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Volume Conjecture and Topological Recursion

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Volume Conjecture and Topological Recursion . Volume Conjecture and Topological Recursion .. Hiroyuki FUJI . Nagoya University Collaboration with R.H.Dijkgraaf (ITFA) and M. Manabe (Nagoya Math.) 6th April @ IPMU 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, to appear. Hiroyuki FUJI Volume Conjecture and Topological Recursion . 1. Introduction Asymptotic analysis of the knot invariants is studied actively in the knot theory. Volume Conjecture [Kashaev][Murakami2] Asymptotic expansion of the colored Jones polynomial for knot K ) The geometric invariants of the knot complement S3nK. S3 Recent years the asymptotic expansion is studied to higher orders. 2~ Sk(u): Perturbative" invariants, q = e [Dimofte-Gukov-Lenells-Zagier]# X1 1 δ k Jn(K; q) = exp S0(u) + log ~ + ~ Sk+1(u) ; u = 2~n − 2πi: ~ 2 k=0 Hiroyuki FUJI Volume Conjecture and Topological Recursion Topological Open String Topological B-model on the local Calabi-Yau X∨ { } X∨ = (z; w; ep; ex) 2 C2 × C∗ × C∗jH(ep; ex) = zw : D-brane partition function ZD(ui) Calab-Yau manifold X Disk World-sheet Labrangian Embedding submanifold = D-brane Ending on Lag.submfd. Holomorphic disk Hiroyuki FUJI Volume Conjecture and Topological Recursion Topological Recursion [Eynard-Orantin] Eynard and Orantin proposed a spectral invariants for the spectral curve C C = f(x; y) 2 (C∗)2jH(y; x) = 0g: • Symplectic structure of the spectral curve C, • Riemann surface Σg,h = World-sheet. (g,h) ) Spectral invariant F (u1; ··· ; uh) ui:open string moduli Eynard-Orantin's topological recursion is applicable. x x1 xh x x x x x x 1 h x x k1 k i1 ij hj q q q = q + Σ l J g g 1 g l l Σg,h+1 Σg−1,h+2 Σℓ,k+1 Σg−ℓ,h−k Spectral invariant = D-brane free energy in top. string [BKMP] Hiroyuki FUJI Volume Conjecture and Topological Recursion Correspondences Heuristically we discuss a relation between the perturbative (g,h) invariants Sk(u) and the free energies F (u1; ··· ; uh) ´ala BKMP. [Dijkgraaf-F.] 3D Geometry Topological Open String { Character variety } { Spectral curve } ∗ ∗ p x ∗ ∗ p x (`; m) 2 C × C jA~ K(`; m) = 0 (e ; e ) 2 C × C jH(e ; e ) = 0 u = log m: Holonomy u: Open string moduli Neumann-Zagier fn. H(u)/2 Disk Free Energy F¯(0,1)(u) Reidemeister Torsion T (M; u) Annulus Free Energy F¯(0,2)(u) ~ q = e2 q = egs In this talk we will explore the following relation: X k−2 1 (g,h) Sk(u) $ Fk(u) = 2 F¯ (u): h! 2g+h=k+1, h≥0 Hiroyuki FUJI Volume Conjecture and Topological Recursion 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 SU(n)CS Top.A-model 2 non-cpt D-branes Large N unknot Transition n cpt D-branes 2 non-cpt D-branes T * L(p,1) p=3 for fig.8 Level-rank? mirror symmetry SU(2)CS Top.B-model ... A K =H spin j→∞ Our Proposal n=2j+1 S3 K Hiroyuki FUJI Volume Conjecture and Topological Recursion CONTENTS 1. Introduction 2. Volume Conjecture and Perturbative Invariants 3. Topological Recursion on Character Variety 4. Summary, Discussions and Future Directions Hiroyuki FUJI Volume Conjecture and Topological Recursion . 