Volume Conjecture and Topological Recursion

Transcription

1 Nagoya University Collaboration with R.H.Dijkgraaf (ITFA) and M. Manabe (Nagoya Math.) 6th IPMU Papers: R.H.Dijkgraaf and H.F., Fortsch.Phys.57(2009), , arxiv: [hep-th] R.H.Dijkgraaf, H.F. and M.Manabe, to appear.

2 . Introduction Asymptotic analysis of the knot invariants is studied actively in the knot theory. Volume Conjecture [Kashaev][Murakami 2 ] Asymptotic expansion of the colored Jones polynomial for knot K The geometric invariants of the knot complement S 3 \K. S 3 Recent years the asymptotic expansion is studied to higher orders. S k (u): Perturbative invariants, q = e 2 [Dimofte-Gukov-Lenells-Zagier] [ J n (K; q) = exp S 0(u) + δ ] log + k S k+ (u), u = 2n 2πi. 2 k=0

3 Topological Open String Topological B-model on the local Calabi-Yau X X = { (z, w, e p, e x ) C 2 C C H(e p, e x ) = zw }. D-brane partition function Z D (u i )

4 Topological Recursion [Eynard-Orantin] Eynard and Orantin proposed a spectral invariants for the spectral curve C C = {(x, y) (C ) 2 H(y, x) = 0}. Symplectic structure of the spectral curve C, Riemann surface Σ g,h = World-sheet. Spectral invariant F (g,h) (u,, u h ) u i :open string moduli Eynard-Orantin s topological recursion is applicable. x x x h x x x xh x x k x i k h i j j q q g = + Σ q g l J g x q l l Σ g,h+ Σ g,h+2 Σ l,k+ Σ g l,h k Spectral invariant = D-brane free energy in top. string [BKMP]

5 Correspondences Heuristically we discuss a relation between the perturbative invariants S k (u) and the free energies F (g,h) (u,, u h ) á la BKMP. [Dijkgraaf-F.] 3D Geometry Topological Open String Character variety Spectral curve { (l, m) C C Ã K (l, m) = 0 } { (e p, e x ) C C H(e p, e x ) = 0 } u = log m: Holonomy Neumann-Zagier fn. H(u)/2 Reidemeister Torsion T (M; u) q = e 2 u: Open string moduli Disk Free Energy F (0,) (u) Annulus Free Energy F (0,2) (u) q = e gs In this talk we will explore the following relation: S k (u) F k (u) = 2 k 2 2g+h=k+, h 0 h! F (g,h) (u).

6 Motivation of Our Research Realization of 3D quantum gravity in top. string Dual description of quantum CS theory as free boson on character variety Large n duality not for rank but for level Novel class of duality

7 CONTENTS. Introduction 2. Volume Conjecture and Perturbative Invariants 3. Topological Recursion on Character Variety 4. Summary, Discussions and Future Directions

8 2. Volume Conjecture and Perturbative Invariants Volume Conjecture [Kashaev][Murakami 2 ] In 997 Kashaev proposed a striking conjecture on the asymptotic expansion of the colored Jones polynomial J n (K; q). log J n (K; e 2πi/n ) 2π lim n n = Vol(S 3 \K). 3 S Knot complement = S 3 \N(K) N(K): Tubular neighborhood of a knot K. The hyperbolic knot complement admits a hyperbolic structure.

9 Generalized Volume Conjecture In 2003, Gukov generalized the volume conjecture to -parameter version. log J n (K; e (u+2πi)/n ) (u + 2πi) lim n n H(u): Neumann-Zagier s potential function H(u) u = v + 2πi. u and v satisfies an algebraic equation. A K (l, m) = 0, l = e v, m = e u. = H(u), u C. A K (l, m): A-polynomial for a knot K. incomplete Up to linear term of u and v, the Neumann-Zagier potential H(u) yields to H(u) = u+2πi 2πi du v(u) + linear terms.

10 AJ conjecture and higher order terms In 2003 Garoufalidis proposed a conjecture on q-difference equation for the colored Jones polynomial. (Quantum Riemann Surface) A K (ˆl, ˆm; q)j n (K; q) = 0, A K (l, m; q = ) = (l )A K (l, m). ˆlf(n) = f(n + ), ˆmf(n) = q n/2 f(n), ˆl ˆm = q /2 ˆmˆl. Commutation relation of the Chern-Simons gauge theory [ ] [ ρ(µ) = P exp, ρ(ν) = P exp A µ {A a α (x), Ab β (y)} = 2π k δab ϵ αβ δ 2 (x y). ν ] A, Meridian µ and longitude ν intersect at one point. ˆl ˆm = q /2 ˆmˆl [û, ˆv] = 2π. (θ = vdu, ω = dθ.) k

11 q-difference Equation for Fig.8 Knot Example: Figure 8 knot 4 [Garoufalidis] A 4 ( ˆl, ˆm; q) q 5 ˆm 2 ( q 3 + q 3 ˆm 2 ) = (q 2 + q 3 ˆm 2 )( q 5 + q 6 ˆm 4 ) (q2 q 3 ˆm 2 )(q 8 2q 9 ˆm 2 + q 0 ˆm 2 q 9 ˆm 4 + q 0 ˆm 4 q ˆm 4 + q 0 ˆm 6 2q ˆm 6 + q 2 ˆm 8 ) ˆl q 5 ˆm 2 (q + q 3 ˆm 2 )(q 5 q 6 ˆm 4 ) + ( q + q3 ˆm 2 )(q 4 + q 5 ˆm 2 2q 6 ˆm 2 q 7 ˆm 4 + q 8 ˆm 4 q 9 ˆm 4 2q 0 ˆm 6 + q ˆm 6 + q 2 ˆm 8 ) ˆl 2 q 4 ˆm 2 (q 2 + q 3 ˆm 2 )( q + q 6 ˆm 4 ) + q4 ˆm 2 ( + q 3 ˆm 2 ) ˆl 3. (q + q 3 ˆm 2 )(q q 6 ˆm 4 ) AJ conjecture for Wilson loop W n (K; q) := J n (K; q)w n (U; q), Ã K (ˆl, ˆm; q)w n (K; q) = 0. q-difference equation is factorized. Ã 4 (ˆl, ˆm; q) = (q /2 ˆl )Â 4 (ˆl, ˆm; q), q ˆm 2 Â 4 (ˆl, ˆm; q) = ( + q ˆm 2 )( + q ˆm 4 ) ( + q ˆm2 )( q ˆm 2 (q + q 3 ) ˆm 4 q 3 ˆm 6 + q 4 ˆm 8 ) ˆl q /2 ˆm 2 ( + q ˆm 4 )( + q 3 ˆm 4 ) q 2 ˆm 2 + ( + q ˆm 2 )( + q 3 ˆm 4 ) ˆl 2.

