Volume Conjecture and Topological String

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. .

.. .

.

Volume Conjecture and Topological String

Hiroyuki FUJI

Nagoya University

Collaboration with R.H.Dijkgraaf (ITFA& KdVI) and M. Manabe (Nagoya Math.)

18th Dec. @ Taiwan String Theory Workshop 2011 Papers:

R.H.Dijkgraaf and H.F., Fortsch.Phys.57(2009),825-856, arXiv:0903.2084 [hep-th]

R.H.Dijkgraaf, H.F. and M.Manabe, arXiv:1010.4542 [hep-th].

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1. Introduction

Classiﬁcation of knots ⇒Knot invariants

Borromean Ring

Trefoil (3,2)-torus knot Solomon s Seal knot (5,2)-torus knot Figure eight knot

Hopf link

Knot invariants are invariant underReidemeister move:

Move I Move IIHiroyuki FUJI Volume Conjecture and Topological StringMove III

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Jones polynomial

One of the most famous knot invariant will be the

Skein relation:

K :+ K :- K :0

V_{K +}(q)

q^-1 -qV_{K -}(q) = (q - q )^1/2 ^-1/2 V_{K 0}(q) unknot

V :=1(q)

m unknots ...

V ... =(q) (-q - q )^1/2 ^{-1/2 m-1} Jones polynomial: V (q)_K

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Computation of the Jones polynomial fortrefoil knot 31 :

K+

K- K₀

q2 q(q - q )^1/2 ^-1/2

-q - q ^1/2 ^-1/2

V =(q) V =(q) 1 V =(q) 1

K+

K- K₀

V₃₁(q) = q²V_⃝(q)

+q(q^1/2− q^−1/2)[

q²V_⃝⃝(q) + q(q^1/2− q^−1/2)V_⃝(q)]

=−q⁴+ q³+ q.

Hiroyuki FUJI Volume Conjecture and Topological String

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Jones polynomial and Chern-Simons gauge theory

[Witten]

The Jones polynomial is related with the expectation value of the Wilson loop operatoralong the knot K with the fundamental representation forSU(2) Chern-Simons gauge theoryonS³.

S_CS[A] = k 4π

∫

S³Tr [

AdA + 2

3A∧ A ∧ A ]

.

S³

Wilson loop operator W_˜(K ; q)

⟨

W_˜(K ; q)

⟩ :=

⟨

Tr_˜P exp [I

K

A ] ⟩

, q := exp ( 2πi

k + 2 )

, VK(q) =

⟨

W_˜(K )

⟩/⟨

W_˜(⃝)

⟩ .

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Colored Jones polynomial

Wilson loop operator (R: Spin j representation n := 2j + 1) W_R(K ) = Tr_RPexp

[I

K

A ]

.

S³

Spin j-rep.

J_n(K ; q): Colored Jones polynomial

J_n(K ; q) =

⟨

W_j(K ; q)

⟩/⟨

W_j(⃝; q)⟩ .

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Surgery and knot complement

FromTQFTpoint of view the colored Jones polynomial is also related with the partition function onS³\N(K).

S3

a U a

M1

j

M₂

Z(M ;a)=δ(log m )₂ ^2π(2j+1)

k+2

=

S3

<W (K;q)>_(j) Z(M ,m)₁ M1

μ

⟨

W_j(K ; q)

⟩

=

∫

Da Z_k^{SU(2) CS}(S³\N(K); a)Z_k^{SU(2) CS}(N(K ); a)

= Z_k^{SU(2) CS}(S³\N(K); ρ). ρ: holonomy along meridian cycle µ← Pexp[H

µa]

ρ =

( m ∗

0 m⁻¹ )

, m = q^{2j +1}=: e^u.

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Knot complement and hyperbolic manifold

For the “most of” knots the complement ofS³ admits the hyperbolic structure. ^[Thurston] → Section 2

S3

Borromean Ring Trefoil (3,2)-torus knot Solomon s Seal knot (5,2)-torus knot

Non-hyperbolic Knots : Vol(S \K)=0³ Hyperbolic Knots : Vol(S \K)≠0³

Figure eight knot

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Analytic continuation of the Chern-Simons gauge theory

[Witten]

SL(2;C) Chern-Simons gauge theory is equivalent to the 3D Euclidean gravity with negative cosmological constant.

