The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
aa r X i v : . [ h e p - t h ] O c t ITFA-2010-05
The Volume Conjecture,Perturbative Knot Invariants,andRecursion Relations for Topological Strings
Robbert Dijkgraaf ∗ , Hiroyuki Fuji † and Masahide Manabe ‡ Institute for Theoretical Physics & KdV Institute for MathematicsUniversity of Amsterdam, Spui 21, 1012 WX Amsterdam, The Netherlands Department of Physics, Nagoya University, Nagoya 464-8602, Japan Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan
Abstract
We study the relation between perturbative knot invariants and the free energies de-fined by topological string theory on the character variety of the knot. Such a corre-spondence between SL (2; C ) Chern-Simons gauge theory and the topological open stringtheory was proposed earlier on the basis of the volume conjecture and AJ conjecture. Inthis paper we discuss this correspondence beyond the subleading order in the perturbativeexpansion on both sides. In the computation of the perturbative invariants for the hy-perbolic 3-manifold, we adopt the state integral model for the hyperbolic knots, and thefactorized AJ conjecture for the torus knots. On the other hand, we iteratively computethe free energies on the character variety using the Eynard-Orantin topological recursionrelation. We check the correspondence for the figure eight knot complement and the oncepunctured torus bundle over S with the holonomy L R up to the fourth order. For thetorus knots, we find trivial the recursion relations on both sides. ∗ [email protected] † [email protected] ‡ [email protected] Introduction
Three-dimensional Chern-Simons gauge theory is one of the most widely studied topolog-ical quantum field theories. It has found many applications in physics and mathematics.In the celebrated paper by Witten [1] a relation between the Chern-Simons gauge theoryand knot invariants was discovered, and it was shown that the expectation value of theWilson loop operator along the knot on 3-sphere S and the colored Jones polynomial areequivalent.Another remarkable aspect of the Chern-Simons gauge theory is its relation withtheories of three-dimensional quantum gravity [2]. When gravity in three dimensions isreformulated in the first order formalism, it also yields a Chern-Simons gauge theory. Inparticular, in this approach SL (2; C ) Chern-Simons gauge theory becomes equivalent tothe Euclidean signature quantum gravity with a negative cosmological constant. In theclassical limit this corresponds to the study of the hyperbolic structures. Connecting thesetwo aspects of the Chern-Simons gauge theory, led to the volume conjecture as originallyproposed by Kashaev [3].The volume conjecture concerns the asymptotic behavior of the colored Jones polyno-mial [4]. Let J n ( K ; q ) be the n -colored Jones polynomial for a hyperbolic knot K . Theclaim of the volume conjecture is [3, 4]:1 π lim n →∞ | log J n ( K ; q = e πi/n ) | n = Vol( S \ K ) , (1.1)where Vol( S \ K ) is the hyperbolic volume of the knot complement. The volume conjec-ture has been extended to the complexified version [4], and further generalized to knotcomplements with the deformed hyperbolic structure [5]. The volume conjecture beyondthe leading order was discussed firstly in [6]. The subleading term in the asymptoticexpansion of the colored Jones polynomial coincides with the Reidemeister torsion [7].In previous work [8], two of the authors proposed a correspondence between SL (2; C )Chern-Simons gauge theory and the topological open string theory. This correspondencewas suggested by a similar set-up in the two problems. Let us briefly review this argument.If M is the three-manifold obtained by removing a tubular neighbourhood of the knot K , then any quantum field theory on M will produce a quantum state Z ( M ) in theHilbert space associated to the boundary ∂M of M . In the case of a knot complement,the boundary has the topology ∂M ∼ = T . Semi-classically, the state Z ( M ) is described bya Lagrangian sub-manifold in the phase space associated to the boundary. For SL (2; C )Chern-Simons gauge theory the classical phase space can be identified with the space ofgauge equivalence classes of the flat SL (2; C ) connections on T . The Lagrangian corre-sponding to M is the character variety C of the knot K , defined as the set of connections1n T that extend as flat connections over M [9]. The character variety is an algebraiccurve in (a quotient of) C ∗ × C ∗ equipped with its canonical symplectic structure. We canpick a local coordinate u on the curve C , which can be identified with the free monodromyaround the meridian of the knot. The full CS partition function or knot invariant nowcorresponds to the full quantum wave function Z ( M ; u ).This set-up of an algebraic curve C ⊂ C ∗ × C ∗ appears also in the topological stringtheory. In this case we considered the toric Calabi-Yau 3-fold X which can be regardedas a fibration over the character variety of the knot. In this case the Riemann surface isusually called the spectral curve. If we add a topological D-brane in this CY variety, theopen string partition function will also be a wave function Z ( X ; u ). So, in both the Chern-Simons gauge theory and the topological string theory we quantize the character variety.The conjecture that we study further in this paper is that, with a suitable identification,these quantizations are equivalent, i.e. , we have Z ( M ; u ) = Z ( X ; u ) . (1.2)This correspondence is summarized in more detail in the following table. The variousnotations that we use here, will become clear in the subsequent.
3D Chern-Simons Topological Open String u = 2 πi (cid:0) nk − (cid:1) : Meridian holonomy u : Area of holomorphic disk q = e πi/k q = e g s Vol+ i CS Disk free energy F (0 , Reidemeister torsion Annulus free energy F (0 , AJ conjecture Quantum Riemann Surface A K ( ˆ m, ˆ ℓ ; q ) J n ( K ; q ) = 0 ˆ H ( e − u − g s / , e g s ∂ u ; q ) Z open ( u ; q ) = 0ˆ m ˆ ℓ = q / ˆ ℓ ˆ m e − u − g s / e g s ∂ u = q / e g s ∂ u e − u − g s / This correspondence is checked explicitly for two examples, the figure eight knot com-plement and the once punctured torus bundle over S with the holonomy L R , whichis isomorphic to the SnapPea census manifold m
009 [10], which is the complement of aknot in a three-manifold of different topology than the three-sphere [11, 12, 13]. Up tosubleading order, one can find coincidence for these examples.On the other hand, in [14, 15] it is conjectured that the free energies of the topological(A-type) open string theory on toric Calabi-Yau 3-folds with toric branes are iterativelyobtained by the Eynard-Orantin topological recursion relation [16]. The recursion relationis applicable for any complex plane curve, and in the topological string theory, via themirror symmetry, the open string moduli are described by the mirror curve which is acomplex plane curve in C ∗ × C ∗ . These recursions can be derived as the Schwinger-Dysonequations in two dimensional Kodaira-Spencer theory, the theory of a chiral boson on the2irror curve [17]. The techniques of the computation are developed in [18, 19, 20]. In thispaper, we discuss our correspondence to the higher orders beyond the subleading orderin the topological expansion of the free energies of the topological open string theory.To this end we define the BKMP’s free energies according to remodeling the B-model[14, 15], and compute them for the character variety of the hyperbolic manifold up to thefifth order in the recursions, in the case of the above two examples.On the Chern-Simons gauge theory side, the higher order perturbative invariants arecomputed in [21, 22]. For the figure eight knot complement we can compute the pertur-bative invariant from the colored Jones polynomial. But for the once punctured torusbundle the complete form of the colored Jones polynomial is not known. To analyze suchmanifolds, we adopt the state integral model which is constructed in [23, 24, 25, 21]. Thepartition function of the state integral model for a simplicially decomposed hyperbolic3-manifold gives topological invariants like the Ponzano-Regge [26] and the Turaev-Viromodels [27]. In this paper, we compute the perturbative invariants from the state integralmodel, and compare them with the free energy on the character variety.As a result of these computations, we find some discrepancies between the perturbativeinvariants and the BKMP’s free energy of the topological string in the higher order. Thesediscrepancies may come from the choice of the integration path in the computation of thefree energy in the B-model or O ( g s ) modification of the Calabi-Yau geometry [28, 29,30]. To remedy this point, we consider some regularization for the constant G whichappears in the Bergman kernel, because the constant G changes under the monodromytransformation of the genus one character variety. Although the regularization would bead-hoc, we find the regularization rules for G n terms in the recursions up to the fourthorder. After the regularization, we recover the perturbative invariants of the state integralmodel for the above two examples non-trivially.For the case of torus knots, the colored Jones polynomial is well-studied. We canextract the perturbative invariants adopting the q -difference equation which is called theAJ conjecture [31, 32, 33, 34, 35]. Compared to the state integral model, there exist twobranches which correspond to abelian and non-abelian representations of the P SL (2; C )holonomy along the meridian for the colored Jones polynomial. In this paper we willcompare the perturbative invariants for the non-abelian branch and the BKMP’s freeenergy on the character variety. In this class of knots we find that the perturbativeinvariants and BKMP’s free energies are trivial as the u -dependent functions, and we cancheck our correspondence to all orders in the perturbative expansion.The organization of this paper is as follows. In section 2, after a short summaryof the state integral model, we show the explicit computation for the figure eight knotcomplement and the once punctured torus bundle over S with the holonomy L R . The3omputation of the figure eight knot complement was already given in [21], and the secondexample is novel. In section 3, we turn to the computation of the BKMP’s free energieson the character variety. In this section, we firstly derive the general solution of thetopological recursion relations for the two branched plane curve with genus one up tothe fourth order. And then we apply the formula to the character varieties for the figureeight knot complement and the once punctured torus bundle over S with the holonomy L R . The correspondence for the torus knots is discussed in section 4. In appendix A, wediscuss the AJ conjecture. We explicitly see the factorization of the q -difference equationfor the figure eight knot, and summarize the computation of the perturbative invariantsfor some torus knots in the abelian branch. In appendix B, we summarize the details ofthe derivations of the general formula for the fourth order terms W (0 , and W (1 , . Inappendix C, we show the computation on the annulus free energy F ( p ). In appendix D,the computational result of the fifth order free energy F ( p ) is summarized. The volume conjecture describes the asymptotic behavior of the colored Jones polynomial.To evaluate the saddle point value perturbatively, the information of the cyclotomic ex-pansion of the colored Jones polynomial is necessary because we use a q -difference equationin the analysis. In general it is not an easy task to obtain such an expansion. In particularfor the once punctured torus bundle over S , the original colored Jones polynomial cannot be found, although the simplicial decomposition is realized explicitly.The partition function of the state integral model [23, 24, 25] on the hyperbolic 3-manifold gives the topological invariant. The asymptotic expansion of the partition func-tion is computable only from the information of the simplicial decomposition via idealtetrahedra. For the figure eight knot, the asymptotic expansions of the partition functionof the state integral model and the colored Jones polynomial are computed thoroughlyin [21], and both of them give the same expansions. In this section, we evaluate theasymptotic expansion of the state integral model based on the method which is developedin [21]. Here we briefly summarize about the state integral model. For a hyperbolic knot com-plement, the simplicial decomposition with the ideal tetrahedra can be performed [36].There are two kinds of ideal tetrahedra with an orientation ε = ±
1. For each face of thetetrahedron, a vector space V or its dual V ∗ is assigned corresponding to the orienta-tion. The vector space V is the Hilbert space of the Heisenberg algebra with continuum4igenvalue for the momentum operator:[ˆ q, ˆ p ] = 2 ~ , (2.1)ˆ p | p i = p | p i , | p i ∈ V. (2.2)Thus a tensor product of the vector spaces V ⊗ V ⊗ V ∗ ⊗ V ∗ is assigned for an idealtetrahedron.As a weight of the state integral model, one can choose the matrix element of theoperator S ∈ Hom C ( V ⊗ V, V ⊗ V ). To acquire topological invariance for the partitionfunction, the operator should satisfy the pentagon relation on V ⊗ V ⊗ V : S , S , = S , S , S , , (2.3)where S i,j acts on V i ⊗ V j . In state integral model [23, 25], the operator S which satisfythe above pentagon relation on the infinite dimensional momentum space | p i is definedon V ⊗ V as: S , = e ˆ q ˆ p / ~ Φ ~ (ˆ p + ˆ q − ˆ p ) , (2.4)where Φ ~ ( p ) is the quantum dilogarithm function. For i ~ ∈ R and | Im p | < π , the functionΦ ~ ( p ) is given by the Faddeev’s integral formula [37]:Φ ~ ( p ) = exp (cid:20) Z R + i dxx e − ipx sinh( i ~ x ) sinh( πx ) (cid:21) . (2.5)Because this Faddeev integral fulfills the pentagon relation:Φ ~ (ˆ p )Φ ~ (ˆ q ) = Φ ~ (ˆ q )Φ ~ (ˆ p + ˆ q )Φ ~ (ˆ p ) , (2.6)the pentagon relation (2.3) for S operators is implied.The S -matrix elements for the ideal tetrahedra of ε = ± h p ( − )1 , p ( − )2 | S | p (+)1 , p (+)2 i = δ (cid:0) p ( − )1 + p ( − )2 − p (+)1 (cid:1)p π ~ /i Φ ~ ( p (+)2 − p ( − )2 + iπ + ~ ) × e ~ h p ( − )1 ( p (+)2 − p ( − )2 )+ iπ ~ − π − ~ i , (2.7) h p ( − )1 , p ( − )2 | S − | p (+)1 , p (+)2 i = δ (cid:0) p ( − )1 − p (+)1 − p (+)2 (cid:1)p π ~ /i ~ ( p ( − )2 − p (+)2 − iπ − ~ ) × e ~ h − p (+)1 ( p ( − )2 − p (+)2 ) − iπ ~ + π − ~ i . (2.8)The partition function for the state integral model on a 3-manifold M with the com-plete hyperbolic structure is defined by Z ~ ( M ; ~ ,
0) = √ Z d p δ C ( p ) δ G ( p ) N Y i =1 h p ( − )2 i − , p ( − )2 i | S ε i | p (+)2 i − , p (+)2 i i , (2.9)5 ( ) p ( ) p ( ) p ( ) ε=+ ε=− p ( ) p ( ) p ( ) p ( ) Figure 1: Ideal tetrahedra of different orientationwhere p is a set of ( p ( ε i )2 i − , p ( ε i )2 i ). The delta functions δ G ( p ) := Q δ (cid:0) p ( − ) j − p (+) k (cid:1) and δ C ( p ) imply the gluing condition for the faces and the complete gluing condition forthe triangles on the boundary ∂M ≃ T respectively. The asymptotic behavior of thisinvariant is studied for various knot complements [23, 25].The partition function for the 3-manifold with the deformed complete structure is alsoconsidered [21]. The deformation of the completeness condition changes one of the deltafunctions in δ C ( p ) with δ ( P p i = 2 u ). In this delta function, the sum is taken for themomenta which appear in the shape parameters z i = e p (+)2 i − p ( − )2 i − of the ideal tetrahedraalong the meridian of the boundary torus [24]. As a result of this deformation, one findsa holonomy representation along the meridian µ : ρ ( µ ) = (cid:18) m ∗ m − (cid:19) , m = e u . (2.10)Under this deformation, the partition function yields Z ~ ( M ; ~ , u ) = √ e − u Z d p δ C ( p ; u ) δ G ( p ) N Y i =1 h p ( − )2 i − , p ( − )2 i | S ε i | p (+)2 i − , p (+)2 i i = √ Z N − Y j =1 dp j √ π ~ N Y i =1 Φ ~ ( g i ( p , u ) + ε i ( iπ + ~ )) ε i e ~ f ( p , u, ~ ) − u , (2.11)where g i ( p , u ) is a linear function of p , and f ( p , u, ~ ) is a quadratic polynomial.Although the contour of the integrations of the partition function is not defined ex-plicitly, the WKB expansion around the saddle point is computable [21]. In the limit ~ →
0, the partition function (2.11) is approximated by Z ~ ( M ; ~ , u ) ∼ Z d p e ~ V ( p ,u ) , ~ → , (2.12) V ( p , u ) = 12 N X i =1 ε i Li ( − exp( g i ( p , u ) + iπε i )) + 12 f ( p , u, ~ = 0) . (2.13)6he saddle point conditions ∂V ( p , u ) ∂p j = 0 (2.14)for all j specify the saddle point p . There may exist several saddle points for a hyperbolic3-manifold M , and the value of the partition function may differ for each branch. Inthe following, we discuss only the geometric branch. For the general saddle point values,the face angles of the ideal tetrahedra which is determined by a shape parameters maybecome non-geometric [36, 38]. In the geometric branch, all of the ideal tetrahedra in thesimplicial decomposition are geometric and completely glued. The saddle point value ofthe potential V ( p , u ) will satisfy V ( p , u = 0) = i Vol( M ) − CS( M ). This property is thesame as the volume conjecture [3, 4], and it is expected that the perturbative invariantsfor the state integral model will coincide with those of colored Jones polynomial.In [25] it is proposed that the potential function V ( p , u ) is identified with the Neumann-Zagier potential [39]. The Neumann-Zagier potential satisfies a relation: ∂∂u V ( p , u ) = v, (2.15)where l := e v is an eigenvalue of the holonomy representation ρ ( ν ) along the longitude ν of the boundary ∂M . This condition imposes a non-trivial constraint on ( l, m ) ∈ C ∗ × C ∗ .By the saddle point equation (2.14) one can eliminate p and find an algebraic equation : A K ( l, m ) = 0 . (2.16)The polynomial A K ( l, m ) coincides with the A-polynomial which is reciprocal A K ( l, m ) = l a m b A K ( l − , m − ). Then the saddle point value of the potential V ( p , u ) will satisfy thegeneralized volume conjecture V ( p , u ) = i Vol(
M, u ) − CS(
M, u ) [5].The higher order terms in the expansion around the saddle point is evaluated by thefollowing expansion of the quantum dilogarithm function:Φ ~ ( p + p ) = exp (cid:20) ∞ X n =0 B n (cid:16) p ~ + 12 (cid:17) Li − n ( − e p ) (2 ~ ) n − n ! (cid:21) , (2.17)where the Bernoulli polynomial B n ( x ) = P nk =0 n C k B k x n − k satisfies B ′ n ( x ) = nB n − ( x ).Plugging this expansion into (2.11), one can expand the partition function Z ~ ( M ; ~ , u )around the saddle point p as: Z ~ ( M ; ~ , u ) = exp (cid:20) ~ S ( u ) + ∞ X n =0 S n +1 ( u ) ~ n (cid:21) , S ( u ) = V ( p , u ) . (2.18) There also exists the conjugate branch which satisfies v (geom) = − v (conj) . In particular for the fullyampherical knot S (geom) n +1 = ( − n +1 S (conj) n +1 is also satisfied. In this algebraic equation, a factor l −