12 Perturbative Invariants WKB expansion of the Wilson loop expectation value: [ W n (K; q) = exp S 0(u) + δ ] log + k S k (u), 2 k= q := e 2, q n = m = e u. Applying this expansion into q-difference equation, one finds a hierarchy of differential equations : Â K (l, m; q) = d l j k a j,k (m). k=0 k=0 d e js 0 a j,0 = 0, A polynomial j=0 d j=0 d j=0 e js 0 e js 0 [ ( )] a j, + a j,0 2 j2 S 0 + js = 0, [ ( ) ( a j,2 + a j, 2 j2 S 0 + js + a j,0 2 ( 2 j2 S 0 + js )2 + 6 j3 S 0 + )] 2 j2 S + js 2 = 0,.

13 Computational Results top top2 Solving q-difference equation, one obtains the expansion of the expectation value of the Wilson loop around a non-trivial flat connection. Figure eight knot: [DGLZ] l(m) = 2m2 2m 4 m 6 + m 8 + ( m 4 ) 2m 2 + m 4 2m 6 + m 8 2m 4, S 0 (u) = log l(m), [ S (u) = ] 2 log σ0 (m), σ 0 (m) := m 4 2m 2 + 2m 2 + m 4, 2 S 2 (u) = 2σ 0 (m) 3/2 m 6 ( m2 2m 4 + 5m 6 2m 8 m 0 + m 2 ), 2 S 3 (u) = σ 0 (m) 3 m 6 ( m2 2m 4 + 5m 6 2m 8 m 0 + m 2 ). S (u) coincides with the Reidemeiser torsion. [Porti] [ T(M; u) = exp ] 3 n( ) n log det E ρ n. 2 E ρ : flat line bdle, n=0 E ρ n : Laplacian on n-forms.

14 3. Topological Recursion on Character Variety BKMP s Free Energy Topological B-model amplitudes are computed in the similar way as the matrix models. The general structure of the amplitudes is capctured by the symplectic structure of the spectural curve C. C = { (x, y) C C H(y, x) = 0 }. Free energies for closed world-sheet: Symplectic invariants F (g,0) Free energies for world-sheet with boundaries: Spectral invariants F (g,h) (u,, u h ), u i : open string moduli These free energies are integrals of the meromorphic forms W (g) F (g,h) (u,, u h ) = e u e u h. e u h [Bouchard-Klemm-Marino-Pasquetti] dx e u h dx h W (g) h (x,, x h ).

15 D-brane Partition Function The D-brane partition function is defined by Z D (ξ,, ξ n ) = R Z R Tr R V log Z D = h! g2g 2+h s F (g) g=0 h= w,,w h w,,w h TrV w TrV w h, V = diag(ξ i, ξ n ) ξ i (i =, n) are location of non-compact D-brane in X. non-compact D-branes 3 2

16 BKMP s Free Energy and D-brane Partition Function D-brane partition function is related with BKMP s free energies F (g,h) (u,, u h ). [Marino] Dictionary TrV w TrV w h x w xw h h, x i = e u i. Identification of the free energy: F (g,h) (u, u h ) = w,,w h h! F(g) w, w h x w xw h h.

17 Topological Recursion Eynard-Orantin s topological recursion (2g + h 3): W (g) h+ (x, x,, x h ) = x i Res q=q i + de q (x) y(q) y( q) g l=0 J H [ W (g ) h+2 (q, q, x,, x h ) W (g l) J + (q, p J)W (l) H J + ( q, p H\J) q, q: points x = q on the st sheet and 2nd sheet of the spectral curve q i : End points of cuts in the double covering of C. x x xh x x x x x x h g q x k x i h i j k j q q = + Σ g de q (x): Meromorphic -form w/ properties. Simple pole at x = q with residue + Zero A-period on the spectral curve C. l J g q ]. l l

18 Disk invariant The initial condition for W (0) (x): W (0) (x) = 0. x The top. string free energy is determined independently. [Aganagic-Vafa] F (0,) (u) = u u d log x log y(x), H(y, x) = 0, u := log x. Annulus invariant The initial condition for W (0) 2 (x): W (0) 2 (x, y) = B(x, y). B(x, y): Bergmann kernel on the spectral curve B(x, y) x y dx dy (x y), q B(x, y) = 0, B(x, p) = de 2 q (p). A I 2 q The top. string free energy is regularized. [Marino][F.-Mizoguchi] x x2 [ ] F (0,2) (u, u 2 ) = B(x, y), x (x y) 2 i = e u i x x 2 x y

19 Topological Recursion in Lower Orders W W W W (0) (x,x,x) 3 () (0) 4 () 2 (x) (x,x,x,x) (x,x) x x2 x3 = = = = x2 x q q x x x x2 x3 x4 x x2 x2 x3 q q q q 2 x3 + x x x x x x x x x x q q x x x x x x q + q q q 3 4 q q 2 +2 W (0) 3 (x, x 2, x 3 ) = de q (x ) Res q=q i y(q) y( q) B(x 2, q)b(x 3, q) q i W () (x) = de q (x) Res B(q, q), q=q i y(q) y( q) q i W (0) 4 (x, x 2, x 3, x 4 ) = de q (x ) Res q=q i y(q) y( q) q i W () 2 (x, x 2 ) = de q (x ) Res q=q i y(q) y( q) q i ( ) B(x 2, q)w (0) 3 (x 3, x 4, q) + perm(x 2, x 3, x 4 ), ( ) W (0) 3 (x 2, q, q) + 2W () (q)b(x 2, q).