⇒ On-shell geometry ishyperbolic manifold . The analytic continuation of the gauge group

SU(2)→ SL(2; C)

⇒ Geometic aspects of knot complement is more manifest.

I = t 8π

∫

M

Tr(AdA + 2

3A∧ A ∧ A) + ¯t

8π

∫

M

Tr(¯Ad ¯A + 2

3A¯∧ ¯A ∧ ¯A), t, ¯t ∈ C SU(2) Chern-Simons theory−→ SL(2; C) Chern-Simons theory

Zk(M3; ρ) → Z_t,¯^SL(2;C)_t (M3; ρ)

= Z_t^SU(2)(M3; ρ)Z_¯_t^SU(2)(M3; ¯ρ)

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Quantization of the Chern-Simons gauge theory Classical ﬂat connection moduli spaceMK of SL(2;C) Chern-Simons gauge theory on M3 =S³\K.

MK is a Riemann surface−→ Quantization Character variety of knot Detail⇒ Section 3 Non-commutative deformation of M [Garoufalidis et.al.][DGLZ]

Quantum Hamiltonian constraint from non-commutative Riemann surface

Aˆ_K(ˆℓ, ˆm)Z_t^SU(2)CS= 0, mˆˆℓ = q ˆℓ ˆm.

Topological Recursion [Eynard-Orantin][Dijkgraaf-F.-Manabe]

The moduli space MK as the matrix model curve

1 1 h

1 x x x

g

h

1 x x

x

g x

g

q

q q q

k x_k

j j i x_i

J

= + Σ

x₁ x_h

l

l l

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Correspondences

WKB expansion of the Chern-Simons gauge theory Z_k^SU(2)CS(S³\K; ρu) = exp

[ 1

~S0(u) + S1(u) +

∑∞ k=1

~^kSk+1(u) ]

.

3D Geometry Topological Open String Character variety Spectral curve {(ℓ, m)∈ C^∗× C^∗|˜AK(ℓ, m) = 0} {

(e^p, e^x)∈ C^∗× C^∗|H(e^p, e^x) = 0} u = log m: Holonomy u: Spectral parameter Leading order invariant S0(u) Disk Free Energy ¯F^(0,1)(u) Subleading order invariant S1(u) Annulus Free Energy ¯F^(0,2)(u)

q = e^2~ q = e^g^s

In this talk we will explore the following relation:

¶ ³

Sk(u)↔ Fk(u) = 2^k⁻² ∑

2g +h=k+1, h≥0

1

h!F¯^{(g ,h)}(u).

µ ´

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CONTENTS 1. Introduction

2. Geometric Aspects of the Knot Complement 3. Quantization of the Chern-Simons Gauge Theory 4. The Topological Recursion on the Character Variety 5. Summary, Discussions and Future Directions

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2. Geometric Aspects of the Knot Complement

Hyperbolic 3-manifold

The hyperbolic 3-manifold admits a geodesically complete hyperbolic metric R_ij =−2gij.

M₃=H³/Γ, Γ: discrete subgrp of PSL(2;C)

The hyperbolic 3-manifold is simplicially decomposed into the ideal tetrahedra .

Geodesic Line

z

Conformal Transformation

Upper half space model

α α

β β

γ γ Conformal ball model

α β

γ

0 1

z

α＋β＋γ＝π

The ideal tetrahedron is speciﬁed by the dihedral angles α, β, γ.

They are toggled into a shape parameterz ∈ C.

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Figure Eight Knot Complement

As an example, let us discuss the ﬁgure eight knot complement.

Consider a surface S with ∂S = K .

Francis, "A Topological Picturebook"

Intuitively, this surface is created by dipping the knot into viscous liquid.

Viscous Liquid Dip

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Simplicial Decomposition

Decompose the bounded surface into 4 pieces.

W

E

W

E

S

N S

N

W S N

E

W' S' N'

E'

• The knot complement is decomposed into 2 tetrahedra.

• Knot is localized at the tip of tetrahedra.

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Simplicial decomposition as ideal tetrahedra

The ﬁgure eight knot can be decomposed into 2 tetrahedra.

← This decomposition is still onlytopological .

S3

∼=

¹¹^w

w 1 z w

z z

1 1

1 z 1

z 1 z

A B C

D

w

w 1 w

1

w 1 w

A' B' D'

C'

The hyperbolic structure is introduced, if the tetrahedra are glued consistently asideal tetrahedra.

