\frac{-16+32at-8t^{2}+32a^{2}t^{2}-16at^{3}-t^{4}}{48a^{4}t^{8}}$

.

(91)

where these are unique solution that is given by the rational function ofthese geometric data of the super-character variety, one can calculate
whi $ch$ appears in
term. Here we consider the $(2|2)-BA$ kernel
which is invariant under the deformed reciprocal symmetry (80). The terms
andare treated independently, and they are given as the Abel map and discriminant ofas follows $[6, 10]:\P$$a,$ $t.$$\mathbb{R}om$$C_{3_{1}}^{\sup er}$

$G_{n|\kappa}^{h,(d)}(p, 0;\iota(p), \iota(0))$$G_{k}$$G_{-1}$$G_{0}$$C_{K}^{\sup er}$

$G_{-1}= \int_{0,\iota(0)}^{p,\iota(p)}\frac{dx}{x}\log y(x) , G_{0}=\frac{1}{2}\log\frac{\gamma}{\sqrt{S(x)/x^{2}}}$

,

(92)

where $\gamma=\sqrt{-(3+4at+t^{2})}/2at$ . The next order $G_{1}(68)$ is found via topological recursion (55):$G_{1|\kappa(\tau)}^{1(0)}=(-16-16t^{2}-24at^{3}-11t^{4}-18at^{5}-2t^{6}-8a^{2}t^{6}-4at^{7}-2a^{2}t^{8}+(48t^{2}+24t^{4}$$+32at^{5}+27t^{6}-16a^{2}t^{6}+24at^{7}+6t^{8}-16a^{2}t^{8}+12at^{9}-16a^{3}t^{9}+6a^{2}t^{10})x+(-128t^{2}$$-80at^{3}-208t^{4}-336at^{5}-80t^{6}-152a^{2}t^{6}-207at^{7}-33t^{8}-178a^{2}t^{8}-20at^{9}-72a^{3}t^{9}$$-6t^{10}+40a^{2}t^{10}-12at^{11}+30a^{3}t^{11}-6a^{2}t^{12})x^{2}+(-224at^{5}-112t^{6}-128a^{2}t^{6}-560at^{7}$$-24t^{8}-768a^{2}t^{8}-302at^{9}-288a^{3}t^{9}+9t^{10}-664a^{2}t^{10}-16at^{11}-544a^{3}t^{11}+2t^{12}$$-52a^{2}t^{12}-160a^{4}t^{12}+4at^{13}-28a^{3}t^{13}+2a^{2}t^{14})x^{3}+(128at^{5}+80a^{2}t^{6}+208at^{7}$$+336a^{2}t^{8}+80at^{9}+152a^{3}t^{9}+207a^{2}t^{10}+33at^{11}+178a^{3}t^{11}+20a^{2}t^{12}+72a^{4}t^{12}+6at^{13}$$-40a^{3}t^{13}+12a^{2}t^{14}-30a^{4}t^{14}+6a^{3}t^{15})x^{4}+(48a^{2}t^{8}+24a^{2}t^{10}+32a^{3}t^{11}+27a^{2}t^{12}$$-16a^{4}t^{12}+24a^{3}t^{13}+6a^{2}t^{14}-16a^{4}t^{14}+12a^{3}t^{15}-16a^{5}t^{15}+6a^{4}t^{16})x^{5}+(16a^{3}t^{9}$$+16a^{3}t^{11}+24a^{4}t^{12}+11a^{3}t^{13}+18a^{4}t^{14}+2a^{3}t^{15}+8a^{5}t^{15}+4a^{4}t^{16}+2a^{5}t^{17})x^{6})$$/(12(1+at)(1+t^{2}+at^{3})(16+8t^{2}+16at^{3}+t^{4})$$\cross(1-2t^{2}x+(4t^{2}+2at^{3}+t^{4})x^{2}+2at^{5}x^{3}+a^{2}t^{6}x^{4})^{3/2})$

,

(93)