20 2-cut Solutions W (0) 3 (x, x 2, x 3 ) = 2 W () (x) = χ (2) (x) + 6 q 4 i W (0) 4 (x, x 2, x 3, x 4 ) = 4 W () 2 (x, x 2 ) = 32 + j i q i [ M i M j σ j 2 M 2 i σ i χ() (x i )χ () (x i 2 )χ () (x i 3 ), q i ( ) G q σ σ i 2σ i i i χ () (x), i { 3M 2 ( i G + 2 q 3 σ i + 3σ M ) i i χ () (x i )χ () (x i 2 )χ () (x i 3 )χ () (x i 4 ) M i i ( M i M j G + 2f(q i, q j ) )( (q i q j ) 2 χ () (x i )χ () (x i 2 )χ () (x j 3 )χ () j + ) (x 4 ) + perm(x 2, x 3, x 4 ) j i +3M 2 ( i σ i χ () (x i )χ () (x i 2 )χ () (x i 3 )χ (2) (x i 4 ) + perm(x, x 2, x 3, x 4 )) }, {{ 8G 2 σ 2 i ( 2σ i 3σ i 2 M i σ i M i )G σ 2 i 2σ 2 i σ i σ ( j 3(q i q j ) 2 + 4G 2 3 σ j σ j (G + 2f(q i, q j ) ) 2χ () (q i q j ) 2 i + 4 σ j i i σ j +3χ (2) i (x )χ (2) ( i (x 2 ) + 5 χ () i (x )χ () ( 2G (x j 2 ) + σ i (x )χ (3) i (x 2 ) + (x x 2 )) }. 5σ i 8σ i 7σ i M i 6σ i M i M )( j G + 2f(q i, q j ) )]} M j (q i q j ) 2 σ i 2σ i + 2M ) ( i χ () i M i + 5M i 2M i 3M 2 i M 2 i χ () i (x )χ () i (x 2 ) (x )χ (2) ) (x i 2 ) + (x x 2 ) Notations: σ(x; q i ) := σ(x)/(x q i ), σ i := σ(q i ; q i ) σ i := 2σ (q i ; q i ), σ i := 3σ (q i ; q i ), ( ) χ (n) deq (x) (x) := Res i q=q i y(q) y( q) (q q i ) n, M i := M(q i ).

21 Our Set-up: Character variety as spectral curve We choose the character variety as the spectral curve. character variety of knot K. C = { (l, m) C C Ã K (l, m) = 0 }, Ã K (l, m 2 ) := A K (l, m). i.e. H(y, x) = Ã K (y, x). Location of D-brane On the information of D-brane we identify V = diag(ξ, ξ 2 ) ρ(µ) = diag(m, m ), m = e u. Actually this choice of D-brane locus is computed as F (g,h) (u) := F (g,h) (±u,, ±u). All signs

22 Our Set-up: 2 Expansion Parameters We identify the expansion parameters 2 g s. Therefore we compare the free energies with a fixed Euler number. Fixed Euler Number We discuss the following correspondence: S k (u) F k (u) := 2 k 2 2g+h=k+ h! F (g,h) (u)

23 Computational Results perturbative In the following, we summarize the spectral invariants on the character variety for the figure eight knot. Disk invariant: u F 0 (u) = d log x log y, A K (y, x) = 0. u This satisfies the Neumann-Zagier s relation up to constant shift. F 0 (u)/ u = log y = v, H(u) = F 0 (u) + (linear terms). Essential u-dependence is consistent with the perturbative invariant S 0 (u). Annulus invariant: 2! F (0,2) (x) = log σ(x), σ(x) = x 2 2x 2x + x 2, x = m 2. Comparing with F (u) = F (0,2) (u)/2, we recover the subleading term of the perturbative invariant S (u). (Reidemeister torsion)

24 2nd order term: perturbative The spectral invariants F (0,3) and F (,) are 3! F (0,3) (x) = 2w2 2w + 7, w := x + x, 2σ(x) 3/2 2 F (,) (x) = 8( + 6G)w3 4( + 2G)w w G 80σ(x) 3/2. G: Constant in the Bergmann kernel on 2-cut curve G = (q + q 2 )(q 3 + q 4 ) 2(q q 2 + q 3 q 4 ) 2 The function F 2 yields to E(k) K(k) (q q 2 )(q 3 q 4 ). F 2 = G 50w + 36w2 84Gw 2 + 8w Gw 3 80σ(x) 3/2.

25 3rd order term: The spectral invariants F (0,4) and F (,2) are 4! F (0,4) (x) = 25 67w + 44w2 + 24w 3 32w 4 + 6w 5 2σ(x) 3, 2! F (,2) (x) = 280w6 9088w w w w w σ(x) 3 +G 64w4 232w w w (4w 3)2 080σ(x) 2 + G 3600σ. Summing these contributions, we find F 3. [ 280w 6 448w 5 444w w w w F 3 = σ(x) 3 ] +G 64w4 232w w w (4w 3)2 080σ(x) 2 + G. 3600σ

26 .. Change of G n Unfortunately both of the contributions does not coincide because of the constant G C in the Bergmann kernel. Bergmann kernel G = (q + q 2 )(q 3 + q 4 ) 2(q q 2 + q 3 q 4 ) 2 E(k) K(k) (q q 2 )(q 3 q 4 ). But the coincidence is found by the following small changes. Change : We discard the red part which consists of the elliptic functions. G () reg = (q + q 2 )(q 3 + q 4 ) 2(q q 2 + q 3 q 4 ). 2 2 Change 2: We regularize G 2 independent of G. G 2 G (2) reg = (G() reg )2 ( k 2 )(q q 3 ) 2 (q 2 q 4 ) 2, Conjecture : By changing G n independently we will find G n G (n) reg, S k(u) = F (reg) (u). k

27 Comparing Results perturbative y(x) = 2x 2x2 x 3 + x 4 + ( x 2 ) 2x + x 2 2x 3 + x 4 2x 2 F 0 = d log x log y(x), F = 2 log 3 + 4w + 4w 2, w = x + x, 2 F (reg) G() reg 50w + 36w 2 84G () regw 2 + 8w G () regw 3 2 = 80σ(x) 3/2. F (reg) 3 = 280w6 448w 5 444w w w w σ(x) 3 We find +G () 64w 4 232w w w 243 reg 080σ(x) 2 + G (2) (4w 3) 2 reg 3600σ. S 0 = F 0 + linear, S = F, S 2 = F (reg) 2, S 3 = F (reg) 3. We also checked this coincidence for SnapPea census manifold m009 under the same assumption.