α 0

β γ

1 z

β α

γ

0 1

z-1z

0 1

γ 1 1-z

α β

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Gluing Condition 1

There are two kinds of gluing conditions for ideal tetrahedra.

• Gluing conditions (bulk):

z1

z1z2

1 z1z2z3

z1z2...zk-1

Gluing Condition

Π^{z = 1}ⁱ

k i=1

Red edge: zw^z−1_z ^w−1_w zw = 1

Blue edge : ₁_−z¹ ₁_−w¹ ^z⁻¹_z ^w_w⁻¹₁_−z¹ ₁_−w¹ = 1

⇒ (z²− z)(w²− w) = 1.

w

1 1

w 1

z w

z z

1 1

z

1 1

z 1

z

z 1

z

A B C

D

w

w w

1 1

w 1

w A'

B' D'

C'

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Gluing Condition 2

• Gluing conditions (boundary ∂M ≃ T²):

Boundary is realized bychopping oﬀ small tetrahedra.

⇒ Each triangles are glued each other, and the simplicially decomposed boundaryT² can be constructed.

z

z z 1

1 z 1

1

z 1 z

z 1 z A

B C

D

w

w w 1

1 w 1

1

w 1 w

w 1 w a

b

c

d

A' B' D'

C' e

f

h

g

Meridian µ: ^w_z¹

3 = w (1− z) = 1

Longitude ν: (z1z2z3)²(z1)²(w2w3)² = (z/w )²= 1

z₁ z₃ z₂ z₃ z₂ z₁

a h

b g

c f

z₁ z₁

w₃

w₃ w₂ w₂

d e

ν μ

w₁

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Solution and Hyperbolic Structure

Solving gluing conditions, one ﬁnds all dihedral angles.

z = w = e^{πi /3}, α_i = β_i = γ_i = π/3, i = 1, 2.

S3

For ﬁgure eight knot complement the hyperbolic structure can be admitted.

⇒ SL(2; C) Chern-Simons partition function can give the geometric invariants for the knot complement.

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.

3. Quantization of Chern-Simons Gauge Theory

Moduli space of the ﬂat connection

Moduli spaceM of the ﬂat connection for SL(2; C) Chern-Simons gauge theory on M3:

M ={

A, ¯A¯¯FA= 0 = FA¯

}/

Gauge equiv.

Mathematically the ﬂat connection moduli space is rewritten by the the honolomy representation ρ:

ρ : π1(M3)−→ SL(2; C)

∈ ∈

C 7→ ρ(C ) = Pexp[

I

C

AC].

M = Hom(

π₁(M₃), SL(2;C))/

Gauge equiv.

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The 3-manifold M₃ with a torus boundary ∂M₃≃ T²

• M: The moduli space of the ﬂat connection on M3

• P: The moduli space of the ﬂat connection on T²

M3

T²

μ ν

The fundamental group ofT² is π₁(T²)≃ Z × Z. ρ(µ) =

( m 0

0 m⁻¹ )

, ρ(ν) =

( ℓ 0

0 ℓ⁻¹ )

,

⇒ P ={

(m, ℓ)∈ C^∗× C^∗} . M is a submanifold in P: M ⊂ P .

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For the knot complementS³\K the fundamental group is computed viaWirtinger algorithm.

.

Wirtinger

The fundamental group for the knot complement S³\K : π₁(S³\K) =

{

x , y|xω = ωy }

, ω₄₁ := xy⁻¹x⁻¹y , ω₃₁ := xy .

ν

μ^meridian

longitude

The meridian and longitude holonomies are identiﬁed as µ = x ,

ν₄₁ = xy⁻¹xyx⁻²yx⁻²yxy⁻¹x⁻¹, ν₃₁ = yx²yx⁻⁴,

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Holonomy representations cannot be diagonalized simultaneously:

ρ(µ) =

( m ∗

0 m⁻¹ )

, ρ(ν) =

( ℓ ∗

0 ℓ⁻¹ )

. Nontrivial π1 structure

⇒ Constraint on (ℓ, m): A-polynomial ^[CCGLS]

A4₁(ℓ, m) = (ℓ− 1)(ℓ + ℓ⁻¹+ (m⁴− m²− 2 − m⁻²+ m⁻⁴)) = 0, A₃₁(ℓ, m) = (ℓ− 1)(ℓ + m⁶) = 0.