$G_{3|\kappa(\mathcal{T})}^{0(0)}=(1+(3t^{2}+6at^{3})x+(-12t^{2}-15at^{3}-9t^{4}-12at^{5})x^{2}+(4t^{4}-10at^{5}+5t^{6}$$-20a^{2}t^{6}+6at^{7})x^{3}+(12at^{5}+15a^{2}t^{6}+9at^{7}+12a^{2}t^{8})x^{4}+(3a^{2}t^{8}+6a^{3}t^{9})x^{5}-a^{3}t^{9}x^{6})$$/(12(1+at)(1-2t^{2}x+(4t^{2}+2at^{3}+t^{4})x^{2}+2at^{5}x^{3}+a^{2}t^{6}x^{4})^{3/2})$ ,(94)

In[10] the Bergman kernel is calculated in the variable $w=(x+x^{-1})/2$ that is invaraint under the involution . In
this case, we use the variable $w= \frac{1}{2}(x-1/(at^{3}x))$ to evaluateterm.
$\eta$$\iota$$G_{0}$

62$G_{2|\kappa(\tau)}^{0,(1)}=0,$$c_{01_{\kappa(\tau)}^{(3)}}^{0}=0$

(95)

,

$G_{1|\kappa(\tau)}^{0,(2)}\cdot(2\varpi_{A})^{2}=(a^{2}t^{4}(-4-3t^{2}-4at^{3}+(4t^{2}+5t^{4}+6at^{5})x+(4at^{3}+3at^{5}+4a^{2}t^{6})x^{2})$$/((1+at)(16+8t^{2}+16at^{3}+t^{4})(1-2t^{2}x+(4t^{2}+2at^{3}+t^{4})x^{2}+2at^{5}x^{3}+a^{2}t^{6}x^{4})^{1/2})$

.

(96)

Applying these$G_{n|\kappa(\tau)}^{h,(d)}$and (90) to (68), one finds the$G_{1}$

term:$G_{1}(x)=(1+2at+2t^{2}+4at^{3}+2a^{2}t^{4}+(3t^{2}-2t^{4}-4at^{5}-2a^{2}t^{6})x$$+(-16t^{2}-11at^{3}-21t^{4}+2a^{2}t^{4}-28at^{5}-2t^{6}-8a^{2}t^{6}-4at^{7}+2a^{3}t^{7}-2a^{2}t^{8})x^{2}$$+(-30at^{5}-7t^{6}-24a^{2}t^{6}-48at^{7}+2t^{8}-68a^{2}t^{8}+4at^{9}-28a^{3}t^{9}+2a^{2}t^{10})x^{3}$$+(16at^{5}+11a^{2}t^{6}+21at^{7}-2a^{3}t^{7}+28a^{2}t^{8}+2at^{9}+8a^{3}t^{9}+4a^{2}t^{10}-2a^{4}t^{10}+2a^{3}t^{11})x^{4}$$+(3a^{2}t^{8}-2a^{2}t^{10}-4a^{3}t^{11}-2a^{4}t^{12})x^{5}+(-a^{3}t^{9}-2a^{4}t^{10}-2a^{3}t^{11}-4a^{4}t^{12}-2a^{5}t^{13})x^{6})$$/(48(1+at)(1+t^{2}+at^{3})(1-2t^{2}x+4t^{2}x^{2}+2at^{3}x^{2}+t^{4}x^{2}+2at^{5}x^{3}+a^{2}t^{6}x^{4})^{3/2})$$= \frac{-1}{48(at+1)}+\frac{1-5x^{2}+x^{4}}{16(1-x+x^{2})^{2}}+\mathcal{O}(a-1,t+1)$

.