7n the following, we will compute the higher order terms in the saddle point approximationfor the figure eight knot complement [21] and the once puncture torus bundle over S withthe holonomy L R . As the first example we summarize the computation for the figure eight knot complement[25, 21]. The figure eight knot complement can be decomposed into two ideal tetrahedrawith the different orientations [36]. The partition function for the state integral modelyields Z ~ ( S \ ; ~ , u ) = √ e − u Z d p δ C ( p , u ) h p , p | S | p , p ih p , p | S − | p , p i . (2.19)The shape parameters z = e p − p and z = e p − p satisfies the meridian condition z z − = e − u , and the delta function δ C ( p ; u ) has the support on p − p − ( p − p ) = − u. (2.20)Evaluating some of the integrals in (2.19) one obtains Z ~ ( S \ ; ~ , u ) = 1 √ π ~ Z C dp Φ ~ ( p + iπ + ~ )Φ ~ ( − p − u − iπ − ~ ) e − ~ u ( u + p ) − u . (2.21)Since the figure eight knot is fully amphichiral, the term iπ + ~ in the argument of thequantum dilogarithm can be removed by shifting p → p − u − iπ − ~ . This shift of thevariable changes the integration path C . Although such shift gives rise to the correctionsof order e − const / ~ to Z ~ , the higher order terms S n ( n = 1 , . . . ) are not affected.Under the shift of the variable p → p − u − iπ − ~ , the partition function of the stateintegral model simplifies Z ( S \ ; ~ , u ) = 1 √ π ~ e πiu ~ + u Z C dp Φ ~ ( p − u )Φ ~ ( − p − u )= 1 √ π ~ e πiu ~ + u Z C dp e Υ( ~ ,p,u ) , (2.22)where one can compute the expansion of the function Υ( ~ , p, u ) adopting (2.17) as:Υ( ~ , p + p, u ) = ∞ X j =0 ∞ X k = − Υ j,k ( p, u ) p j ~ k , (2.23)Υ j,k = B k +1 (1 / k ( k + 1)! j ! (cid:2) Li − j − k ( − e p − u ) − ( − j Li − j − k ( − e − p − u ) (cid:3) . (2.24) This computation is already shown in [21].
8n the following, we will evaluate Z ( S \ ; ~ , u ) on the geometric branch. In this branch,the saddle point value of p is p = p (geom) ( u ) = log (cid:20) − m − m − m ∆( m )2 m (cid:21) , (2.25)∆( m ) = √ m − − m − − − m + m . (2.26)The perturbative invariants S n ( u ) are computed systematically by evaluating theGaussian integrals, Z ~ ( S \ ; ~ , u ) = e u + ~ V ( u ) √ ~ Z C dp e − b ( u )2 ~ p exp (cid:20) ~ ∞ X j =3 Υ j, − p j + ∞ X j =0 ∞ X k =1 Υ j,k ~ k p j (cid:21) . (2.27)In the geometric branch, the first two terms yield S = V ( u ) = 12 h Li ( − e p (geom) − u ) − Li ( − e − p (geom) − u ) − p (geom) u + 4 πiu i , (2.28) S = −
12 log b ( u ) m = −
12 log[ i ∆( m ) / , (2.29)The result S ( u ) is consistent with the Reidemeister torsion of S \ [7, 6]. The A-polynomial is computed from the equations (2.14) and (2.15) with S ( u ) = V ( u ), A ( l, m ) = − m + l (1 − m − m − m + m ) − l m . (2.30)The higher order terms are computed in the same way: S = 152 b Υ , − + 3 b Υ , − + Υ , , (2.31) S = 34658 b Υ , − + 9452 b Υ , − Υ , − + 1052 b (cid:0) , − Υ , − + Υ , − (cid:1) + 152 b (cid:0) , − + Υ , Υ , − (cid:1) + 3 b (Υ , Υ , − + Υ , Υ , − )+ 1 b Υ , + Υ , + 12 Υ , − S , (2.32) S = 124 b (cid:0) , − Υ , − + 540540Υ , − Υ , − (cid:1) + 34658 b (cid:0) Υ , Υ , − + 4Υ , − + 24Υ , − Υ , − Υ , − + 12Υ , − Υ , − (cid:1) + 3152 b (cid:0) Υ , Υ , − + 3Υ , Υ , − Υ , − + 3Υ , − + 6Υ , − Υ , − + 6Υ , − Υ , − (cid:1) + 1052 b (cid:0) Υ , Υ , − + 2Υ , Υ , − Υ , − + Υ , Υ , − + 2Υ , Υ , − Υ , − + 2Υ , − (cid:1) + 154 b (cid:0) Υ , Υ , − + 2Υ , Υ , − + 4Υ , − Υ , + 4Υ , Υ , + 4Υ , Υ , − + 4Υ , Υ , − (cid:1) + 32 b (cid:0) , Υ , Υ , − + 2Υ , Υ , − + Υ , Υ , − + 2Υ , Υ , − + 2Υ , (cid:1) + 1 b (cid:0) Υ , + 2Υ , Υ , + 2Υ , (cid:1) + 16 (cid:0) Υ , + 6Υ , Υ , + 6Υ , (cid:1) − S − S S . (2.33)9pplying this expansion, one finds the perturbative invariants in the geometric branchyields S ( u ) = − m (1 − m − m + 15 m − m − m + m ) , (2.34) S ( u ) = 2∆ m (1 − m − m + 5 m − m − m + m ) , (2.35) S ( u ) = 190∆ m (1 − m − m + 36 m + 1074 m − m + 5782 m +7484 m − m + 7484 m + 5782 m − m + 1074 m +36 m − m − m + m ) . (2.36)In [21] the perturbative invariants are computed up to the eighth order. S with holonomy L R The next example is the once punctured torus bundle over S . This class of manifolds isstudied in the Jorgensen’s theory on the space of quasifuchsian (once) punctured torusgroups from the view point of their Ford fundamental domains [40, 41]. In particular,the complete hyperbolic structure of this class of manifolds is studied well, and the idealtriangulation is found explicitly [42].Let T ϕ be a once punctured torus bundle over S [40]: T ϕ = F × I/ ∼ , (2.37) F = T \{ } , I = [0 , , ( x, ∼ ( ϕ ( x ) , , where T ϕ admits a hyperbolic structure, if the monodromy matrix ϕ has two distincteigenvalues [43, 44]. Such a monodromy matrix is specified by a sequence of 2 p positiveintegers ( a , b , a , b , . . . a p , b p ) and two basis matrices L and R as: ϕ = L a R b L a R b · · · L a p R b p , (2.38) L = (cid:18) (cid:19) , R = (cid:18) (cid:19) . (2.39)For the simplest choice ϕ = LR , the manifold T LR is isomorphic to the figure eight knotcomplement.The next simplest choice is ϕ = L R . The manifold T L R appears in the table of theSnapPea census manifolds as m
009 [10, 12, 13]. m
009 is also described as an arithmeticknot complement in RP [11]. T L R can be decomposed into three ideal tetrahedra. The In [45] it is shown that the figure eight knot is the unique arithmetic knot in S . Z ~ ( T L R ; ~ , u ) = √ e − u Z d p δ C ( p ; u ) h p , p | S − | p , p ih p , p | S − | p , p ih p , p | S | p , p i . (2.40)The shape parameters for each tetrahedra are z = e p − p , z = e p − p and z = e p − p ,and the meridian condition is given by p − p − p + p = 2 u. (2.41)Integrating out the extra parameters, one obtains the partition function for T L R with anincomplete structure: Z ~ ( T L R ; ~ , u ) = √ π ~ Z C dp dp Φ ~ ( − p − u + iπ + ~ )Φ ~ ( − p − p − u − iπ − ~ )Φ ~ (2 p + p + 2 u − iπ − ~ ) × e − ~ h u ( u + p + p )+ ( p + p ) − π − ~ − π ~ i − u . (2.42)In this case the term iπ + ~ in the argument of the quantum dilogarithm functions cannotbe removed by the shift of the parameters as the figure eight knot case.Expanding the integrand as above one obtains Z ~ ( T L R ; ~ , u ) = √ π ~ Z C dp dp e Υ( ~ ,p ,p ,u ) , (2.43)Υ( ~ , p , p , u ) = ∞ X n =0 h B n ( 12 + − p − ~ ~ )Li − n ( 1 m x ) − B n ( 12 + − p − p + ~ ~ )Li − n ( 1 m xy ) − B n ( 12 + 2 p + p + ~ ~ )Li − n ( m x y ) i (2 ~ ) n − n ! − ~ log( m )[log( m x y ) + 2 p + 2 p ] − ~ h log( xy / ) + p + 12 p i − u = ∞ X k = − ∞ X i =0 ∞ X j =0 Υ i,j,k ( x, y, m ) p i p j ~ k , (2.44)where ( x, y ) = ( e p , e p ). At the critical point ( e p , e p ) = ( x ( u ) , y ( u )), the coefficientsΥ , , − and Υ , , − vanishes. The solution for Υ , , − = 0 and Υ , , − = 0 which corre-sponds to the geometric branch is x ( u ) = − m + m − √ − m − m − m + m m + m ) , (2.45) y ( u ) = 2 m (1 + 2 m + m − √ − m − m − m + m )( − m + m − √ − m − m − m + m ) . (2.46)11round the critical point, Z ~ ( T L R ; ~ , u ) is expanded as: Z ~ ( T L R ; ~ , u ) = e u + ~ V ( u ) √ π ~ Z C dp dp e − b u ) p b u ) p b u ) p p ~ × exp (cid:20) ~ ∞ X i + j =3 Υ i,j, − p i p j + ∞ X i,j =0 ∞ X k =0 Υ i,j,k p i p j ~ k (cid:21) , (2.47) V ( u ) = Li (cid:16) m x (cid:17) − Li (cid:16) m xy (cid:17) − Li ( m x y ) − log( m ) log( m x y ) − xy / )] + π . (2.48)From the equations (2.14) and (2.15) the A-polynomial [46]: A T L R ( l, m ) = m + l ( − m + 2 m − m ) + m l (2.49)is found from the above potential V ( u ).In the geometric branch, the coefficients b αβ of the quadratic term yield b = 116 m h − m + 7 m + m − (8 + 3 m ) √ − m − m − m + m i ,b = − m m h − m − m − √ − m − m − m + m i ,b = − m + m + m − (2 + m ) √ − m − m − m + m m . (2.50)The constant term Υ , , is e Υ , , = s (1 + m + m − √ − m − m − m + m )2 . (2.51)Then the 1-loop term S ( u ) obeysexp[ S ( u )] = 1 p b b − b / e u +Υ , , = im m ) , (2.52)∆( m ) = √ − m − m − m + m , (2.53)and this result coincides with the Reidemeister torsion [7, 8].The higher order terms are obtained iteratively by expanding (2.48) and adopting aformula for the Gaussian integral Z f ( ~x ) e − A ij x i x j d n x = r (2 π ) n det A exp (cid:18)
12 ( A − ) ij ∂∂x i ∂∂x j (cid:19) f ( ~x ) (cid:12)(cid:12)(cid:12)(cid:12) ~x =0 . (2.54)12fter some computations, one obtains the perturbative invariant S ( u ): S ( u ) = Υ , , + 1 b [ b Υ , , + 2 b ( α )11 Υ , , + b Υ , , + 2 b ( α )22 Υ , , − b Υ , , Υ , , − b Υ , , ]+ 1 b [12 b (Υ , , Υ , , − + Υ , , − ) + 12 b (Υ , , Υ , , − + Υ , , − ) − b b (Υ , , − Υ , , + Υ , , − Υ , , + Υ , , Υ , , − + Υ , , − ) − b b (Υ , , − Υ , , + Υ , , − Υ , , + Υ , , Υ , , − + Υ , , − )+2(2 b b + b )(Υ , , Υ , , − + Υ , , Υ , , − + Υ , , − )]+ 1 b [60 b (Υ , , − − Υ , , − Υ , , − ) + 60 b (Υ , , − − Υ , , − Υ , , − )+12( b b + b b )(Υ , , − + 2Υ , , − Υ , , − )+12( b b + b b )(Υ , , − + 2Υ , , − Υ , , − )+(36 b b b + 6 b )(Υ , , − Υ , , − + Υ , , − Υ , , − )] , (2.55)where b = 4 b b − b . Plugging the explicit form of Υ i,j,k in the geometric branch, onefinds S ( u ) = − − m + 22 m + 105 m + 22 m − m + 5 m − m − m − m + m ) / + 116 , (2.56)where in this case S ( u = πi ) has real and imaginary part.The further higher order terms S ( u ), S ( u ), and S ( u ) are also computed in the samemanner. Plugging the explicit form of Υ i,j,k ( x, y, m ) for the geometric branch into thisexpansion, one finds S ( u ) = m (1 − m + m )(1 + 9 m + 4 m − m + 4 m + 9 m + m )2(1 − m − m − m + m ) , (2.57) S ( u ) = m (1 − m − m + 137 m − m − m + 104390 m + 20753 m − m + 20753 m + 104390 m − m − m + 137 m − m − m + m ) / (720(1 − m − m − m + m ) / ) , (2.58) S ( u ) = m (1 + 86 m + 179 m + 3870 m + 7447 m − m + 51914 m + 60396 m − m − m + 311325 m − m − m + 60396 m +51914 m − m + 7447 m + 3870 m + 179 m + 86 m + m ) / (24(1 − m − m − m + m ) ) . (2.59)13 Free energy on character variety via topological re-cursion relations