The moduli spaceMK for S³\K is given by MK =

{

(m, ℓ)∈ C^∗× C^∗¯¯A_K(ℓ, m) = 0 }

. M is a Lagrangian submanifold in P: M ,→ P .

(Riemann surface)

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Quantization of the moduli space Quantum Chern-Simons gauge theory

⇒ Quantization of the gauge ﬁelds on T² {A^ai(x ), A^b_j(y )} = 4π

t δâbϵ_ijδ²(x− y), {Aâi(x ), ¯A^b_j(y )} = 0, {¯Aâi(x ), ¯A^b_j(y )} = 4π

¯t δ^abϵ_ijδ²(x− y).

ν

μ^meridian

longitude

µ and ν intersect at only one point: m = exp[u], ℓ = exp[v ] ˆ

mˆℓ = q ˆℓ ˆm, q = exp[4π/t], {ˆu, ˆv} = 4π

t , {ˆ¯u, ˆ¯v} = 4π

¯t .

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The quantum structure ofMK is inherited fromP.

⇒ Non-commutative Riemann surface [Garoufalidis et.al.]

Aˆ4₁(ˆℓ, ˆm) = (q^1/2ˆl− 1) (

q ˆm²

(1 + q ˆm²)(−1 + q ˆm⁴)

−(−1 + q ˆm²)(1− q ˆm²− (q + q³) ˆm⁴− q³mˆ⁶+ q⁴mˆ⁸) q^1/2mˆ²(−1 + q ˆm⁴)(−1 + q³mˆ⁴) ˆl + q²mˆ²

(1 + q ˆm²)(−1 + q³mˆ⁴)ˆl² )

, Aˆ3₁(ˆℓ, ˆm) = (ˆl(1− q⁻¹mˆ⁴)− q^1/2(1− q³mˆ⁴))(ˆl + q^3/2mˆ⁶).

These quantum operators satisﬁes

.

Colored Jones Polynomial

Aˆ_K(ˆℓ, ˆm)Jn(K ; q) = 0, ˆ

mf (n) = qⁿf (n), ℓf (n) = f (n + 1).ˆ

• Inq → 1 limitthe non-commutative A-polynomial reduces to the commutative one.

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WKB expansion[Dimofte-Gukov-Lenells-Zagier]

WKB expansion of the partition function of SL(2;C) Chern-Simons gauge theory:

Z_t^SU(2)CS(S³\K; ρ) = exp [

1

~S₀(u) + S₁(u) +

∑∞ k=1

~^kS_k+1(u) ]

,

The parturbative invariants are found from ˆA_KJ_n(K ; q) = 0:

J_n(K ; q) = Z_t^SU(2)CS(S³\K; ρ)/Zt^SU(2)CS(unknot; ρ), Z_t^SU(2)CS(unknot; ρ) = m− m⁻¹

q− q⁻¹ , q := e^~. The partition function is expaned in the double scaling limit:

~ → 0, n → ∞, qⁿ= eû : finite, q = e^~. Quantum A-polynomial Â_K is expanded as:

AˆK(ˆℓ, ˆm; q) =

∑d k=0

∑∞ k=0

ℓˆ^j~^kaj ,k( ˆm).

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Leading solution in~ expansion

The leading order equation in~ expansion:

∑d j =0

e^jS⁰^′aj ,0(m) = AK(e^S⁰^′^(u), e^u) = 0.

a_{j ,0}(m): Coeﬃcient of A-polynomial⇒ ℓ = e^S⁰^′^(u) The leading order term is given by

S0(u)≃

∫ _u

u_∗

log ℓ(m)d log m, A_K(ℓ(m), m) = 0.

This function is known asNeumann-Zagier function of the hyperbolic manifold. This function is the 1-parameter generalization ofhyperbolic volume (Volume conjecture).

S0(0) = Vol(S³\N(K)).

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Saddle points and braches

In general the A-polynomial is factorized as A_K(ℓ, m) =(ℓ− 1)B_K(ℓ, m).

⇒ 2 kinds of saddle points for Jn(K ; q):

ℓ = 1: abelian branch, ℓ = ℓ(m): non-abelian branch. Abelian branch ⇒ S0(u) = 0 [Melvin-Morton][Rozanski]

Z_t^SU(2)CS(S³\K; ρ) behaves as polynomial growth . Non-abelian branch ⇒ S0(u)̸= 0 ^[Kashaev]

Z_t^SU(2)CS(S³\K; ρ) behaves as exponetial growth.