28 4. Summary Conclusions & Discussions: On hyperbolic 3-manifold side, we have shown the systematic computation of the WKB expansion of the Jones polynomial. On topological string side, we have computed the free energies on the basis of Eynard-Orantin s topological recursion. We compared S k and F k explicitly for figure eight knot case. For disk (NZ function) and annulus (Reidemeister torsion), we find the exact correspondence under our set-up. We expect that coincidence is found, if the regularizations for constant G n is assumed.

29 Future Direcrtions The higher order terms in topological recursion. [Brini] Stokes phenomenon with higher order terms (Exact WKB) [Witten],[F-Manabe-Murakami-Terashima] Abelian branch: There is another expansion point with trivial holonomy representation at l =. Different expansion is found: [ ] J n (K; q) = exp S (abel) (u) + k= k S (abel) k+ (u) K (m): Alexander polynomial Investigations on the arithmeticity conjecture [DGLZ] S (geom) n (0) K = Q(trΓ). e.g. Fig.8 case K = Q( 3)., S (abel) (u) = Toward the free fermion realization of SU(2) Chern-Simons gauge theory. K (m).

30 Back-ups

31 On Hyperbolic Geometry

32 Non-Euclidean Geometry Hyperbolic Geometry: One of non-euclidean Geometry Gauss, Boyai, and Lobachevsky found in 9th century. The parallel postulate of Euclidean geometry is not imposed. Poincaré s disk model The line is an arc of a circle orthogonal to the horocircle. If two lines are not intersecting, they are called parallel. The area A of triangle is determined by three inner angles. A = π α β γ. Vertices are located at horocircle ideal triangle (A = π)

33 Hyperbolic 3-manifold The hyperbolic 3-manifold admits a complete hyperbolic metric R ij = 2g ij. Volume w.r.t. the hyperbolic metric is finite. The hyperbolic 3-manifold is simplicially decomposed into the ideal tetrahedra. Conformal ball model Upper half space model z Geodesic Line Conformal Transformation The ideal tetrahedron is specified by the dihedral angles α, β, γ. They are toggled into a shape parameter z C.

34 . Simplicial Decomposition Simplicial decomposition of the knot complement [Yokota] Assign octahedron on each crossing of a knot Assign Octahedron 2 Reduce the number of ideal tetrahedra by Pachner 2-3 moves.

35 Example: Figure Eight Knot Complement: For figure eight knot one can assign 4 octahedron for each crossings. Octahedron Reducing the number of ideal tetrahedra, one finds that the complement is decomposed into 2 ideal tetrahedra. 3 S z z A z z C D B z z z z w w w w A' D' C' w w B' w w

36 Hyperbolic Volume The volume of each ideal tetrahedra is determined w.r.t. hyperbolic metric on H 3. (x, y) R 2, z R + ds 2 H 3 = dx2 + dy 2 + dz 2 z 2. After some elementary computations, one obtains [Milnor] Vol(T αβγ ) = Λ(α) + Λ(β) + Λ(γ) Λ(x): Lobachevsky s function Dihedral angles for each ideal tetrahedra hyperbolic volume Mostow s rigidity theorem All topological informations are determined by π (M) Dihedral angles are determined uniquely, if we solve gluing conditions. Unique hyperbolic volume.

37 Gluing Conditions There are two kinds of gluing conditions for ideal tetrahedra. Gluing conditions (bulk): Gluing conditions (boundary M T 2 ): Boundary is realized by chopping off small tetrahedra. Each triangles are glued together completely. z z A z z C D B z z z z w w w w A' C' D' w w w B' Cut around ideal points by horospheres w

38 Volume of Fig.8 Knot Complement Solving two conditions Hyperbolic volume z w a' w w z z c z z w c z z z z w z b' z a w w z w z d' w z d z w z w w z w w w c' z w z a' b w z w w z w z c z z Gluing condition along edge Red edge : zw z w z w zw = z w Blue edge : z w z w z w = (z 2 z)(w 2 w) =. Completeness condition Meridian µ: w( z) = Longitude ν: (z 2 z) 2 = Solution: α i = β i = γ i = π/3, i =, 2, Vol(S 3 \4 ) = 6Λ(π/3) = 2,

39 Incomplete Structure Generalized Neumann and Zagier discussed the deformation of the hyperbolic structure by changing the gluing condition of the boundary. (Edge condition z(z )w(w ) = is not deformed.) Meridian µ: w( z) = m 2 Longitude ν: (z/w) 2 = l 2 Dehn surgery M u has non-trivial SL(2; C) holonomy. ( ) ( ) m l ρ(µ) = 0 m, ρ(ν) = 0 l. A 4 (m, l) = m 4 m 2 2 m 2 + m 4 l l = 0. A K (m, l): A-polynomial for knot K { (l, m) C C A K (l, m) = 0 } : Character variety for knot K.

40 Complete Structure Developing map of the boundary torus: Completeness condition: p i = 0, i µ p i = 0. i ν µ: Meridian cycle, ν: Longitude cycle Deformation of the completeness condition: p i = 2u, p i = v. i µ i ν

41 Explicit Gluing Processes

42 Knot is localized at the tip of ideal tetrahedra.

43 Knot Invariants

44 Colored Jones Polynomial Volume Conjecture SU(2) Chern-Simons gauge theory: S CS [A] = k Tr(AdA + 2 4π 3 A A A). The Wilson loop operator with spin j (n = 2j + ) representation: ( )] W n (K; S 3 ) = tr n [P exp A. The colored Jones polynomial J n (K; q) is related with the Wilson loop expectation value. J n (K; q = e 4πi/(k+2) ) = W n (K; S 3 ) / W n (U; S 3 ), W n (U; S 3 ) = qn q n q q, K U : unknot.