In [16] Eynard and Orantin defined a collection of symplectic invariants F ( g, , g ∈ N ∪{ } for any complex plane curve by means of a set of topological recursion relations. In thecontext of matrix models, the complex plane curve is the spectral curve, and the symplecticinvariant F ( g, is the free energy for genus g [47, 48]. In this section using the recursionrelations, we define free energies F ( g,h ) (for genus g with h boundaries in “world sheetlanguage”) on the character variety: C = { x, l ∈ C ∗ | A ( l, x ) = 0 } ⊂ C ∗ × C ∗ (3.1)defined as the zero locus of the A-polynomial A K ( l, m ) reviewed in section 2, wherewe redefined the parameters as A K ( l, m ) = A ( l, m ) = A ( l, x ). The topological recursionrelations iteratively determine the free energies F ( g,h ) (order by order in the Euler number χ = 2 − g − h in world sheet language). We compute the free energies up to χ = − χ = − In this subsection we summarize the Eynard-Orantin topological recursion relation, andits computation. Assuming that the branching number at each ramification point q i , i = 1 , . . . , n on the character variety C is one, and then on neighborhood of q i , one findstwo distinct points q, ¯ q ∈ C such that x ( q ) = x (¯ q ) on the projected coordinate. Themultilinear meromorphic differentials W ( g,h ) ( p , . . . , p h ) on C are defined by the Eynard-Orantin topological recursion relation: W (0 , ( p ) := 0 , W (0 , ( p, q ) := B ( p, q ) ,W ( g,h +1) ( p, p , . . . , p h ) := X q i ∈C Res q = q i dE q, ¯ q ( p ) ω ( q ) − ω (¯ q ) n W ( g − ,h +2) ( q, ¯ q, p , . . . , p h )+ g X ℓ =0 X J ⊂ H W ( g − ℓ, | J | +1) ( q, p J ) W ( ℓ, | H |−| J | +1) (¯ q, p H \ J ) o , (3.2)where ω ( p ) = log l ( p ) dx ( p ) /x ( p ), and H = { , . . . , h } , J = { i , . . . , i j } ⊂ H, p J = { p i , . . . , p i j } . The Bergman kernel B ( p, q ), which should be a planar two-point function14 pp p g h ppp gp g (cid:15) (cid:15) (cid:15) (cid:15) q qq (cid:15) q k p k jj i p i J = + Σ pp h l l l Figure 2: Structure of the Eynard-Orantin topological recursion relation (3.2)of a chiral boson on C as [17], is defined by the conditions: • B ( p, q ) ∼ p → q dpdq ( p − q ) + finite . • Holomorpic except p = q. • I A i B ( p, q ) = 0 , i = 1 , . . . , g = the genus of C , (3.3)where A i are the A -cycles in a canonical basis ( A i , B i ) of one-cycles on C , and dE q, ¯ q ( p )is the third type differential which is a one-form on p and a multivalued function on q defined by the conditions: • dE q, ¯ q ( p ) ∼ p → q − dp p − q ) + finite . • dE q, ¯ q ( p ) ∼ p → ¯ q dp p − q ) + finite . • I A i dE q ( p ) = 0 . (3.4)The topological recursion relation (3.2) is diagrammatically described as in Fig.2.In the following we consider the case that the character variety C is a genus g (distin-guished from the genus g of the world sheet) curve with two sheets. From the one-form ω ( p ) = log l ( p ) dx ( p ) /x ( p ), one defines y ( p ) dp = ( ω ( p ) − ω (¯ p )) / y ( p ) = M ( p ) p σ ( p ) , σ ( p ) = g +2 Y i =1 ( p − q i ) , M ( p ) = 1 p p σ ( p ) tanh − h p σ ( p ) f ( p ) i , (3.5)where Re( q ) ≤ Re( q ) ≤ . . . ≤ Re( q g +2 ), f ( p ) is a rational function in p , and M ( p ) iscalled the moment function in the context of matrix models [47, 48]. In this case in [49]it is found that the third type differential dE q, ¯ q ( p ) has the form: dE q, ¯ q ( p ) = − p σ ( q )2 p σ ( p ) (cid:16) p − q − g X i =1 C i ( q ) L i ( p ) (cid:17) dp, C i ( q ) = 12 πi I q A i dp ( p − q ) p σ ( p ) , (3.6)where we introduced the (normalized) basis of the holomorphic differentials L i ( p ) dp/ p σ ( p )on C by I A j L i ( q ) p σ ( q ) dq = 2 πiδ i,j , L i ( q ) = g X j =1 L j,i q j − , i = 1 , . . . , g. (3.7)15 p p q ==== (cid:15) q p p p p p p p p p p p p q (cid:15) q (cid:15) q p p p qq (cid:15) q p p p q (cid:15) q p p p p p p + + + q (cid:15) q p + p q (cid:15) q p p p + + + + WWWW (0,3)(1,1)(0,4)(1,2) ( p , p , p )( p )( p , p , p , p )( p , p ) :: :: Figure 3: Diagrammatic representation of (3.11), (3.12), (3.13), and (3.14)Note that when q approaches a branch point q i , (3.6) cannot be used, i.e. if a contour A i contains the point q , instead one must replace C j ( q ) with C j ( q ) + δ j,i / p σ ( q ). Using therelation B ( p, q ) = dq ∂∂q (cid:16) dp p − q ) − dE q, ¯ q ( p ) (cid:17) , (3.8)between the Bergman kernel B ( p, q ) and the third type differential dE q, ¯ q ( p ), one findsthat the Bergman kernel has the form: B ( p, q ) = dpdq p σ ( p ) σ ( q ) (cid:16) p σ ( p ) σ ( q ) + F ( p, q )2( p − q ) + H ( p, q )4 (cid:17) , (3.9) F ( p, q ) := 12 (cid:0) σ ( p ) + σ ( q ) (cid:1) − p − q (cid:0) ∂ p σ ( p ) − ∂ q σ ( q ) (cid:1) , (3.10)where H ( p, q ) is a symmetric polynomial in p and q [49].Let us compute the multilinear meromorphic differentials f W ( g,h ) ( p , . . . , p h ) up to theEuler number χ = − W (0 , ( p , p , p ) = X q i ∈C Res q = q i dE q, ¯ q ( p ) ω ( q ) − ω (¯ q ) B ( p , q ) B ( p , ¯ q ) , (3.11) W (1 , ( p ) = X q i ∈C Res q = q i dE q, ¯ q ( p ) ω ( q ) − ω (¯ q ) B ( q, ¯ q ) , (3.12) W (0 , ( p , p , p , p ) = X q i ∈C Res q = q i dE q, ¯ q ( p ) ω ( q ) − ω (¯ q ) (cid:8) B ( p , ¯ q ) W (0 , ( p , p , q ) + perm( p , p , p ) (cid:9) , (3.13) W (1 , ( p , p ) = X q i ∈C Res q = q i dE q, ¯ q ( p ) ω ( q ) − ω (¯ q ) (cid:8) W (0 , ( p , q, ¯ q ) + 2 W (1 , ( q ) B ( p , ¯ q ) (cid:9) , (3.14)where these differentials are represented in Fig.3. One can expand these differentials by16he kernel differentials [15], χ ( n ) i ( p ) : = Res q = q i (cid:16) − dE q, ¯ q ( p ) y ( q ) 1( q − q i ) n (cid:17) = dp n − p σ ( p ) ∂ n − ∂q n − q = q i M ( q ) (cid:16) p − q − g X i =1 C i ( q ) L i ( p ) (cid:17) , (3.15)where in the second equality (3.6) is utilized. Using the relation (3.8), one can expand B ( p, q ) around s = q − q i = 0 as B ( p, q ) ≃ M i p σ ′ i ds n χ (1) i ( p ) + 3 s (cid:16) χ (2) i ( p ) + (cid:16) M ′ i M i + σ ′′ i σ ′ i (cid:17) χ (1) i ( p ) (cid:17) +5 s (cid:16) χ (3) i ( p ) + (cid:16) M ′ i M i + σ ′′ i σ ′ i (cid:17) χ (2) i ( p ) + 12 (cid:16) M ′′ i M i + M ′ i σ ′′ i M i σ ′ i + σ ′′′ i σ ′ i − σ ′′ i σ ′ i (cid:17) χ (1) i ( p ) (cid:17) + O ( s ) o , (3.16)where M i := M ( q i ) , σ ′ i := σ ′ ( q i ) etc., and the odd terms in s are ignored because theseterms are irrelevant in the computation of the topological recursion (3.2). Therefore from(3.11) one obtains [15], W (0 , ( p , p , p ) = 12 X i M i σ ′ i χ (1) i ( p ) χ (1) i ( p ) χ (1) i ( p ) . (3.17)When q = ¯ p the Bergman kernel (3.9) yields B ( p, ¯ p ) = lim q → p dpdp p − q ) (cid:16) − F ( p, q ) p σ ( p ) σ ( q ) (cid:17) − H ( p ) dpdp σ ( p ) = dpdp (cid:16) σ ′′ ( p )2 σ ( p ) − σ ′ ( p ) σ ( p ) − H ( p ) σ ( p ) (cid:17) , (3.18)where H ( p ) := H ( p, p ), and then this can be expanded around a branch point p = q i as: B ( p, ¯ p ) ≃ − dpdp p − q i ) n p − q i ) + (cid:16) H ( q i ) σ ′ ( q i ) − σ ′′ ( q i )4 σ ′ ( q i ) (cid:17) + O ( p − q i ) o . (3.19)Using this expansion, from (3.12) one finds [15]: W (1 , ( p ) = 116 X i χ (2) i ( p ) + 14 X i (cid:16) H ( q i ) σ ′ i − σ ′′ i σ ′ i (cid:17) χ (1) i ( p ) . (3.20)To compute (3.13) and (3.14) let us write the kernel differentials in terms of thepolynomials F ( p, q ) and H ( p, q ) by comparing the expansion (3.16) with the expansionof (3.9) around s = q − q i = 0, B ( p, q ) = dpdq s p σ ( p ) V ( p, q ; q i ) , V ( p, q ; q i ) := 1 p σ ( q ; q i ) (cid:16) H ( p, q ) + 2 F ( p, q )( p − q ) (cid:17) , ≃ dpdq s p σ ( p ) n V ( p, q i ; q i ) + s ∂ q V ( p, q i ; q i ) + s ∂ q V ( p, q i ; q i ) + O ( s ) o , (3.21)17here σ ( q ; q i ) := σ ( q ) / ( q − q i ), and we have removed the term dpdq/ p − q ) in theexpansion which is irrelevant in the computation of the topological recursion (3.2). Someof the kernel differentials are χ (1) i ( p ) = dp M i σ ′ i p σ ( p ) e V ( p, q i ) , e V ( p, q ) := H ( p, q ) + 2 F ( p, q )( p − q ) , (3.22) χ (2) i ( p ) = dp M i σ ′ i p σ ( p ) ∂ q e V ( p, q i ) − (cid:16) M ′ i M i + σ ′′ i σ ′ i (cid:17) χ (1) i ( p ) , (3.23) χ (3) i ( p ) = dp M i σ ′ i p σ ( p ) ∂ q e V ( p, q i ) − (cid:16) M ′′ i M i + 2 M ′ i σ ′′ i M i σ ′ i + σ ′′′ i σ ′ i (cid:17) χ (1) i ( p ) − (cid:16) M ′ i M i + 2 σ ′′ i σ ′ i (cid:17) χ (2) i ( p ) . (3.24)In the following for simplicity we only discuss the cases of g = 1. In this case theBergman kernel is concretely given by the Akemann’s formula [48, 18]: B ( p, q ) = dpdq p σ ( p ) σ ( q ) (cid:16) p σ ( p ) σ ( q ) + f ( p, q )2( p − q ) + G ( k )4 (cid:17) , (3.25) f ( p, q ) := p q − pq ( p + q ) S + 16 ( p + 4 pq + q ) S −
12 ( p + q ) S + S , (3.26) G ( k ) := − S + ( q q + q q ) − E ( k ) K ( k ) ( q − q )( q − q ) , (3.27) K ( k ) = Z dt p (1 − t )(1 − k t ) , E ( k ) = Z dt r − k t − t , (3.28)where K ( k ), (resp. E ( k )) is the complete elliptic integral of the first, (resp. second) kindwith the modulus k = ( q − q )( q − q )( q − q )( q − q ) , and S k = P ≤ j 2. We have defined e σ ( w ) := σ ( p ) p = 4 w − S w + ( S − 2) = 4( w − α )( w − α ) , (3.44)where α = ( q + q − ) / q + q − ) / α = ( q + q − ) / q + q − ) / 2. Using (3.43)by replacing χ ( n ) i ( p ) with e χ ( n ) i ( p ), we define averaged multilinear meromorphic differentials f W ( g,h ) ( p , . . . , p h ) for ( g, h ) = (0 , , (0 , F ( g,h ) ( p ) (according toremodeling the B-model [14, 15]) for the two toric branes on the character variety C as: F ( g,h ) ( p ) := 1 h ! F ( g,h ) ( p, . . . , p ) , (3.45) F ( g,h ) ( p , . . . , p h ) := Z p Z p · · · Z p h W ( g,h ) ( p ′ , . . . , p ′ h ) , (3.46) W (0 , ( p ) := ω ( p ) + ω ( p − ) := log l ( p ) dpp + log l ( p − ) dp − p − , (3.47) W (0 , ( p , p ) := 2 B ( p , p ) + 2 B ( p , p − ) − dw dw ( w − w ) , w i = p i + p − i , (3.48) W ( g,h ) ( p , . . . , p h ) := f W ( g,h ) ( p , . . . , p h ) for ( g, h ) = (0 , , (0 , , (3.49)where the factor h ! in (3.45) is the symmetric factor. In (3.48) the factor 2 comes from B ( p − , p − ) + B ( p − , p ) where B ( p , p ) = B ( p − , p − ), and the term dw dw / ( w − w ) needs for the regularization (exclusion of the double pole) of the Bergman kernel at p = p . By introducing a coupling constant g s , we define F ( p ) := 12 ∞ X g =0 ,h =1 g g − hs F ( g,h ) ( p ) = ∞ X n =0 ˜ ~ n − F n ( p ) (3.50)on the character variety C , where to express “chiral part” of the free energy we insist the21ecessity of the factor 1 / We also introduced a new coupling constant ˜ ~ = g s / ~ in the Chern-Simons gauge theory.Note