One of the saddle point in this branch contains the geometric information as the hyperbolic geometry.

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Perturbative Invariants

Applying this expansion into q-difference equation, one finds a hierarchy of differential equations:

AˆK(ℓ, m; q) = Xd k=0

X∞

k=0

ℓ^j~^kaj ,k(m).

Xd j =0

e^jS⁰^′a_{j ,0}= 0, ← A− polynomial

Xd j =0

e^jS⁰^′

»

aj ,1+ aj ,0

„1

2j²S₀^′′+ jS₁^′

«–

= 0,

Xd j =0

e^jS⁰^′

»

aj ,2+ aj ,1

„1

2j²S₀^′′+ jS₁^′

« + aj ,0

„1 2(1

2j²S₀^′′+ jS₁^′)²+1

6j³S₀^′′′+1

2j²S₁^′′+ jS₂^′

«–

= 0,

· · · .

Solving these equations iteratively, one obtains the perturbative invariants S_k(u).

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Computational Results

.

top1

.

top2

.

top3

.

top4

From the q-diﬀerence equation, one obtains the expansion of the Wilson loop expectation value aroud thenon-abelian branch.

• Figure eight knot: [Dimofte-Gukov-Lenells-Zagier]

ℓ(m) =1− 2m²− 2m⁴− m⁶+ m⁸+ (1− m⁴)√

1− 2m²+ m⁴− 2m⁶+ m⁸

2m⁴ ,

S₀^′(u) = log ℓ(m), S1(u) =−1

2log

" p σ0(m)

2

#

, σ0(m) := m⁻⁴− 2m⁻²+ 1− 2m²+ m⁴, S2(u) = −1

12σ0(m)^3/2m⁶(1− m²− 2m⁴+ 15m⁶− 2m⁸− m¹⁰+ m¹²), S3(u) = 2

σ0(m)³m⁶(1− m²− 2m⁴+ 5m⁶− 2m⁸− m¹⁰+ m¹²).

S₁(u) coincides with the Reidemeiser torsion. [Porti][Gukov-Murakami]

T (M; u) = exp [

−1 2

∑3 n=0

n(−1)ⁿlog det^′∆^En^ρ

] .

Eρ: ﬂat line bdle, ∆^En^ρ: Laplacian on n-forms.

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.

4. Topological Recursion on Character Variety

TheSL(2;C) Chern-Simons gauge theoryandmatrix model/topological B-modelhave similar quantum structure.

Fact [Dijkgraaf-Hollands-Sulkowski]

Non-commutative Riemann surface

↕

D-module structure in top. string

⇒ Quantum invariants can also be calculated via WKB method developed in the matrix model analysis.

3D Geometry Topological String Character variety Spectral curve {(ℓ, m)∈ C^∗× C^∗|BK(ℓ, m) = 0} {

(e^p, e^x)∈ C^∗× C^∗|H(e^p, e^x) = 0} u = log m: Holonomy u: Spectral parameter Leading free energy S0(u) Disk Free Energy ¯F^(0,1)(u) Subleading free energy S1(u) Annulus Free Energy ¯F^(0,2)(u)

q = e^2~ q = e^g^s

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BKMP’s Free Energy

In the computations of the matrix model and topological B-model amplitudes, thespectral curveplays the central role.

C ={

(x , y )∈ C^∗× C^∗|H(y, x) = 0} .

Free energies for the B-model world-sheet with boundaries:

Spectral invariants F^{(g ,h)}(u1,· · · , uh), ui: spectral parameter These free energies are integrals of the meromorphic forms W_h^{(g )}.

[Bouchard-Klemm-Marino-Pasquetti]

F^{(g ,h)}(u₁,· · · , uh) =

∫ _eu1

e^u∗¹

dx₁· · ·

∫ _euh

e^u∗^h

dx_hW_h^{(g )}(x₁,· · · , xh).

In the hermitian matrix model, W_h^{(g )}is nothing but the genus g correlation function of h resolvents:

W_h^{(g )}∼⟨∏^h

i =1

1 z_i − M

⟩(g )

, e^uⁱ ∼ zi.