45 Examples of Colored Jones Polynomial Trefoil 3 and figure eight knot 4 J n(3 ; q) = J n (4 ; q) = n k=0 j= n k=0 j= k ( ) k q k(k+3)/2 (q (n j)/2 q (n j)/2 )(q (n+j)/2 q (n+j)/2 ), k (q (n j)/2 q (n j)/2 )(q (n+j)/2 q (n+j)/2 ). Volume Conjecture

46 . Colored Jones Polynomial J n (K; q) Assign i h = 0,, a for each segments in K. 2 Assign R-matrix for each crossings: (a) q := q a/2 q a/2 i j i j = R ij kl ( R ) = - ij kl k l i + j= k + l l i, k j k l i + j= k + l l i, k j min(n i,j) R ij kl = m=0 min(n i,j) (R ) ij kl = m=0 3 Sum all possible i h s δ l,i+m δ k,j m (l) q!(n k) q! (i) q!(m) q!(n j) q! q (i (n )/2)(j (n )/2) m(m+)/4. δ l,i m δ k,j+m (k) q!(n l) q! (j) q!(m) q!(n i) q! ( ) m q (i (n )/2)(j (n )/2) m(i j)/2+m(m+)/4.

47 Surgery and holonomy In the topological field theory, the partition function is computed via the surgery procedure. [Atiyah] Z CS (M) = Da Z(M ; a)z(m 2 ; a). a: Gauge field on the boundary M = M 2. 3 S 3 S The holonomy ρ(µ) on M is related with( the SL(2; C) ) holomony around Wilson loop: m 0 = exp 4πj. [Murayama] k+2 W n (K; S 3 ) = DA Z k (n; S D 2 )[A] Z k (S 3 \N(K))[A] M ( M = du δ u n π ) Z k (M)[u] = Z k (M)[u 0 ]. k

48 Computation of Volume Conjecture [Kashaev],[Murakami 2 ] log J n (K, q = e 2π n ) lim = n n 2π Vol(S3 \N(K)). J n (K; q): n-colored Jones Polynomial Example: Figure 8 knot J n (4, q) = n k=0 j= Specialize to q = e 2π /n k (q (n+j)/2 q (n+j)/2 )(q (n j)/2 q (n j)/2 ). J n(4, e 2π n /n ) = (q) k 2, (q) k := L(qk+/2 ; q) k=0 L(q /2 ; q) S π n ( π n π) S π n (π 2πk/n) [ S γ (p) := exp ] e px : Faddeev integral of q-dilog. 4 sinh(πx) sinh(γx)

49 Asymptotic behavior n (γ = π n 0) [ ] S γ (p) exp 2 γ Li 2( e p ), [ n J n (4 ; e 2π ( ) dz exp Li2 (z) Li 2 (z ) )] 2π z := q k The saddle point of log J n (4, e 2π /n) z 0 = e π /3 Asymptotic value of Jones polynomial log J n (4 ; q = e 2π /n ) 2π lim n n = 2Im[Li 2 (z 0 )] = 2, = Vol(S 3 \N(K))

50 Fundamental Group and A-polynomial Generalized Wirtinger A-polynomial is determined by the fundamental group π (S 3 \K). { } π (S 3 \K) = x, y xω = ωy, ω 4 := xy x y, ω 3 := xy. The meridian and longitude holonomies are identified as µ = x, ν 4 = xy xyx 2 yx 2 yxy x, ν 3 = yx 2 yx 4, Holonomy rep. of hyperbolic mfd. ρ PSL(2; C)/Γ, Γ: discrete subgp. ( ) ( ) m l ρ(µ) = 0 m, ρ(ν) = 0 l.

51 Examples of A-polynomial Applying these holonomy representations, one finds the constraint eqnation on (l, m). A 4 (l, m) = l + l + (m 4 m 2 2 m 2 + m 4 ) = 0, A 3 (l, m) = l + m 6 = 0. Generalized Volume Conjecture

52 . Wirtinger Presentation of Knot Group Generalized A-polynomial The fundamental group for the knot complement is computed via Wirtinger presentation. The algorithm is briefly summarized as follows: For each intervals, non-commuting operators are assigned. y x xy= zx x z 2 Assign non-commuting operators x, y, z, w,... for each line segments. y x y x x z x z. z w z y w 3 Eliminate extra operators except for x and y by crossing rule.

53 Meridian and Longitude in Wirtinger Algorithm Generalized y x y x x z x z z w z y w The meridian is identified with the operator at the base point on the knot. µ = x. The longitude ν is identified with the ordered product of x i s which are assigned for the transversal interval at each crossings. ν = x ϵ i i, i:under crossings ν 4 = wx yz, ν 3 = yxzx 3.

54 Properties of A-polynomial [CCGLS] Reciprocal A K (m, l) = ±A K (/m, /l) Under the change of π ( M) basis (γ m, γ l ) l ( γl γ m m ) ( ) ( a b γl c d γ m ), A K (m, l) A K (m a l c, m b l d ) ( a b c d ) SL(2; C) Tempered Face of Newton polygon define cyclotomic polynomial in -variable

55 Logarithmic Mahler Measure The logarithmic Mahler measure for the polynomial P(z,, z n ) are defined as follows: m(p) = (2πi) n z = dz dz n log P(z,, z n ). z z n = z n Jensen s formula Let P(z) be a -parameter polynomial with complex coefficients. dz 2πi z = z log P(z) = log a 0 + log + a i, d P(z) = a 0 (z a i ), where log + x = i= { log x for x >, 0 for x <. Applying the Jensen s formula for each variable z i, one can evaluate the logarithmic Mahler measure.

56 Logarithmic Mahler Measure for A-polynomial A-polynomial is a reciprocal polynomial with 2-variable A(l, m ) = m a l b A(l, m). This property simplifies the logarithmic Mahler measure m(a) πm(a) = d i= π A(l, m) = l p m q 0 log + l k (e 2π u ) du, d (l l k (m)). i= Examples: Logarithmic Mahler measure [Boyd] where πm(a 4 ) = 2πd 3, πm(a m009 ) = 2 πd 7. d f = L (χ f, ), L(χ f, s) = χ f (n) n. s n= χ f : real odd primitive character for the discrimiant f.

57 Bianchi manifold M f : M f = H 3 /Γ, Γ = PSL(2; O Q( f) ) VolM f = f f 24 L(χ f, 2).

58 Once Punctured Torus Bundle over S Once punctured torus bundle over S is classified by the holonomy group. M(φ) = (T 2 \{0})/(x, 0) (φ(x), ). The holonomy φ has two distinct eigenvalue M(φ) admit hyperbolic structure. φ = L s R t L s 2 R t2 L sn R tn, ( ) ( 0 L = R = 0 s i, t i N ). φ = LR M(LR) = S 3 \4. φ = L 2 R M(L 2 R) =SnapPea census manifold m009.