that in the definition (3.46), there are ambiguities of the integration constants.In this paper we claim that, by taking the universal part which does not depend on thechoice of the integration constants, we obtain F n ( p ) ≃ S (geom) n ( u ) , p = m = e u . (3.51)Here S (geom) n ( u ) is the perturbative invariant on the geometric branch discussed in section2. In this claim we neglect the constant term in S (geom) n ( u ) which does not depend on u .In the left hand side of the above claim, regularizations of G ( k ) n in (3.27), as explainedin the following, are needed.In the rest of this section, to check this claim we compute F ( p ) = F (0 , ( p ) + F (1 , ( p ) , (3.52) F ( p ) = 2 F (0 , ( p ) + 2 F (1 , ( p ) , (3.53)for the two examples of section 2.2 and 2.3, and find that the different regularizations for G ( k ) in the Bergman Kernel (3.25), and its square G ( k ) are needed as: G ( k ) = − S + 2 − E ( k ) K ( k ) ( q − q )( q − q ) ⇒ G := − S + 2 , (3.54) G ( k ) ⇒ G := G − (1 − k )( q − q ) ( q − q ) = G − (cid:0) S − S − (cid:1) , (3.55)for the examples, where k is the modulus of the elliptic integrals K ( k ) and E ( k ) definedin (3.28). The constant G is determined uniquely by imposing zero A-period condition.But we have to impose some ad-hoc regularizations to G n terms in the free energy. Thisregularization may be compensating the some subtleties of the correspondence in thehigher order terms of ~ expansion. The subtleties may come from the choice of theintegration contour for the BKMP’s free energy or O ( ~ / 2) shift of the moduli of theA-polynomial. Although we do not know the general rule for this regularizations, weheuristically find the rules which are applicable to some lower order terms in the WKBexpansion. In appendix D we compute the free energy F ( p ), and for the figure eight knotcomplement we find that different regularizations for G ( k ) in F (1 , ( p ) and in F (2 , ( p ) areneeded as in (D.10). The meaning of “chiral part” may come from SL (2; C ) Chern-Simons gauge theory. The partitionfunction of SL (2; C ) Chern-Simons gauge theory is holomorphically factorized as Z SL (2; C ) ( M ; t, ¯ t ) = Z ( M ; t ) ¯ Z ( M ; ¯ t ) where t, ¯ t are coupling constants. The factor 1 / The similar problem occurs in the inner toric brane computation to realize the 2 D/ D instantonpartition function of the four dimensional N = 2 gauge theory in the AGT context [30]. l ( p ) l ( p − ) = 1.Therefore we obtain F ( p ) = 14 F (0 , ( p ) = 12 Z p log l ( p ) dpp = Z m log l ( m ) dmm , (3.56)and this is nothing but S (geom)0 ( u ), where p = m , except a constant shift [39, 50]. Thesubleading term F ( p ) = F (0 , ( p ) / As the first example, from the A-polynomial (2.30) of the figure eight knot, we obtain thedata of the curve: σ ( p ) = p − p − p − p + 1 , e σ ( w ) = 4 w − w − , (3.57) f ( p ) = p − p − p − p + 1 p − , (3.58)where w = ( p + p − ) / 2, and e σ ( w ) = σ ( p ) /p .The free energy F ( p ) defined in (3.52) is computed from (3.30) and (3.31): W (0 , ( p , p , p ) dw dw dw = 8 (cid:2) (2 w − w − w − 3) + (2 w + 1)(2 w + 1)(2 w + 1) (cid:3)e σ ( w ) / e σ ( w ) / e σ ( w ) / , (3.59) W (1 , ( p ) dw = − (2 w − + (2 w + 1) e σ ( w ) / + − G + 2)(2 w − 3) + 3(3 G − w + 1)180 e σ ( w ) / . (3.60)Thus from (3.45) and (3.52) we obtain F (0 , ( p ) = − w − w + 712 e σ ( w ) / , (3.61) F (1 , ( p ) = − e σ ( w ) / − G + 1) w − G + 1)180 e σ ( w ) / , (3.62) F ( p ) = − e σ ( w ) / n G + 1)15 w − G − w − w + 9 G + 645 o . (3.63)Using the regularization (3.54), by replacing G with G = 7 / F ( p ) = − e σ ( w ) / (8 w − w − w + 17) . (3.64)23his coincides with the perturbative invariant (2.34) by identifying the parameter w =( m + m − ) / F ( p ) defined in (3.53). Asthe result by (3.45) and (3.53) we obtain F (0 , ( p ) = 1 e σ ( w ) (cid:16) w − w + 2 w + 113 w − w + 2512 (cid:17) , (3.65) F (1 , ( p ) = 1 e σ ( w ) (cid:16) w − w + 821405 w + 15445 w − w − w + 859240 (cid:17) + G e σ ( w ) (cid:16) w − w + 1390 w + 720 w − (cid:17) + G (4 w − e σ ( w ) , (3.66) F ( p ) = 2 e σ ( w ) (cid:16) w − w − w + 24445 w + 910 w − w + 45380 (cid:17) + 2 G e σ ( w ) (cid:16) w − w + 1390 w + 720 w − (cid:17) + G (4 w − e σ ( w ) , (3.67)and using the regularization (3.54) and (3.55), by replacing G with G = 7 / G with G = − / F ( p ) = 2 e σ ( w ) (8 w − w − w + 7) . (3.68)This also coincides with the perturbative invariant (2.35). S with holonomy L R As the second example, from the A-polynomial (2.49) of the once punctured torus bundleover S with holonomy L R , we obtain the data of the curve: σ ( p ) = p − p − p − p + 1 , e σ ( w ) = 4 w − w − , (3.69) f ( p ) = p − p − p + 1 p − , (3.70)where e σ ( w ) = σ ( p ) /p .As same as the computation in section 3.3, the free energies F (0 , ( p ) and F (1 , ( p ) areobtained as: F (0 , ( p ) = − w + 36 w + 6 w + 1948 e σ ( w ) / , (3.71) F (1 , ( p ) = − (72 G − w − (156 G − w − (42 G − w + 147 G + 217336 e σ ( w ) / , (3.72)and using the regularization (3.54), by replacing G with G = 11 / F ( p ) = − e σ ( w ) / (40 w − w + 14 w + 127) . (3.73)24his coincides with the perturbative invariant (2.56) by identifying the parameter w =( m + m − ) / F (0 , ( p ) and F (1 , ( p ) are also computed. Using the regularization(3.54) and (3.55), by replacing G with G = 11 / G with G = − / F ( p ) = − e σ ( w ) (64 w − w − w − w + 2300 w + 2996 w − . (3.74)For the free energy F ( p ) we should consider an imaginary term corresponding to theChern-Simons term of the partition function for the state integral model as in (2.56).Such the contribution, if we add a constant term 1 / F ( p ) + 1128 = 12 e σ ( w ) (16 w + 64 w − w − w + 27) , (3.75)then this coincides with the perturbative invariant (2.57). In this section, we will further discuss the correspondence for torus knots. The pertur-bative invariants S k ( u ) and the BKMP’s free energies F k are computed exactly for thiscase. Although the results are rather trivial on both sides, we are able to check thecorrespondence exactly for this example. A torus knot is described as a curve on a two-torus T , and a pair of coprime integers( p, q ) specifies the number of windings around each cycle of T . For the ( p, q ) torus knotthe colored Jones polynomial is found explicitly [51].Although a torus knot does not admit a hyperbolic structure on the S complement,the asymptotic behavior of the the colored Jones polynomials is studied in the contextof the Melvin-Morton-Rozansky conjecture [52, 51, 53, 54, 55] and the volume conjecture[56, 57, 58]. In the analysis of the volume conjecture, the volume of the torus knotcomplement vanishes but the Chern-Simons invariant [59] is realized as the asymptoticlimit of the colored Jones polynomial around the exponential growth point.Furthermore, the q -difference equation for the ( p, q ) torus knots have been found ex-plicitly [31, 32, 33, 34, 35]. Adopting the technique of [21], we will perform the WKBexpansion iteratively from the q -difference equation.25 .2 AJ conjecture for torus knots The q -difference equations for (2 , m + 1) torus knots T , m +1 are found from the inhomo-geneous difference equation [35]: J n ( T , m +1 ; q ) = q m ( n − − q n − − q n − q (2 m +1) n − m − q n − − q n J n − ( T , m +1 ; q ) , (4.1) J n ( T s,t ; q ) = q ( s − t − n − − q n (1 − q s ( n − − q t ( n − + q ( s + t )( n − )+ 1 − q n − − q n q st ( n − J n − ( T s,t ; q ) . (4.2)As was firstly calculated in [31, 32], one can obtain the homogeneous q -difference equa-tion from the inhomogeneous one by adopting Mathematica packages ‘ qZeil.m ’ and‘ qMultiSum.m ’ developed by Paule and Riese [60].For the trefoil knot the q -difference equation for the colored Jones polynomial is[34]: P ( E, Q ) J n ( ; q ) = 0 , (4.3) P ( E, Q ) = q Q ( q − q Q ) q − q Q + ( q − q Q )( q + q Q )( q − q Q + q Q − q Q − q Q + q Q ) q Q ( q − q Q )( q − q Q ) E + − q QQ ( q − q Q ) E , (4.4)where ( Qf )( n ) = q n f ( n ) , ( Ef )( n ) = f ( n + 1) , q = e ~ . (4.5)In ~ → P yields P ( L, M ) = − ( L − L + M ) M (1 + M ) . (4.6)The numerator of P ( L, M ) is the A-polynomial for the trefoil knot.The expectation value of the Wilson loop operator W n ( K ; q ) is different from thecolored Jones polynomial J n ( K ; q ) by the unknot factor J n (unknot; q ). By factoring out q j/ Q − q − j/ in the coefficient of E j , one obtains the q -difference equation for W n ( K ; q ).The Wilson loop expectation value W n ( K ; q ) is identified with Z ( S \ K ; ~ , u ) for q = e ~ and m = q n/ , if K is the hyperbolic knot. 26e rewrite the q -difference equation for W n ( ; q ) following the notation of [21] as Q = ˆ m : X j =0 a j ( q n/ ; q ) W n + j ( ; q ) = 0 , (4.7) a ( ˆ m, q ) = q ˆ m q ˆ m − ,a ( ˆ m, q ) = − q / (1 + q ˆ m )[1 − q ˆ m − ( q − q ) ˆ m − q ˆ m + q ˆ m ]ˆ m (1 − q ˆ m )(1 − q ˆ m ) ,a ( ˆ m, q ) = 1ˆ m (1 − q ˆ m ) . Furthermore q -difference operator P is factorized:ˆ m (1 − q ˆ m )(1 − q ˆ m ) P (ˆ l, ˆ m )= (1 − q ˆ m )ˆ l − q / (1 − q ˆ m − q ˆ m + q ˆ m ) − q (1 − q ˆ m ) ˆ m = ˆ l (1 − q − ˆ m )ˆ l − q / (1 − q ˆ m )ˆ l + q / ˆ l (1 − q − ˆ m ) ˆ m − q (1 − q ˆ m ) ˆ m = (cid:2) ˆ l (1 − q − ˆ m ) − q / (1 − q ˆ m ) (cid:3) (ˆ l + q / ˆ m ) . (4.8)There are two branches for the solution of this difference equation. One branch cor-responds to the solution l = − m for P ( l, m ) = 0 in q → non-abelian branch . Another branch corresponds to a solution l = 1, and we callthis the abelian branch . In the following, we will discuss the non-abelian branch for thetrefoil knot. The results of abelian branch and the other torus knots are summarized inappendix A.The perturbative invariants S ( α ) n ( u ) for the branch α are defined as follows: W n ( K ; q ) = exp " ~ S ( α )0 ( u ) − δ ( α ) log ~ + ∞ X n =0 S ( α ) n +1 ( u ) ~ n . (4.9)In the non-abelian