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D-brane Free Energy

The D-brane free energy F_D(u) is deﬁned by FD(u) =∑

g

∑

h≥1

gs^2g^−2+h

h! F^{(g ,h)}(u,· · · , u).

For n D-brane insertions, we should consider F_D(u) =∑

g

∑

h≥1

g_s^2g^−2+h h!

∑

u_i=u⁽¹⁾,··· ,u⁽ⁿ⁾

F^{(g ,h)}(u₁,· · · , uh).

u(1)

e

u(2)

e

u(3)

e

V = diag(e^u⁽¹⁾,· · · , e^u⁽ⁿ⁾): Set of the location of non-compact D-branes on the spectral curveC

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Topological Recursion

Eynard-Orantin’s topological recursion (2g + h≥ 3):

W_h+1^{(g )}(x , x₁,· · · , xh)

=∑

x_i

Resq=q_i

dEq(x ) y (q)− y(¯q)

[

W_h+2^(g⁻¹⁾(q, ¯q, x1,· · · , xh) +

∑g ℓ=0

∑

J⊂H

W_|J|+1^(g^−ℓ)(q, p_J)W_|H|−|J|+1^(ℓ) (¯q, p_H_\J) ]

.

q, ¯q: points x = q on the 1st sheet and 2nd sheet of the spectral curve qi: End points of cuts in the double covering ofC.

1 h

1

1 x x x

g

h

1 x x

x

g x

g

q

q q q

k x_k

j j i x_i

J

= + Σ

x

x1 _h

l

l l

dE_q(x ): Meromorphic 1-form w/ properties.

• Simple pole at x = q with residue +1

• Zero A-period on the spectral curve C.

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Disk invariant

The initial condition for W₁⁽⁰⁾(x ):

W₁⁽⁰⁾(x ) = 0.

x

The top. string free energy is determined independently. [Aganagic-Vafa]

F^(0,1)(u) =

∫ _u

u_∗

d log x log y (x ), H(y , x ) = 0, u := log x . Annulus invariant

The initial condition for W₂⁽⁰⁾(x ):

W₂⁽⁰⁾(x , y ) = B(x , y ).

x y

B(x , y ): Bergmann kernel on the spectral curve B(x , y )^x∼^∼y dx dy

(x− y)², I

A_I

B(x , y ) = 0, 1 2

∫ ¯q q

B(x , p) = dE_q(p).

The top. string free energy is regularized. [Marino][F.-Mizoguchi]

F^(0,2)(u₁, u₂) =

∫ _x₁

x₁^∗

∫ _x₂

x₂^∗

[

B(x , y )− 1 (x− y)²

]

, x_i = e^uⁱ

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Topological Recursion in Lower Orders

x₁

q

=

q x₁ x₂

x₂ x₃ x₃

x₁

1

q q

q

q q

q

2 2

4

q q

2 3

1 1

1

+

+ + ² ^q ^q

4

q q

1 3 2

+

2

+² +² +²

W W W W

(0) (x ,x,x ) (x ) (x ,x ,x ,x ) (x ,x )1

1 1 1

2 2 2

3 3

4

: :

3

(1)

(0)

(1) 1

4

2

x

x₁x₂x₃x₄

x₁ x₂

x₂x₃ x₁ x₄ x x x₁ x x

x x x

x x x x

x x

W₃⁽⁰⁾(x1, x2, x3) =X

q_i q=qRes_i

dEq(x1)

y (q)− y(¯q)B(x2, q)B(x3, ¯q) W₁⁽¹⁾(x ) =X

q_i

Resq=q_i

dEq(x )

y (q)− y(¯q)B(q, ¯q), W₄⁽⁰⁾(x1, x2, x3, x4) =X

q_i q=qRes_i

dEq(x1) y (q)− y(¯q)

“

B(x2, ¯q)W₃⁽⁰⁾(x3, x4, q) + perm(x2, x3, x4)” ,

W₂⁽¹⁾(x1, x2) =X

q_i q=qRes_i

dEq(x1) y (q)− y(¯q)

“

W₃⁽⁰⁾(x2, q, ¯q) + 2W₁⁽¹⁾(q)B(x2, ¯q)” .

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Our Set-up: 1

Character variety as spectral curve

We choose the character variety as the spectral curve.

character varietyof knot K . C ={

(ℓ, m)∈ C^∗× C^∗|˜AK(ℓ, m) = 0}

, A˜_K(ℓ, m²) := B_K(ℓ, m).

i.e. H(y , x ) = ˜AK(y , x ).