59 3D Gravity

60 Physical meaning of the volume conjecture Einstein-Hilbert action for 3D Euclidean gravity I EH [g ij ] = d 3 x g(r 2Λ). 4π M Normalizing the cosmological constant to Λ =, the Einstein equation yields to R ij = 2g ij. (Hyperbolic 3 manifold) The same equation is also derived from the equation of motion of the following action. (st order formulation) I grav [e, ω] = Tr (e R(ω) 3 ) 2π e e e, g ij := M e a i :dreibein, ωa i :spin connection (a, i =,, 3) 3 3 e a i ea j, R(ω) := dω + ω ω, e = e a i Ta SU(2)adj dxi. a= a,i=

61 There is a topological term which gives rise to the same equation of motion. I CS [e, ω] = ( Tr ω dω e de 4π In general, the st order action yields to M I gcs = ki CS + σi grav. Let A, Ā and t, t be the linear combinations ω ω ω 2ω e e ). A := ω + e, Ā := ω e, t := k + σ, t := k σ. 3D grav.w/ neg. c.c. I gcs = t 8π + t 8π SL(2; C) Chern-Simons gauge theory Tr [A da + 23 ] A A A M M [ Tr Ā dā + 2 ] 3Ā Ā Ā.

62 Under on-shell condition, the value of the action yields to I grav [e, ω] Tre e e = Vol(M). M I CS [e, ω] CS(M). Leading terms of log Z CS grav. in the WKB expansion gives rise to the volume and Chern-Simons invariants. Classical solution of SL(2; C) Chern-Simons gauge theory The classical solution F = 0 = F is given by the holonomy representation ρ. ρ : π (M) SL(2; C) [ C ρ = P exp C ] A. Moduli space L of the solution for F = F = 0 on M = S 3 \N(K): ( L = Hom C π (S 3 \N(K)); SL(2; C) ) /Gauge equiv. { = (m, l) ( C ) } 2 AK (m, l) = 0.

63 The partition function for SL(2; C) Chern-Simons gauge theory on M is expanded perturbatively as Z gcs (M; m) = exp( S) T K (M; m) + O(/k, /σ). Geometric quantization on L Leading term S [Gukov] S = σ ( log l d(argm) + log m d(argl)), π γ + k (log m d(log m ) + argld(argm)), π γ γ: -dimensional cycle in L In the case of k = σ, the leading term simplifies. S = k log l(m)d(log m) π γ One loop term T K (M; m) is the Reidemeister torsion of the hyperbolic manifold.

64 On Hikami s Invariant

65 State Integral Model Hikami proposed a state integral model which gives a topological invariant for hyperbolic 3-manifold. This model can be seen as the SL(2; C) analogue of Turaev-Viro model. ε=+ ε= p 2 p p p 2 p p p 2 p 2 For each ideal tetrahedra the following factors are assigned. p ( ), p ( ) 2 S p (+), p (+) 2 = p ( ), p ( ) 2 S p (+), p (+) 2 = δ ( ) p ( ) + p ( ) 2 p (+) Φ (p (+) 2 p ( ) 2 + iπ + ) 4π/i [ p ( ) 2 (p (+) 2 p ( ) 2 )+ iπ 2 π2 2 ] 6 e, z = e p(+) p ( ) 2 2. ( ) δ p ( ) p (+) p (+) 2 4π/i Φ (p ( ) 2 p (+) 2 iπ ) [ p 2 e (p( ) 2 p (+) 2 ) iπ 2 + π2 2 ] 6, z = e p( ) p (+) 2 2.

66 Quantum Dilogarithm The function Φ (p) is called quantum dilogarithm [ ] e Φ (p) = exp 4 R+ xz/(πi) dx. sinh x sinh x/(πi) x This function satisfies the pentagon relation Φ (ˆp)Φ (ˆq) = Φ (ˆq)Φ (ˆp + ˆq)Φ (ˆp), [ˆq, ˆp] = 2. The perturbative expansion: [ Φ (p 0 + p) = exp B n ( 2 + p ] 2 )Li 2 n( e p 0 ) (2)n. n! n=0 Li k (z) := z n n=0, B n k n (x) = n k=0 nc k b k x n k.

67 Hikami s Invariant The partition function for the simplicially decomposed hyperbolic 3-manifold is defined by Z (M; u) = N 2 dpδ C (p; u)δ G (p) p ( ) 2i, p( ) 2i S ϵ i p (+) δ G (p) Gluing condition along edges. (z i = e ϵ i(p (+) 2i p ( ) 2i ) ) δ C (p; u): Gluing condition around meridian and longitude. p i = 2u, u = 0 Complete i µ This partition function is invariant under 2-3 Pachner moves by pentagon relation. i= 2i, p(+) 2i,

68 Saddle Point of Hikami s Invariant Now we discuss 0 limit of the partition function of the state integral model. The leading term O(/) is found by the steepest descent method. [ ] Z (M; u) dp i e V(p i ) 2, Φ (p) exp Li 2( e p ), V(p i ) = 0. z p i z i A z z C D B z z Example: Figure eight knot complement Z (S 3 \4 ; u) = eu+2πiu/ Φ (p + iπ + ) dp 2π Φ ( p 2u πi ) eu+2πiu/ dp e 2 V(p), 2π z z w w V(p) = [ Li 2 (e p ) Li 2 (e p 2u ) 4u(u + p) ]. w w A' D' C' w w B' w w

69 Saddle Point Value of Figure Eight Knot Complement The solution of the saddle point V(p; u)/ p = 0 is [ m 2 m 4 + ] 2m p 0 (u) = log 2 m 4 2m 6 + m 8 2m 3, m := e u. Complete case: For u = 0, the saddle point value yields to p 0 = e πi/3. Plugging this value into the above V(p), one finds V(p 0 ) = Li 2 (e πi/3 ) Li 2 (e πi/3 ) = 2, = Vol(S 3 \4 ). Incomplete case: The saddle point value of the potential V(p 0, u) satisfies the Neumann-Zagier s relation. v := V(p 0(u)), l = e v, u A 4 (l, m) = l + l + m 4 + m m 2 + m 4 = 0,