(97)

The above result and the solution of the -difference equation (21) with fixed variablebehave differently even in the augmentation limit $t=-1$ . This discrepancy would be
related with the subtlety of the -dependence of and in the asymptotic expansion. In
terms of the mirror symmetry proposed in [1], the open string amplitude may be identified
with $(2|2)-BA$ kernel after the pull back by the mirror map. This point will be studied in
the future works.
$a$

$q$

$q$

A.4.1

$G_{2}$$a$$x$

term for the augmentation variety

is also computed via the above data. Here we note the result for $t=-1$
The
specialization:$G_{2}$$G_{4|\kappa(\tau)}^{0,(0)}=-(a-(8+10a-8a^{2})x+(64-101a+58a^{2})x^{2}+(-16-76a+82a^{2}-40a^{3})x^{3}$$+(-160+375a-60a^{2}-65a^{3})x^{4}+(-104+810a-1524a^{2}+644a^{3}+48a^{4})x^{5}+(224$$-1043a+1154a^{2}+306a^{3}-500a^{4})x^{6}+(-104a+810a^{2}-1524a^{3}+644a^{4}+48a^{5})x^{7}$$+(-160a^{2}+375a^{3}-60a^{4}-65a^{5})x^{8}+(-16a^{3}-76a^{4}+82a^{5}-40a^{6})x^{9}+(64a^{4}-101a^{5}$$+58a^{6})x^{10}+(-8a^{5}+10a^{6}-8a^{7})x^{11}+a^{7}x^{12})$$/(128(-1+a)^{2}(1-2x+5x^{2}-2ax^{2}-2ax^{3}+a^{2}x^{4})^{3})$ ,

(98)

63

$G_{2|\kappa(\tau)}^{0,(2)}\cdot(2\varpi_{A})^{2}=a^{2}(125+57a-192a^{2}+64a^{3}+(-100-1248a+1644a^{2}-512a^{3})x$$+(2650-2382a+272a^{3})x^{2}+(3300-20508a+29220a^{2}-15948a^{3}+3072a^{4})x^{3}$$+(-14775+57821a-72418a^{2}+37086a^{3}-6560a^{4}-128a^{5})x^{4}+(3300a-20508a^{2}$$+29220a^{3}-15948a^{4}+3072a^{5})x^{5}+(2650a^{2}-2382a^{3}+272a^{5})x^{6}+(-100a^{3}-1248a^{4}$$+1644a^{5}-512a^{6})x^{7}+(125a^{4}+57a^{5}-192a^{6}+64a^{7})x^{8})$(99)$/(96(-1+a)^{2}(-25+16a)^{2}(1-2x+5x^{2}-2ax^{2}-2ax^{3}+a^{2}x^{4})^{2})$ ,$G_{0|\kappa(\tau)}^{0,(4)} \cdot(2\varpi_{A})^{4}=-\frac{(a^{4}(-1120+1983a-1200a^{2}+256a^{3})}{8(-1+a)^{2}(-25+16a)^{3}}$

(100)