branch α = nab, the leading term yields [59, 61] S (nab)0 ( u ) = CS( ; 0) − Z u du v nab ( u ) + πiu = 3 log m − π − π s, s ∈ Z , (4.10) l = − m . (4.11)The perturbative invariant for this branch satisfies the q -difference equation:(ˆ l + q / ˆ m ) W (nab) n ( ; q ) = 0 . (4.12)27rom this q -difference equation, we obtain the perturbative invariants in the non-abelianbranch: S (nab) ′ ( u ) = 6 log m, (4.13) S (nab) n ( u ) = constant , n ≥ . (4.14)Since the non-abelian branch corresponds to the geometric branch for the hyperbolicknots, we are able to compare our result with the BKMP’s free energy computed fromthe Eynard-Orantin topological recursion. ( p, q ) torusknot The character variety C ( p,q ) corresponding to the non-abelian branch of the ( p, q ) torusknot is given by C ( p,q ) = { x, y ∈ C ∗ | A ( x, y ) = y + x n = 0 , n := pq } . (4.15)By making use of the topological recursion relation (3.2), let us compute the free en-ergies F ( g,h ) ( x , . . . , x h ) defined in (3.46) on the character variety C ( p,q ) . In this ap-pendix, for simplicity at first we do not introduce the averaged meromorphic differentials f W ( g,h ) ( x , . . . , x h ), instead we use W ( g,h ) ( x , . . . , x h ): F ( g,h ) ( x , . . . , x h ) := Z x · · · Z x h c W ( g,h ) ( x ′ , . . . , x ′ h ) , (4.16) c W (0 , ( x ) := log y ( x ) dxx , c W (0 , ( x , x ) := B ( x , x ) − dy dy ( y − y ) , y i := y ( x i ) , (4.17) c W ( g,h ) ( x , . . . , x h ) := W ( g,h ) ( x , . . . , x h ) for ( g, h ) = (0 , , (0 , , (4.18)and after the computation we take the average. Here we treat y ( p ) = y (¯ p ) as the projectedcoordinate on C ( p,q ) . Since (4.15) has no ramification point, we introduce a free parameter µ as : e C ( p,q ) = n x, y ∈ C ∗ | e A ( x, y ) = y + x n e nµx = 0 , n = pq o , (4.19)and from ∂ x e A ( x, y ) = 0, we find that the deformed curve e C ( p,q ) has one ramification point:( x, y ) = ( − µ − , − ( − µ ) − n e − n ) . (4.20) When µ = − 1, and n = 1, the curve e A ( x, y ) = 0 is nothing but the Lambert curve, and then c W ( g,h ) ( x , . . . , x h ) gives a generating function of the Hurwitz numbers [62, 63]. 28n the curve e C ( p,q ) , the Bergman kernel B ( x , x ) and the third type differential dE p, ¯ p ( x )are given by B ( x , x ) = dx dx ( x − x ) , (4.21) − dE p, ¯ p ( x ) = 12 Z p ¯ p B ( x , ξ ) = dx (cid:26) x − x ( p ) − x − x (¯ p ) (cid:27) . (4.22)To solve the recursion (3.2) we have to consider the expansion of W ( g,h ) ( x , . . . , x h ) aroundthe ramification point (4.20), and for the purpose we introduce a parameter p = ζ nearthe ramification point as [15, 62]: x ( ζ ) = − µ − + ζ , x ( ¯ ζ ) = − µ − + S ( ζ ) , S ( ζ ) := − ζ + ∞ X k =2 C k ζ k , (4.23)where C k are iteratively determined by the equation: y ( ζ ) = − x ( ζ ) n e nµx ( ζ ) = − x ( ¯ ζ ) n e nµx (¯ ζ ) = y ( ¯ ζ ) . (4.24)From this equation we obtain the algebraic equation for C k :(1 − µζ ) e µζ = (1 − µS ( ζ )) e µS ( ζ ) . (4.25)Here we rescale the parameters x and ζ as e x := µx and e ζ := µζ respectively, and thenfrom the equation (4.25) we find − ζ S ( ζ ) := − e S ( e ζ ) = e ζ + 23 e ζ + 49 e ζ + 44135 e ζ + 104405 e ζ + 40189 e ζ + 764842525 e ζ + · · · . (4.26)The annulus amplitude F (0 , ( x , x ) yields F (0 , ( x , x ) = Z Z dx dx ( x − x ) − dy ( x ) dy ( x )( y ( x ) − y ( x )) µ → −→ log − P n − k =0 x n − − k x k , (4.27) F (0 , ( x, x ) µ → −→ (1 − n ) log x + const. (4.28)If we consider the averaged Bergman kernel as in (3.48), then the averaged annulus am-plitude has the form F (0 , ( x ) = log x − x − x n − x − n .Next we compute the higher free energies by the recursions. Using y ( ζ ) (cid:16) dy ( ζ ) dζ (cid:17) − = y ( ζ ) (cid:16) dy ( x ) dx dx ( ζ ) dζ (cid:17) − = ζ − µ − nµζ , (4.29)and (4.25), we find dE ζ, ¯ ζ ( x ) ω ( ζ ) − ω ( ¯ ζ ) = ( e ζ − d e x n e ζ ( e ζ − e S ( e ζ )) d e ζ n e x + 1 − e ζ − e x + 1 − e S ( e ζ ) o . (4.30)29sing (3.11) and (4.30), we can compute F (0 , ( x , x , x ) as: F (0 , ( x , x , x )= Z Res e ζ =0 ( e ζ − e S ′ ( e ζ ) d e ζd e x d e x d e x n e ζ ( e ζ − e S ( e ζ ))( e x + 1 − e ζ ) ( e x + 1 − e S ( e ζ )) n e x + 1 − e ζ − e x + 1 − e S ( e ζ ) o = Z n Y i =1 d e x i ( e x i + 1) = − n Y i =1 µx i + 1) µ → −→ − n , (4.31)and thus F (0 , ( x , x , x ) is constant on C ( p,q ) . Using (3.12) and (4.30), we can alsocompute F (1 , ( x ) as: F (1 , ( x ) = Z Res e ζ =0 ( e ζ − e S ′ ( e ζ )2 n e ζ ( e ζ − e S ( e ζ )) n e x + 1 − e ζ − e x + 1 − e S ( e ζ ) o d e ζd e x = − n Z e x ( e x + 4)( e x + 1) d e x = 124 n µ x + 3 µx + 1( µx + 1) µ → −→ n . (4.32)We see that the free energy F (1 , ( x ) is also constant on C ( p,q ) . In the same way, the freeenergies F (0 , ( x , x , x , x ) and F (1 , ( x , x ) on e C ( p,q ) are computed by (3.13) and (3.14),and we find F (0 , ( x , x , x , x ) = F (1 , ( x , x ) = 0 on C ( p,q ) . One can easily find that c W ( g,h ) ( x , . . . , x h ) are expressed by the rescaled variables e x i , and therefore we see that F ( g,h ) ( x , . . . , x h ) are also constant on C ( p,q ) after taking the average. This matches theresult that the asymptotic expansion of the Wilson loop expectation value W n ( K ; q ) alongthe ( p, q ) torus knot is trivial on the non-abelian branch. The constants of the higherorder terms S (nab) n ( n ≥ 2) may also come from the end points of the integration of theBKMP’s free energies, although we do not know the correct prescription to determine themrigorously at present. In this computation, we found the triviality of the u -dependence ofthe perturbative invariants and BKMP’s free energy for the torus knot. In this paper, we have discussed the correspondence between the perturbative invariantsof SL (2; C ) Chern-Simons gauge theory and the free energies of the topological stringdefined `a la BKMP on the character variety for the figure eight knot complement, the oncepunctured torus bundle over S with the holonomy L R , and the ( p, q ) torus knots. On thethree dimensional geometry side, we computed the perturbative expansion of the partitionfunction of the state integral model around the saddle point which corresponds to the For annulus free energy F (0 , ( x ), we find the non-trivial contribution − log( x n − + x n − + · · · + x − n +2 + x − n +1 ) even after taking µ → W n ( T p,q ; q ), the perturbativeinvariant S ( u ) vanishes in the non-abelian branch. Here we consider this discrepancy would come fromthe normalization factor of the partition function. S with the holonomy L R . For the torus knots, we adopted the factorized q -differenceequation for the colored Jones polynomial. On the character variety side, we computedthe free energies on the basis of the Eynard-Orantin topological recursion. We found thecoincidence to the fourth order on both sides under some particular regularization of G n in the Bergman kernel.The most ambiguous point in our discussion is the regularization of the constants G n for each n independently, although we found a nice presentation for the regularization.Without this regularization, we cannot establish an exact coincidence. But in the free en-ergy computations, there exists an ambiguity of the choice of the integration path. In thispaper we have picked-up the end points of the integrations and neglected the contributionfrom the reference points ( u ∗ , v ∗ ). In the context of the volume conjecture, the analyticcontinuation is discussed in detail in the recent work [61]. The Stokes phenomenon is alsoapplicable to determine the higher order terms in the WKB expansion, so further studyalong these lines may fix the ambiguity completely.In [29], the relation between the Chern-Simons gauge theory on 3-manifold M andthe two dimensional N = (2 , 2) theory on R ~ is discussed via five dimensional N = 2supersymmetric gauge theory. The analogous relation is discussed in the AGT correspon-dence which connects four dimensional N = 2 SU (2) supersymmetric gauge theory andtwo dimensional Liouville field theory [64]. In the context of the AGT correspondence,the surface operator in the four dimensional N = 2 gauge theory can be realized bythe non-compact toric brane in the geometric engineering [28, 29, 30]. There may existsome relations between the chiral boson theory [17] on the character variety and the twodimensional N = (2 , 2) [65]. Acknowledgements: The authors would like to thank Andrea Brini, Sergio Cecotti, Tudor Dimofte, SergeiGukov, Marcos Mari˜no, Masanori Morishita, Hitoshi Murakami, Yuji Terashima, andCumrun Vafa for fruitful discussions and useful comments. R.D. wishes to thank theSimons Center for Geometry and Physics for providing a stimulating environment andgenerous hospitality, and the participants of the 2010 Simons Workshop in Mathematicsand Physics for interesting discussions. Two of the authors (H.F. and R.D.) are alsograteful to RIKEN and IPMU for warm hospitality. The work of H.F. and M.M. issupported by the Grant-in-Aid for Nagoya University Global COE Program, Quest forFundamental Principles in the Universe: from Particles to the Solar System and theCosmos . H.F. is also supported by Grant-in-Aid for Young Scientists (B) [ StringTheory and Quantum Gravity . A Perturbative invariants from AJ conjecture In this appendix, we will summarize some computations on AJ conjecture. A.1 Factorization of AJ conjecture The AJ conjecture is the q -difference equation for the colored Jones polynomial. Thefactorization of the q -difference equation will occur for any knots [66]. In the following,we will see such factorization explicitly for the figure eight knot.For the figure eight knot, the q -difference equation yields [33, 34]: A q (ˆ l, ˆ m ) J n ( K ; q ) = 0 , A q (ˆ l, ˆ m ) = X j =0 a j ( ˆ m ; q ) l j , (A.1) a ( ˆ m ; q ) = q ˆ m ( − q + q ˆ m )( q + q ˆ m )( − q + q ˆ m ) ,a ( ˆ m ; q ) = − q − q ˆ m q ˆ m ( q + q ˆ m )( q − q ˆ m ) × ( q − q ˆ m + q ˆ m − q ˆ m + q ˆ m − q ˆ m + q ˆ m − q ˆ m + q ˆ m ) ,a ( ˆ m ; q ) = − q + q ˆ m q ˆ m ( q + q ˆ m )( − q + q ˆ m ) × ( q + q ˆ m − q ˆ m − q ˆ m + q ˆ m − q ˆ m − q ˆ m + q ˆ m + q ˆ m ) ,a ( ˆ m ; q ) = q ˆ m ( − q ˆ m )( q + q ˆ m )( q − q ˆ m ) , where J n ( K ; q ) is the colored Jones polynomial. The q -Weyl operators ( ˆ m, ˆ l ) satisfiesˆ mf ( u ) = e u f ( u ) , ˆ lf ( u ) = f ( u + ~ ) , (A.2)ˆ l ˆ m = q / ˆ m ˆ l, q = e ~ . (A.3)The Jones polynomial is normalized as J (unknot; q ) = 1. Taking into account for thenormalizations of the colored Jones polynomial and the SL (2 , C ) Chern-Simons partitionfunction, one finds the q -difference equation for the SL (2 , C ) Chern-Simons partition32unction Z ~ ( M, u ; q ) as follows [21]:˜ A q (ˆ l, ˆ m ) Z ~ ( M, u ; q ) = 0 , ˜ A q (ˆ l, ˆ m ) = X j =0 ˜ a j ( ˆ m ; q ) l j , (A.4)˜ a ( ˆ m ; q ) = q ˆ m (1 + q ˆ m )( − q ˆ m ) , ˜ a ( ˆ m ; q ) = 1 + ( q − q ) ˆ m − ( q − q − q ) ˆ m − (2 q − q ) ˆ m + q ˆ m q / ˆ m (1 + q ˆ m − q ˆ m − q ˆ m ) , ˜ a ( ˆ m ; q ) = − − (2 q − q ) ˆ m − ( q − q − q ) ˆ m + ( q − q ) ˆ m + q ˆ m q ˆ m (1 + q ˆ m − q ˆ m − q ˆ m ) , ˜ a ( ˆ m ; q ) = − q ˆ m q / (1 + q ˆ m )( − q ˆ m ) . In q → A q (ˆ l, ˆ m ) yields the A-polynomial. But the abelian part ( l − 1) is included.We can show that this abelian part is factorizable even for the q -difference operator as: ˜ A q (ˆ l, ˆ m ) = ( q / ˆ l − 1) ˆ A q (ˆ l, ˆ m ) , (A.5)ˆ A q (ˆ l, ˆ m ) = q ˆ m (1 + q ˆ m )( − q ˆ m ) − ( − q ˆ m )(1 − q ˆ m − ( q + q ) ˆ m − q ˆ m + q ˆ m ) q / ˆ m ( − q ˆ m )( − q ˆ m ) ˆ l + q ˆ m (1 + q ˆ m )( − q ˆ m ) ˆ l . (A.6)From this factorization, we expect that the AJ conjecture will imply the quantum Riemannsurface structure in topological string theory [67, 68]. A.2 Abelian branch We will discuss the perturbative invariants near the abelian branch from AJ conjecture.For the figure eight knot, the abelian branch is studied [21]. In particular for the torusknots, the abelian branch contains rich structure rather than the non-abelian branch.One of the outstanding properties of this branch will be the Melvin-Morton-Rozanskyconjecture [52, 51, 53, 54]. Here we discuss the expansion of W n ( K ; q ) near the abelianbranch point.The leading term of the perturbative invariant (4.9) for the trefoil knot in thisbranch yields l = 1 , (A.7) S (abel) ′ ( u ) = 0 . (A.8) The partition function of the state integral model satisfies the factored q -difference equation [66]. q -difference equation, one finds a non-trivial ex-pansion: S (abel)1 ( u ) = log m ( m − m − m + 1 , (A.9) S (abel)2 ( u ) = 2 m (1 − m + m ) , (A.10) S (abel)3 ( u ) = − m (1 − m + m )(1 − m + m ) , (A.11) S (abel)4 ( u ) = 4 m (1 + 2 m − m − m + 60 m − m − m + 2 m + m )3(1 − m + m ) . (A.12)The partition function in this branch has the polynomial growth, since S (abel)0 ( u ) = 0.The volume conjecture for the torus knots in this branch is studied in [56, 57, 58]. Theperturbative solution above is consistent with [58, 55]. In particular, the subleading term S (abel)1 ( u ) is e S (abel)1 = 2 sinh( u/ T p,q ; m ) , (A.13)where ∆( T p,q ; m ) is Alexander polynomial. In the case of trefoil knot, the Alexanderpolynomial is ∆( K ; m ) = m + m − − 1, and this result is consistent with (A.9). A.3 The other examples of torus knots From (4.1), one can also find the perturbative invariants for the torus knots in each branch.Here we will show some computational results for (2 , 5) and (2 , 7) torus knots. • (2,5) torus knot The q -difference equation for the cinquefoil knot is X j =0 a j ( q n/ ; q ) W n + j ( T , ; q ) = 0 , (A.14) a ( ˆ m, q ) = ˆ m q ( − m q ) ,a ( ˆ m, q ) = q / ( − m q + ˆ m q − ˆ m q ) ,a ( ˆ m, q ) = q − ˆ m q . The q -difference operator ˆ A T , m +1 (ˆ l, ˆ m ) = X j =0 a j ( ˆ m, q )ˆ l j , (A.15)is factorized for m = 2 as follows:ˆ A T , (ˆ l, ˆ m ) = q [(ˆ l (1 − q − ˆ m ) − q / (1 − q ˆ m )](ˆ l + q / ˆ m ) . (A.16)34n the abelian branch, one obtains the perturbative invariants for (2,5) torus knot asfollows: l = 1 , S (abel) ′ ( u ) = log l, (A.17) S (abel)1 ( u ) = log m ( m − − m + m − m + m , (A.18) S (abel)2 ( u ) = 2 m (1 − m + 4 m − m + m )(1 − m + m − m + m ) , (A.19) S (abel)3 ( u ) = − m (1 − m + 12 m − m − m + 32 m − m + 32 m − m − m + 12 m − m + m ) / (1 − m + m − m + m ) , (A.20) S (abel)4 ( u ) = 4 m (1 − m + 25 m − m − m + 602 m − m + 940 m +449 m − m + 3342 m − m + 449 m + 940 m − m + 602 m − m − m + 25 m − m + m ) / − m + m − m + m ) . (A.21)The Alexander polynomial for (2 , 5) torus knot is A ( t ) = 1 − t + t − t + t , (A.22)and the S (abel)1 ( u ) in (A.18) is consistent with the general formula (A.13).In the non-abelian branch, we find the trivial perturbative invariants for the (2 , l = − m , S (nab) ′ ( u ) = log l, (A.23) S (nab) k ( u ) = constant , for k ≥ . (A.24)The factorization of (A.16) indicates the triviality of the higher order terms. • (2,7) torus knot For the (2 , 7) torus knot, the q -difference equation is X j =0 a j ( q n/ ; q ) W n + j ( T (2 , ; q ) = 0 , (A.25) a ( ˆ m, q ) = ˆ m q ( − m q ) ,a ( ˆ m, q ) = q / ( − m q + ˆ m q − ˆ m q ) ,a ( ˆ m, q ) = − q ( − m q ) . The q -difference operator ˆ A T , (ˆ l, ˆ m ) is factorized as follows:ˆ A T , (ˆ l, ˆ m ) = q [ˆ l (1 − q − ˆ m ) − q / (1 − q ˆ m )][ˆ l + q / ˆ m ] . (A.26)35n the abelian branch, we find the perturbative invariants iteratively from (A.25): l = 1 , S (abel) ′ ( u ) = log l, (A.27) S (abel)1 ( u ) = log m ( − m )1 − m + m − m + m − m + m , (A.28) S (abel)2 ( u ) = 2 m (1 − m + 4 m − m + 9 m − m + 4 m − m + m )(1 − m + m − m + m − m + m ) , (A.29) S (abel)3 ( u ) = − [2 m (1 − m + 12 m − m + 58 m − m + 44 m + 24 m − m +224 m − m + 224 m − m + 24 m + 44 m − m + 58 m − m + 12 m − m + m )] / (1 − m + m − m + m − m + m ) , (A.30) S (abel)4 ( u ) = − m ( − m − m + 80 m − m + 314 m + 125 m − m +5186 m − m + 15482 m − m + 9609 m + 5318 m − m + 41862 m − m + 41862 m − m + 5318 m +9609 m − m + 15482 m − m + 5186 m − m +125 m + 314 m − m + 80 m − m + 6 m − m ) / − m + m − m + m − m + m ) . (A.31)The Alexander polynomial for the (2 , 7) torus knot is A ( t ) = 1 − t + t − t + t − t + t , (A.32)and the perturbative invariant S (abel)1 ( u ) is consistent with (A.13).In the non-abelian branch, we find the trivial perturbative invariants for (2 , 7) torusknot as follows: l = − m , S (nab) ′ ( u ) = log l, (A.33) S (nab) k ( u ) = constant , for k ≥ . (A.34) • (2,p) torus knots (Conjecture) From the computations for p = 3 , , 7, we can guess the q -difference equation for (2 , p )torus knots: ˆ A T p, (ˆ l, ˆ m ) = [ˆ l (1 − q − ˆ m ) − q p/ − (1 − q ˆ m )][ˆ l + q p/ ˆ m p ] = 0 . (A.35) B Derivation of (3.32) and (3.33) Here we consider the case that the character variety is genus one curve with two sheetswritten as (3.34), and describe the derivation of (3.32) and (3.33) in detail. For computing363.13) and (3.14) let us expand W (0 , ( p , p , q ) , W (0 , ( p, q, q ) and W (1 , ( q ) around q = q i .At first, from (3.30) and (3.31) we consider the expansion of the kernel differentials χ (1) j ( q )and χ (2) j ( q ) around q = q i . By (3.29) the kernel differentials are obtained: χ (1) j ( q ) = dsM j σ ′ j p σ ( q ; q i ) (cid:16) G + 2 f ( q, q j )( q − q j ) (cid:17) , (B.1) χ (2) j ( q ) = ds M j σ ′ j p σ ( q ; q i ) 4 σ ( q ; q j ) − σ ′ ( q )( q − q j ) − (cid:16) M ′ j M j + σ ′′ j σ ′ j (cid:17) χ (1) j ( q ) , (B.2)where σ ( q ; q i ) := σ ( q ) / ( q − q i ). Using the expressions when j = i , we find the expansions χ (1) j ( q ) ≃ dsM j σ ′ j p σ ′ i n G + 2 f ( q i , q j )( q i − q j ) + O ( s ) o , s := q − q i , (B.3) χ (2) j ( q ) ≃ − dsM j σ ′ j p σ ′ i n σ ′ i q i − q j ) + (cid:16) M ′ j M j + σ ′′ j σ ′ j (cid:17)(cid:16) G + 2 f ( q i , q j )( q i − q j ) (cid:17) + O ( s ) o , (B.4)and when j = i , we find the expansions χ (1) i ( q ) ≃ dsM i p σ ′ i n s + 1 σ ′ i (cid:16) G − σ ′′ i (cid:17) − s σ ′ i (cid:16) σ ′′ i G − σ ′′ i + 13 σ ′ i σ ′′′ i (cid:17) + O ( s ) o , (B.5) χ (2) i ( q ) ≃ dsM i p σ ′ i n s − s (cid:16) M ′ i M i + σ ′′ i σ ′ i (cid:17) − Gσ ′ i (cid:16) M ′ i M i + σ ′′ i σ ′ i (cid:17) + 112 σ ′ i (cid:16) σ ′′ i + σ ′ i σ ′′ i M ′ i M i − σ ′ i σ ′′′ i (cid:17) + O ( s ) o . (B.6)By the expansions, W (0 , ( p , p , q ) , W (0 , ( p, q, q ) and W (1 , ( q ) can be expanded around s = q − q i = 0 as, W (0 , ( p , p , q ) ≃ ds M i p σ ′ i n M i σ ′ i (cid:16) s + 1 σ ′ i (cid:0) G − σ ′′ i (cid:1)(cid:17) χ (1) i ( p ) χ (1) i ( p )+ X j = i M i M j (cid:16) G + 2 f ( q i , q j )( q i − q j ) (cid:17) χ (1) j ( p ) χ (1) j ( p ) + O ( s ) o , (B.7) W (0 , ( p, q, q ) ≃ dsds n(cid:16) s + 2 s σ ′ i (cid:0) G − σ ′′ i (cid:1) + 1 σ ′ i (cid:0) G − σ ′′ i G + 19 σ ′′ i − σ ′ i σ ′′′ i (cid:1)(cid:17) χ (1) i ( p )+ X j = i σ ′ i σ ′ j (cid:16) G + 2 f ( q i , q j )( q i − q j ) (cid:17) χ (1) j ( p ) + O ( s ) o , (B.8) W (1 , ( q ) ≃ ds M i p σ ′ i n s + 1 s σ ′ i (cid:16) G − σ ′′ i − σ ′ i M ′ i M i (cid:17) + 1 σ ′ i (cid:16) G − (cid:0) σ ′′ i + σ ′ i M ′ i M i (cid:1) G + 112 (cid:0) σ ′′ i + σ ′ i σ ′′ i M ′ i M i − σ ′ i σ ′′′ i (cid:1)(cid:17) + X j = i M i M j σ ′ j (cid:16) − σ ′ i σ ′ j q i − q j ) + (cid:0) G − σ ′′ j − σ ′ j M ′ j M j (cid:1)(cid:0) G + 2 f ( q i , q j )( q i − q j ) (cid:1)(cid:17) + O ( s ) o , (B.9)37nd from (3.13) and (3.14), we obtain (3.32) and (3.33). C Computation of the subleading term F ( p ) In this appendix, on the character variety y ( p ) = M ( p ) p σ ( p ) , σ ( p ) = Y i =1 ( p − q i ) = p − S p + S p − S p + 1 , (C.1)we compute (3.48), F ( p ) = 12 F (0 , ( p ) = 14 F (0 , ( p, p ) = 12 Z p Z p B ( p ′ , p ′ )+ B ( p ′ , p ′− ) − dw ′ dw ′ ( w ′ − w ′ ) , (C.2)where w ′ i = ( p ′ i + p ′− i ) / 2. Using (3.25) we get B ( p , p ) + B ( p , p − )= dw dw pe