70 Perturbative Expansion of Hikami s Invariant [Dimofte et.al.] Utilizing the expansion of the quantum dilogarithm function, one can expand the partition function Z (M; u) w.r.t.. Z (M; u) = e 2πiu 2π = e u+v(p 0)/ 2π Υ(, p 0 + p; u) = +u dp e b2 j=0 k= dp e Υ(,p;u) 2 p2 exp Υ j, p j + Υ i,k p j k j=3 Υ j,k (p 0, u)p j k, j=0 k= b(p, u) := 2 V(p; u). p2 Neglecting O(e const/ ) Gaussian integral dp p n e b 2 p2. [ Z (M; u) = exp V(p 0) ] 2 log b + u + k S k+. k=

71 Computational Results l(m) = 2m2 2m 4 m 6 + m 8 + ( m 4 ) 2m 2 + m 4 2m 6 + m 8 2m 4, S 0 (u) = V (p 0 (u)) = log l(m), σ 0 (m) := m 4 2m 2 + 2m 2 + m 4, S (u) = 2 log b(p, u) = 2 log [ σ0 (m) 2 ], u = log m S 2 (u) = 2σ 0 (m) 3/2 m 6 ( m2 2m 4 + 5m 6 2m 8 m 0 + m 2 ). 2 S 3 (u) = σ 0 (m) 3 m 6 ( m2 2m 4 + 5m 6 2m 8 m 0 + m 2 ).

72 Topological Recursion for 2-cut

73 Bergmann Kernel for 2-cut Curve For the curve y 2 = σ(x) with 2-cuts, the Bergmann kernel is given explicitly. [Akemann][BKMP][Manabe] B(x, x 2 ) dx dx 2 = ( dx dx 2 σ(x )σ(x 2 ) + f(x, x 2 ) + G ), σ(x )σ(x 2 ) 2(x x 2 ) 2 4 f(p, q) := p 2 q 2 pq(p + q) 6 (p2 + 4pq + q 2 ) (p + q) +. G: Constant that makes B(x, x 2 ) zero A-period. G = e 3 3 E(k) K(k) (q q 2 )(q 3 q 4 ), e 3 = (q + q 2 )(q 3 + q 4 ) 2(q q 2 + q 3 q 4 ), k = (q q 2 )(q 3 q 4 ) 2 (q q 3 )(q 2 q 4 ). Regularization

74 From curve on C to C One has to change variables from C to C to discuss the mirror curve. [Marino] y(x) = a(x) ± σ(x) 2n, σ(x) := (x q i ). c(p) i= Change of variables: [ ] a(x) + σ(x) y v := log y = log. c(x) The branching structure of v is captured by the following identity: [ ] a + σ log = ( ) c 2 log a2 σ σ + tanh. c 2 a The effective curve is given by y(x) = [ ] σ(x) x tanh =: M(x) σ(x), a(x) [ ] σ(x) M(x) = x σ(x) tanh : Moment fn. a(x)

75 Results for m009

76 Once Punctured Torus Bundle over S [Jorgensen] Once punctured torus bundle over S is classified by the holonomy group. M(φ) = (T 2 \{0})/(x, 0) (φ(x), ). The holonomy φ has two distinct eigenvalue M(φ) admit hyperbolic structure. φ = L s R t L s 2 R t2 L s n R t n, ( ) ( 0 L = R = 0 s i, t i N ). Examples: φ = LR M(LR) = S 3 \4. φ = L 2 R M(L 2 R) =SnapPea census manifold m009.

77 Examples of Simplicial Decomposition The simplicial decomposition of the once punctured torus bundle over circle is performed explicitly. φ = LR case: 3 S z z A z z C D B z z z z w w w w A' D' C' w w B' w w φ = L 2 R case: z z z z w w w w w w w w z z z z w w w w w w w w

78 Gluing Conditions for m009 Gluing condition for edges Gluing conditions (boundary M T 2 ): Boundary is realized by chopping off small tetrahedra. Each triangles are glued together completely. z z z z z z z z w w w w w w w w w w w w w w w w

79 Complete Structure Developing map of the boundary torus: Completeness condition: p i = 0, i µ p i = 0. i ν µ: Meridian cycle, ν: Longitude cycle Deformation of the completeness condition: p i = 2u, p i = v. i µ i ν

80 Example: SnapPea Census Manifold m009 Z (m009; u) 6 = dp i δ C (p; u) p, p 5 S p 6, p 3 p 6, p 4 S p 2, p 5 p 3, p 2 S p 4, p i= = 2π dp dp 2 e 2 [ u(u+p +p 2 )+ 2 (p +p 2 /2) 2 π π 4 ] u Φ ( p 2u + iπ + ) Φ ( p p 2 2u πi )Φ (2p + p 2 + 2u πi ) z z z z w w w w w w w w z z z z w w w w w w w w Shape parameters & Meridian holonomy: z = e p p 2, w = e p 3 p 5, w 2 = e p 5 p 4, p 3 p 4 p + p 2 = 2u.

81 Saddle Point of Hikami s Invariant Now we discuss 0 limit of the partition function of the state integral model. The leading term O(/) is found by the steepest descent method. [ ] Z (M; u) dp i e V(p i ) 2, Φ (p) exp Li 2( e p ), Example: SnapPea census manifold m009 Z (m009; u) dp dp 2 e 2 V(p,p 2 ;u), i V(p, p 2 ) = Li 2 (e p 2u ) Li 2 (e p 2p 2 2u ) Li 2 (e 2p +2p 2 +2u ) 4u(u + p + 2p 2 ) 2(p + p 2 ) 2 + π2 6. A solution of the saddle point V(p j ; u)/ p i = 0 is [ p (0) + m 2 + m 4 + ] 2m (u) = log 2 5m 4 2m 6 + m 8 2m 3, p (0) 2 (u) = 2 log + m 2 e p (0), m := e u. m 2 ( + m 2 )e 2p(0)

82 Saddle Point Value of m009 Complete case: For u = 0 the saddle point is (e p(0), e 2p(0) 2 ) = ( 7+i 7, i 7 ). 4 2 Plugging these values into V(p (0) ), one finds, p(0) 2 V(p (0), p(0) 2 ) = i[2, i2π2 0, ] = i[vol(m009) + 2π 2 ics(m009)]. Incomplete case: The saddle point value of the potential V(p 0, u) satisfies the Neumann-Zagier s relation. Remark v := V(p 0(u), p (u)), l = e v, u A m009 (l, m) = m 2 l + m 4 l + 2m 2 + 2m 4 m 6 = 0. [Boyd-Rodriguez-Villegas] The volume is also given by the logarithmic Mahler measure Vol(m009) = πm(a m009 ) = d 7 /2.