,

$G_{2|\kappa(\tau)}^{1,(0)}=(3850+1186a-11437a^{2}+8008a^{3}-16a^{4}-1280a^{5}+256a^{6}+(-49300$$+103052a-122582a^{2}+128516a^{3}-92528a^{4}+34304a^{5}-4864a^{6})x+(128150$$-72366a-223255a^{2}+258862a^{3}-75964a^{4}-9728a^{5}+6720a^{6}-512a^{7})x^{2}$$+(-1218200+4566948a-8455640a^{2}+9515930a^{3}-6630236a^{4}+2759120a^{5}$$-625920a^{6}+59648a^{7})x^{3}+(3323350-12843994a+22690091a^{2}-22999486a^{3}$$+13992613a^{4}-4981960a^{5}+941328a^{6}-70656a^{7}-256a^{8})x^{4}+(-628500-3175260a$$+15266330a^{2}-24549808a^{3}+19927324a^{4}-8353480a^{5}+1355680a^{6}+141056a^{7}$$-54784a^{8})x^{5}+(-333750+4605350a-7420845a^{2}-4329006a^{3}+22532762a^{4}$$-25674524a^{5}+14219576a^{6}-3962368a^{7}+441728a^{8}+1024a^{9})x^{6}+(-628500a$$-3175260a^{2}+15266330a^{3}-24549808a^{4}+19927324a^{5}-8353480a^{6}+1355680a^{7}$$+141056a^{8}-54784a^{9})x^{7}+(3323350a^{2}-12843994a^{3}+22690091a^{4}-22999486a^{5}$$+13992613a^{6}-4981960a^{7}+941328a^{8}-70656a^{9}-256a^{10})x^{8}+(-1218200a^{3}$$+4566948a^{4}-8455640a^{5}+9515930a^{6}-6630236a^{7}+2759120a^{8}-625920a^{9}$$+59648a^{10})x^{9}+(128150a^{4}-72366a^{5}-223255a^{6}+258862a^{7}-75964a^{8}-9728a^{9}$$+6720a^{10}-512a^{11})x^{10}+(-49300a^{5}+103052a^{6}-122582a^{7}+128516a^{8}-92528a^{9}$$+34304a^{10}-4864a^{11})x^{11}+(3850a^{6}+1186a^{7}-11437a^{8}+8008a^{9}-16a^{10}-1280a^{11}$$+256a^{12})x^{12})$$/(1152(-2+a)^{2}(-1+a)^{2}(-25+16a)^{2}(1-2x+5x^{2}-2ax^{2}-2ax^{3}+a^{2}x^{4})^{3}))$(101)$G_{0|\kappa(\tau)}^{1,(2)}\cdot(2\varpi_{A})^{2}$$=- \frac{a^{2}(-3575-2759a+23055a^{2}-29532a^{3}+16568a^{4}-4512a^{5}+512a^{6})}{48(-2+a)^{2}(-1+a)^{2}(-25+16a)^{3}}$

.

(102)

64
Applying the above results to (69), one finds the$G_{2}$

term:$G_{2}=(2+6a-6a^{2}+a^{3}+(-4-60a+60a^{2}-14a^{3})x+(-194+910a-1038a^{2}$$+455a^{3}-70a^{4})x^{2}+(-696+1988a-3396a^{2}+3000a^{3}-1238a^{4}+192a^{5})x^{3}$$+(2846x^{4}-8254ax^{4}+10940a^{2}-7367a^{3}+2410a^{4}-305a^{5})x^{4}+(-1156-196a$$+3260a^{2}-2942a^{3}+172a^{4}+676a^{5}-192a^{6})x^{5}+(610-790a+6442a^{2}-15635a^{3}$$+15598a^{4}-6934a^{5}+1132a^{6})x^{6}+(-1156a-196a^{2}+3260a^{3}-2942a^{4}+172a^{5}$$+676a^{6}-192a^{7})x^{7}+(2846a^{2}-8254a^{3}+10940a^{4}-7367a^{5}+2410a^{6}-305a^{7})x^{8}$$+(-696a^{3}+1988a^{4}-3396a^{5}+3000a^{6}-1238a^{7}+192a^{8})x^{9}+(-194a^{4}+910a^{5}$$-1038a^{6}+455a^{7}-70a^{8})x^{10}+(-4a^{5}-60a^{6}+60a^{7}-14a^{8})x^{11}+(2a^{6}+6a^{7}$$-6a^{8}+a^{9})x^{12})$$/(48(-2+a)^{2}(-1+a)^{2}(1-2x+5x^{2}-2ax^{2}-2ax^{3}+a^{2}x^{4})^{3})$ .

(103)

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67
1Interdisciplinary Center for Mathematical Sciences,
Graduate School of Mathematical Sciences,
University of Tokyo,
3-8-1 Komaba, Tokyo 153-8914, Japan
2
Institute for Biology and Mathematics of Dynamical Cell Processes (iBMath),
University of Tokyo,
3-8-1 Komaba, Tokyo 153-8914, Japan-mail address: fuj i@ms. -tokyo. ac. jp

$E$

$u$

$\ovalbox{\tt\small REJECT}\overline{F\backslash }\star\not\cong\cdot \mathscr{X}fg\ovalbox{\tt\small REJECT}_{-}^{\backslash }\backslash \}\not\cong Wf_{J^{t}\iota^{i}}^{2}\ovalbox{\tt\small REJECT}^{\backslash }\}\backslash$ma EZ