Location of D-brane

On the information of D-brane we identify

V = diag(eû⁽¹, eû⁽²⁾) ↔ ρ(µ) = diag(m, m⁻¹), m = eû. Actually this choice of D-brane locus is computed as

F¯^{(g ,h)}(u) := ∑

All signs

F^{(g ,h)}(±u, · · · , ±u).

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Our Set-up: 2 Expansion Parameters

We identify the expansion parameters 2~ ↔ gs.

Therefore we compare the free energies with a ﬁxed Euler number.

Fixed Euler Number

We discuss the following correspondence:

S_k(u)↔ Fk(u) := 2^k⁻² ∑

2g +h=k+1

1

h!F¯^{(g ,h)}(u)

F_D(V ) =

∑∞ k=1

g_s^k⁻²F_k.

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Computational Results

.

perturbative

In the following, we summarize the spectral invariants on the character variety for theﬁgure eight knot.

Disk invariant:

F¯^(0,1)(u) = 2

∫ _u

u_∗

d log x log y , B_K(y , x ) = 0.

This satisﬁes the leading order diﬀerntial equation in CS:

∂ ¯F^(0,1)(u)/∂u = log y = v (u).

Annulus invariant:

1 2!

F¯^(0,2)(x ) = log 1

√σ(x ),

σ(x ) = x²− 2x − 1 − 2x⁻¹+ x⁻², x = m². Comparing with F₁(u) = ¯F^(0,2)(u)/2, we ﬁnd the subleading term of the perturbative invariant S₁(u). (Reidemeister torsion)

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2nd order term:

.

perturbative

The spectral invariants ¯F^(0,3) and ¯F^(1,1) are

• 1 3!

F¯^(0,3)(x ) =−12w²− 12w + 7

12σ(x )^3/2 , w := x + x⁻¹ 2 ,

• ¯F^(1,1)(x ) =−8(1 + 6G )w³− 4(11 + 21G)w²+ 30w + 87 + 27G

180σ(x )^3/2 .

G : Constant in the Bergmann kernel on 2-cut curve G =(q1+ q2)(q3+ q4)− 2(q1q2+ q3q4)

12 −E (k)

K (k)(q1− q2)(q3− q4).

The function F2 yields to

F2=−192 + 27G− 150w + 136w²− 84Gw²+ 8w³+ 48Gw³

180σ(x )^3/2 .

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3rd order term:

The spectral invariants ¯F^(0,4) and ¯F^(1,2) are

• 1 4!

F¯^(0,4)(x ) = 25− 67w + 44w²+ 24w³− 32w⁴+ 16w⁵

12σ(x )³ ,

• 1 2!

F¯^(1,2)(x ) = 1280w⁶− 9088w⁵+ 13136w⁴+ 22176w³− 17928w²− 26352w + 23193 6480σ(x )³

+G64w⁴− 232w³+ 156w²+ 378w− 243

1080σ(x )² + G²(4w− 3)² 3600σ .

Summing these contributions, we ﬁnd F3.

F3= 2

"

1280w⁶− 448w⁵− 4144w⁴+ 35136w³+ 5832w²− 62532w + 36693 6480σ(x )³

+G64w⁴− 232w³+ 156w²+ 378w− 243

1080σ(x )² + G²(4w− 3)² 3600σ

# .

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Change of Gⁿ

Unfortunately both of the contributions does not coincide because of the constant G ∈ C in the Bergmann kernel.

.

Bergmann kernel

G =(q1+ q2)(q3+ q4)− 2(q1q2+ q3q4)

12 −E (k)

K (k)(q1− q2)(q3− q4).

But the coincidence is found by the following small changes.

.

1 Change 1:

We discard thered partwhich consists of the elliptic functions.

G_reg⁽¹⁾= (q₁+ q₂)(q₃+ q₄)− 2(q1q₂+ q₃q₄)

12 .

.

2 Change 2:

We regularize G² independent of G .

G²→ Greg⁽²⁾= (G_reg⁽¹⁾)²− (1 − k²)(q1− q3)²(q2− q4)², These procedure will be ad-hoc, but we expect some

resolutions of this point by the detailed study of the Stokes phenomenon . [Witten][Eynard]