83 Perturbative Expansion of Hikami s Invariant Utilizing the expansion of the quantum dilogarithm function, one can expand the partition function Z (M; u) w.r.t.. q-dilog Z (m009; u) = eu+ V(p (0),p(0) 2 ) 2 2π exp 2 i+j=3 b ij (p, p 2 ) := V(p, p 2 ). p i p j dp dp 2 e b p 2 +b 22 p2 2 +2b 2 p p 2 2 Υ i,j, p i pj 2 + Neglecting O(e const/ ) Gaussian integrals dp dp 2 p a b pb 2 e ij p i p j 2. [ Z (M; u) = exp V(p(0) i (u)) 2 log det b + k= i,j=0 k=0 p i pj 2 k k S k+ (u) ],.

84 Computational Results l(m) = + m2 + m 4 + 2m 2 5m 4 2m 6 + m 8 2m 3 S 0 (u) = V (p (0) (u), p(0) 2 (u)) = log l(m), σ 0(m) := m 4 2m 2 5 2m 2 + m 4, S (u) = 2 log det b(p, u) = 2 log [ σ0 (m) 2 ], u = log m S 2 (u) = 48σ 0 (m) 3/2 m 6 (5 m2 + 22m m m 8 m 0 + 5m 2 ). 2 S 3 (u) = σ 0 (m) 3 m 2 m4 ( m 2 + m 4 )( + 9m 2 + 4m 4 9m 6 + 4m 8 + 9m 0 + m 2 ). S (u) coincides with the Reidemeister torsion. [Porti]

85 Computational Results for m009 in top. string

86 2nd order term: The spectral invariants F (0,3) and F (,) are 3! F (0,3) (x) = 8w2 + 36w 2 + 6w σ(x) 3/2, w := x + x, 2 F (,) (x) = (40 72G)w3 + ( G)w 2 + ( G)w 27 47G 336σ(x) 3/2. G: Constant in the Bergmann kernel on 2-cut curve G = (q + q 2 )(q 3 + q 4 ) 2(q q 2 + q 3 q 4 ) 2 The function F 2 yields to E(k) K(k) (q q 2 )(q 3 q 4 ). F 2 = (6 + 72G)w3 + (264 56G)w 2 + (252 42G)w G 336σ(x) 3/2.

87 3rd order term: The spectral invariants F (0,4) and F (,2) are 4! F (0,4) (x) = 64w w 5 44w w w w σ(x) 3, 2! F (,2) (x) = 7862w6 6544w w w w w σ(x) 3 +G 44w4 86w w w 93 2 (6w 7)2 4704σ(x) 2 + G 2544σ(x). Summing these contributions, we find F 3. F 3 = w 09340w w w w w σ(x) 3 +G 44w4 86w w w 93 2 (6w 7)2 4704σ(x) 2 + G 2544σ(x).

88 Comparing Results y(x) = 2x 2x2 x 3 + ( x) 2x 5x 2 2x 3 + x 4 2x 2 F 0 = d log x log y(x), F = 2 log 7 4w + 4w 2, w = x + x, 2 F (reg) (6 + 72G() reg)w 3 + (264 56G () reg)w 2 + (252 42G () reg)w G () reg 2 = 336σ(x) 3/2 F (reg) 3 = w 09340w w w w w σ(x) 3 +G () 44w 4 86w w w 93 reg 4704σ(x) 2 + G (2) (6w 7) 2 reg 2544σ(x). We find S 0 = F 0 + linear, S = F, S 2 = F (reg) 2, S 3 = F (reg) 3. We also checked this coincidence for fig.8 knot complement under the same assumption.

89 Level-Rank Large n Duality Topological Vertex/CS computation Z D (x,, x n ) = R Z R Tr R V, V = diag(x,, x p ). log Z D = We have identified g=0 h= w,,w h h! g2g 2+h s F (g) w,,w h TrV w TrV w h. V = diag(x, x 2 ) ρ(µ) = diag(m, m ) SL(2; C), m = e u.

90 Motivation of Our Research Realization of 3D quantum gravity in top. string Non-perturbative completion (e.g. Witten s ECFT) Integrability (D-module structure) of knot invariants Large n duality not for rank but for level Novel class of duality

91 D-module structure in top. string

92 Large N transition [Gopakumar-Vafa],[Ooguri-Vafa] One of the famous open/closed duality in topological string is large N transition. On N D-branes, U(N) Chern-Simons gauge theory is realized. ( Z open O( ) O( ) P ; u ) = e ur U(N). W R ( ; q) R e g s = q = e 2π k+n : string coupling, t = g s N: volume of P, u: Area of disk.

93 Example: O( ) O( ) P Z(t, u) = exp[ g,h = M(Q; q) g 2g 2+h s F A g,h (t, u)] = Z closed(e t ; q) Z open (e u ; q) L(e u ; q) L(Qe u ; q). q := e g s, g s : string coupling M(x; q) := L(x; q) := ( xq n ) n : McMahon function n= ( xq n /2 ) : Quantum Dilogarithm n=

94 Mirror Symmetry [Candelas et.al.],[hosono et.al.] A-model on X B-model on X H, (X) H 2, (X ), H 2, (X) H, (X ). The bi-rational map between A-model and B-model is so-called mirror map. O B OB n classical B model O A OA n quantum A model Mirror CY of conifold X = {(z, w, x, y) C 2 (C ) 2 zw = H(x, y)}, H(x, y) = Qx y + xy, Σ := {(x, y) C C H(x, y) = 0}

95 Z open is given by a one-point function ( BA-function ) of a free fremion on Σ inside mirror CY. [ADKMV] Z open (X; u) = ψ(e u ) Σ. Schrödinger equation (conjecture): Ĥ(e ˆx, eˆp )Z open (X; u) = 0, ˆx := u g s /2, ˆp := g s u, [ˆx, ˆp] = g s, e ˆx eˆp = qeˆp e ˆx. Actually the open string partition function for conifold satisfies [ ] e gs u Qe u q /2 + (e u q /2 )(e gs u ) Z open (u) = 0.