σ ( w ) e σ ( w )( w − w ) (cid:16)pe σ ( w ) e σ ( w ) + e f ( w , w ) (cid:17) =: e B ( w , w ) , (C.3) e σ ( w ) := σ ( p ) p = 4 w − S w + ( S − 2) = 4( w − α )( w − α ) , (C.4) e f ( w , w ) := 4 w w − ( w + w ) S + ( S − , (C.5)where this is nothing but the Bergman kernel on the genus 0 reduced curve y = e σ ( w ).Here by a change of variable [49],2 w ( λ ) = S γ ( λ + λ − ) , γ := α − α , (C.6)we can rewrite (C.4) and (C.5) as e σ ( w ) = γ ( λ − λ − ) , (C.7) e f ( w , w ) = γ (cid:16) ( λ + λ − )( λ + λ − ) − (cid:17) . (C.8)Therefore from (C.3) we obtain e B ( w , w ) − dw dw ( w − w ) = dλ dλ ( λ − λ ) − ( λ − λ − dλ dλ ( λ − λ ) ( λ λ − = dλ dλ ( λ λ − , (C.9)and then (C.2) is easily computed as14 F (0 , ( p , p ) = 12 Z w Z w e B ( w ′ , w ′ ) − dw ′ dw ′ ( w ′ − w ′ ) = 12 Z λ Z λ dλ ′ dλ ′ ( λ ′ λ ′ − = 12 log λ λ λ − , F ( p ) = 12 log 1 λ − λ − = 12 log γ pe σ ( w ) . (C.10)This result coincides with the computation (2.29) and (2.52) after the identification ofthe parameter w = ( m + m − ) / = p p p p p q (cid:15) q p p p p + + p q (cid:15) q p p p p + + + p q (cid:15) q p p p p + + p q (cid:15) q p p p p + + p q (cid:15) q p p p p + p q (cid:15) q p p p p + + p q (cid:15) q p p p p + Figure 5: Recursive structure of W (0 , ( p , . . . , p ) D Computation of the fifth order free energy F ( p ) By the recursion (3.2), the multilinear meromorphic differentials with the Euler number χ = − W (0 , ( p , . . . , p ) as follows (see also Fig.5): W (0 , ( p , . . . , p ) = X q i ∈C Res q = q i dE q, ¯ q ( p ) y ( q ) dq n(cid:0) B ( q, p ) W (0 , (¯ q, p , p , p ) + perm( p , p , p , p ) (cid:1) + (cid:0) W (0 , ( q, p , p ) W (0 , (¯ q, p , p ) + ( p ↔ p ) + ( p ↔ p ) (cid:1)o = 38 X i n M i σ ′ i (cid:0) E i + 10(3 A i − a i ) E i + 24 A i + 20 B i + 103 b i (cid:1) χ i i i i i + X j = i M i M j (cid:0) H ij ( E j + 3 A j − a j ) − σ ′ i q i − q j ) (cid:1)(cid:2) χ i ijjj + perm( C ) (cid:3) + X j,k = i M j M k H ij H ik σ ′ i (cid:2) χ ijjkk + perm( 5!2!2!2 ) (cid:3) + 2 M i σ ′ i (cid:0) E i + 10 A i (cid:1)(cid:2) χ i i i i i + perm( C ) (cid:3) + X j = i M i M j H ij (cid:2) χ i ijjj + perm( C × (cid:3) + 6 M i σ ′ i (cid:2) χ i i i i i + perm( C ) (cid:3) + 5 M i σ ′ i (cid:2) χ i i i i i + perm( C ) (cid:3)o . ( D.1)We have defined χ n n ...n h i i ... i h := χ ( n ) i ( p ) χ ( n ) i ( p ) · · · χ ( n h ) i h ( p h ) , (D.2) a i := σ ′′ i σ ′ i + M ′ i M i , A i := σ ′′ i σ ′ i + M ′ i M i , b i := σ ′′ i σ ′ i − σ ′′′ i σ ′ i ,B i := M ′′ i M i + σ ′′ i M i σ ′ i M i − b i , E i := 1 σ ′ i (cid:0) G − σ ′′ i (cid:1) , H ij := G + 2 f ( q i , q j )( q i − q j ) , (D.3)where G = G ( k ), and f ( p, q ) are defined in (3.26), and (3.27) respectively. In (D.1),“perm” denotes the permutation of p , . . . , p so that the result becomes symmetric forthese variables, for example, χ ijjkk + perm( 5!2!2!2 ) = χ ijjkk + χ jijkk + χ jjikk + χ kjjik + χ kjjki + χ ikjjk + χ kijjk + χ kjijk + χ kkjij + χ kkjji + χ ijkjk + χ kijkj + χ jkijk + χ jkjik + χ jkjki . ( D.4)39 = p q (cid:15) q p p p p p p p q (cid:15) q p p p + + + q (cid:15) q p p p + + + Figure 6: Recursive structure of W (1 , ( p , p , p )In the same way we obtain the meromorphic differential W (1 , ( p , p , p ) as follows (seealso Fig.6): W (1 , ( p , p , p ) = X q i ∈C Res q = q i dE q, ¯ q ( p )2 y ( q ) dq n W (0 , ( q, ¯ q, p , p ) + 2 W (1 , ( q ) W (0 , (¯ q, p , p )+ 2 (cid:0) B ( q, p ) W (1 , (¯ q, p ) + ( p ↔ p ) (cid:1)o = 164 X i n(cid:0) E i + (252 A i − a i ) E i + (522 A i − A i a i + 318 a i + 120 B i + 92 b i ) E i − A i B i + 332 A i b i − a i b i + 35( σ ′′ i σ ′ i − M ′ i M i ) b i + 3 σ ′′′′ i σ ′ i + 35 σ ′′ i M ′′ i σ ′ i M i + 35 M ′′′ i M i (cid:1) χ i i i + X j = i M i σ ′ j M j (cid:0) (3 E i + 6 A i − σ ′′ i σ ′ i )(4 E j − a j ) H ij − σ ′ i ( E i + 2 A i ) + σ ′′ i + σ ′ j (4 E j − a j )( q i − q j ) + 2 σ ′ i ( q i − q j ) (cid:1) χ i i i + X j = i (cid:0) E j + 3 A j − a j ) H ij σ ′ i σ ′ j + (8 E i + (36 A i − a i ) E i + 6 A i − A i a i + 9 a i + 283 b i ) M j H ij σ ′ i M i − H ij σ ′ j ( q i − q j ) − (4 E i + 2 A i − a i ) σ ′ j M j σ ′ i M i ( q i − q j ) + σ ′ j M j σ ′ i M i ( q i − q j ) (cid:1)(cid:2) χ ijj + perm( C ) (cid:3) + X j,k = i M j H ij σ ′ i σ ′ k M k (cid:0) (4 E k − a k ) H ik − σ ′ i q i − q k ) (cid:1)(cid:2) χ ijj + perm( C ) (cid:3) + X j = i,k = j M k H ij H jk σ ′ i σ ′ j M j (cid:2) χ ikk + perm( C ) (cid:3) + X j = i,k = i,j H ij H jk H ki σ ′ i σ ′ j σ ′ k χ ijk + 2 (cid:0) E i + (121 A i − a i ) E i − A i + 25 B i + 25 b i (cid:1)(cid:2) χ i i i + perm( C ) (cid:3) + X j = i M i σ ′ j M j (cid:0) (4 E j − a j ) H ij − σ ′ i q i − q j ) (cid:1)(cid:2) χ i i i + perm( C ) (cid:3) + X j = i H ij σ ′ i σ ′ j (cid:2) χ ijj + perm( P ) (cid:3) + X j = i M j σ ′ i M i (cid:0) (12 E i + 2 A i − a i ) H ij − σ ′ j ( q i − q j ) (cid:1)(cid:2) χ ijj + perm( C ) (cid:3) + 5 (cid:0) Gσ ′ i + 2 σ ′′ i σ ′ i + 7 M ′ i M i (cid:1)(cid:2) χ i i i + perm( C ) (cid:3) + X j = i M i H ij σ ′ j M j (cid:2) χ i ij + perm( C ) (cid:3) + 24 (cid:0) Gσ ′ i + M ′ i M i (cid:1)(cid:2) χ i i i + perm( C ) (cid:3) + 18 χ i i i + 30 (cid:2) χ i i i + perm( P ) (cid:3) + 35 (cid:2) χ i i i + perm( C ) (cid:3)o . ( D.5)40 p p (cid:15) qq q (cid:15) q p + Figure 7: Recursive structure of W (2 , ( p )The meromorphic differential W (2 , ( p ) is obtained as follows (see also Fig.7): W (2 , ( p ) = X q i ∈C Res q = q i dE q, ¯ q ( p )2 y ( q ) dq n W (1 , ( q, ¯ q ) + W (1 , ( q ) W (1 , (¯ q ) o = 164 X i σ ′ i M i n(cid:0) E i + (48 A i − a i ) E i + ( 3812 A i − A i a i + 376916 a i − B i + 1916 b i ) E i + (135 A i − A i a i + ( 39758 a i − B i + 211 b i ) A i − a i + 15 a i B i − a i b i − σ ′′′′ i σ ′ i ) E i + 4172 A i b i − A i a i b i − σ ′′′′ i σ ′ i A i + 267 σ ′′′′ i σ ′ i a i + 21098 a i b i − B i b i + 1909 b i (cid:1) χ i + X j = i (cid:0) M i σ ′ i σ ′ j M j (12 E i + 24 A i − a i ) H ij + M i σ ′ j M j ( 18 (4 E j − a j ) + (8 E j + 36 A j − a j ) E j + 15 A j − A j a j + 21 a j − B j + 313 b j ) H ij − M i H ij q i − q j ) σ ′ i σ ′ j M j − σ ′ i M i q i − q j ) σ ′ j M j ( 253 E j + 4 A j − a j + 2 q i − q j ) H ij + M i σ ′ j M j (4 E j − a j )(2 E i + (9 A i − a i ) E i + 32 A i − A i a i + 94 a i + 73 b i ) H ij + 25 σ ′ i M i q i − q j ) σ ′ j M j − σ ′ i M i q i − q j ) σ ′ j M j ( E i + 6( A i − a i ) E i + 2 b i ) + σ ′ i M i q i − q j ) σ ′ j M j ( − E i + (12 A i − a i ) E i − A i + 3 A i a i + 149 b i − (4 E i + 2 A i − a i )(4 E j − a j ) σ ′ j σ ′ i + 1 q i − q j (8 E i − A i + 10 a i + (4 E j − a j ) σ ′ j σ ′ i ) − q i − q j ) ) (cid:1) χ i + X j = i,k = i,j M i σ ′ j σ ′ k M j M k (cid:0) (4 H ik H jk + σ ′ k (4 E k − a k ) H ij H jk − σ ′ j σ ′ k H ij q j − q k ) ) H ij σ ′ j σ ′ k (( E j − a j ) H ij − σ ′ i q i − q j ) )(( E k − a k ) H ik − σ ′ i q i − q k ) ) (cid:1) χ i + (cid:0) E i + (61 A i − a i ) E i + (45 A i − A i a i + 5078 a i − B i + 31 b i ) E i + 1213 A i b i − a i b i − σ ′′′′ i σ ′ i (cid:1) χ i + X j = i M i σ ′ j M j (cid:0) H ij σ ′ i σ ′ j + ( 32 E i + 14 A i − a i )(4 E j − a j ) H ij + σ ′ i q i − q j ) − σ ′ i ( q i − q j ) ( 12 E i − A i + 38 a i + (4 E j − a j ) σ ′ j σ ′ i ) (cid:1) χ i + (cid:0) E i + ( 352 A i − a i ) E i + 4916 A i − B i + 24524 b i (cid:1) χ i + X j = i M i σ ′ j M j (cid:0) 58 (4 E j − a j ) H ij − σ ′ i q i − q j ) (cid:1) χ i + (cid:0) E i − A i (cid:1) χ i + 10516 χ i o . ( D.6)41 .1 Figure eight knot complement The free energies (3.45) with χ = − F (0 , ( p ) = 1 e σ ( w ) / (cid:16) − w + 103 w − w − w + 227360 w − w − w + 6357160 w − (cid:17) , ( D.7) F (1 , ( p ) = − (4 w − G e σ ( w ) / − (4 w − G e σ ( w ) / (cid:16) w − w + 139 w + 72 w − (cid:17) − G e σ ( w ) / (cid:16) w − w + 2944330375 w + 10681 w − w − w + 218236000 w − (cid:17) − e σ ( w ) / (cid:16) w − w + 13962418225 w + 10301210935 w − w − w + 859011350 w + 4151300 w − w + 1408633600 (cid:17) , ( D.8) F (2 , ( p ) = − w − e G e σ ( w ) / + e G e σ ( w ) / (cid:16) w − w + 100354 w + 13813180 w − w − (cid:17) + e G e σ ( w ) / (cid:16) w − w + 106688135 w + 349636405 w − w − w − w + 1131110 (cid:17) + 15625 e σ ( w ) / (cid:16) w − w + 196744481 w + 3066698243 w − w + 1322354 w − w + 331862972 w + 78620316 w − (cid:17) , ( D.9)where by distinguishing G in F (1 , ( p ) from G in F (2 , ( p ), we put G = G, G = G , G = G in (D.8), and e G = G, e G = G , e G = G in (D.9). If we regularize these parametersas G = 73 , G = − , G = (cid:16) (cid:17) , e G = 73 , e G = − , e G = − , (D.10)then the free energy F ( p ) = 4 (cid:0) F (0 , ( p ) + F (1 , ( p ) + F (2 , ( p ) (cid:1) = 115 e σ ( w ) / (cid:16) w − w − w + 10243 w + 148963 w − w + 212 w + 8194 w − (cid:17) (D.11)coincides with the perturbative invariant (2.36), where w = ( m + m − ) / D.2 Once punctured torus bundle over S with holonomy L R The free energies (3.45) with χ = − S with holonomy L R are summarized as follows:42 (0 , ( p ) = 1 e σ ( w ) / (cid:16) − w − w − w + 31920 w − w − w + 13651480 w − (cid:17) , ( D.12) F (1 , ( p ) = − (6 w − G e σ ( w ) / − (6 w − G e σ ( w ) / (cid:16) w − w + 176 w + 7112 w − (cid:17) − G e σ ( w ) / (cid:16) w − w + 275954116 w + 426593528 w − w + 334992016 w + 1344114032 w − (cid:17) + 1 e σ ( w ) / (cid:16) w − w + 88701029 w + 95653427783 w − w + 38095784 w + 37477324 w − w − w + 267223141472 (cid:17) , ( D.13) F (2 , ( p ) = 81(6 w − e G e σ ( w ) / + e G e σ ( w ) / (cid:16) w − w − w + 6605196 w − w − (cid:17) − e G e σ ( w ) / (cid:16) w − w + 209131372 w + 1940772744 w − w − w + 425364 w + 21273128 (cid:17) + 198 e σ ( w ) / (cid:16) w − w + 1233832205 w + 23966334410 w − w − w + 751187720 w + 80097011440 w − w − (cid:17) , ( D.14)where as in the case of the figure eight knot complement, we distinguished G in F (1 , ( p )from G in F (2 , ( p ). 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