2. Volume Conjecture and Perturbative Invariants Volume Conjecture [Kashaev][Murakami2] In 1997 Kashaev proposed a striking conjecture on the asymptotic expansion of the colored Jones polynomial Jn(K; q). 2πi/n log jJn(K; e )j 2π lim = Vol(S3nK): n→∞ n S 3 Knot complement = S3nN(K) N(K): Tubular neighborhood of a knot K. The hyperbolic knot complement admits a hyperbolic structure. Hiroyuki FUJI Volume Conjecture and Topological Recursion . Generalized Volume Conjecture In 2003, Gukov generalized the volume conjecture to 1-parameter version. (u+2πi)/n log Jn(K; e ) (u + 2πi) lim = H(u); u 2 C: n→∞ n H(u): Neumann-Zagier's potential function @H(u) = v + 2πi: @u u and v satisfies an algebraic equation. v u AK(`; m) = 0; ` = e ; m = e : AK(`; m): A-polynomial for a knot K. incomplete ) Up to linear term of u and v, the Neumann-Zagier potential H(u) yields to Z u+2πi H(u) = du v(u) + linear terms: 2πi Hiroyuki FUJI Volume Conjecture and Topological Recursion AJ conjecture and higher order terms In 2003 Garoufalidis proposed a conjecture on q-difference equation for the colored Jones polynomial. (Quantum Riemann Surface) AK(`;^ ^m;q)Jn(K; q) = 0; AK(`; m; q = 1) = (` − 1)AK(`; m): `^f(n) = f(n + 1); ^mf(n) = qn/2f(n); `^^m= q1/2 ^m`:^ ν longitude μmeridian Commutation relation of the Chern-Simons gauge theory [I ] [I ] ρ(µ) = P exp A ; ρ(ν) = P exp A ; µ ν a b 2π ab 2 fA (x); A (y)g = δ ϵαβδ (x − y): α β k Meridian µ and longitude ν intersect at one point. 2π `^^m= q1=2 ^m`^ ) [^u; ^v] = : (θ = vdu;! = dθ:) k Hiroyuki FUJI Volume Conjecture and Topological Recursion q-difference Equation for Fig.8 Knot Example: Figure 8 knot 41 [Garoufalidis] A (ℓ,^ ^m;q) 41 q5 ^m2(−q3 + q3 ^m2) = (q2 + q3 ^m2)(−q5 + q6 ^m4) (q2 − q3 ^m2)(q8 − 2q9 ^m2 + q10 ^m2 − q9 ^m4 + q10 ^m4 − q11 ^m4 + q10 ^m6 − 2q11 ^m6 + q12 ^m8) − ℓ^ q5 ^m2(q + q3 ^m2)(q5 − q6 ^m4) (−q + q3 ^m2)(q4 + q5 ^m2 − 2q6 ^m2 − q7 ^m4 + q8 ^m4 − q9 ^m4 − 2q10 ^m6 + q11 ^m6 + q12 ^m8) + ℓ^2 q4 ^m2(q2 + q3 ^m2)(−q + q6 ^m4) q4 ^m2(−1 + q3 ^m2) + ℓ^3. (q + q3 ^m2)(q − q6 ^m4) AJ conjecture for Wilson loop Wn(K; q) := Jn(K; q)Wn(U; q); A~K(`;^ ^m;q)Wn(K; q) = 0: q-difference equation is factorized. ~ ^ 1/2 ^− ^ ^ A41 (ℓ, ^m;q) = (q ℓ 1)A41 (ℓ, ^m;q), q ^m2 (−1 + q ^m2)(1 − q ^m2 − (q + q3) ^m4 − q3 ^m6 + q4 ^m8) A^4 (ℓ,^ ^m;q) = − ℓ^ 1 (1 + q ^m2)(−1 + q ^m4) q1/2 ^m2(−1 + q ^m4)(−1 + q3 ^m4) q2 ^m2 + ℓ^2. (1 + q ^m2)(−1 + q3 ^m4) Hiroyuki FUJI Volume Conjecture and Topological Recursion Perturbative Invariants WKB expansion of the Wilson loop expectation value: " # X∞ 1 δ k−1 Wn(K; q) = exp S0(u) + log ~ + ~ Sk(u) ; ~ 2 k=1 q := e2~; qn = m = eu: Applying this expansion into q-difference equation, one finds a hierarchy of differential equations : ∑d ∑∞ j k A^K(ℓ, m; q) = ℓ ~ aj,k(m). k=0 k=0 d ∑ 0 jS e 0 aj,0 = 0, ← A − polynomial j=0 d [ ( )] ∑ 0 jS 1 2 ′′ ′ e 0 aj,1 + aj,0 j S + jS = 0, 2 0 1 j=0 d [ ( ) ( )] ∑ 0 jS 1 2 ′′ ′ 1 1 2 ′′ ′ 2 1 3 ′′′ 1 2 ′′ ′ e 0 aj,2 + aj,1 j S + jS + aj,0 ( j S + jS ) + j S + j S + jS = 0, 2 0 1 2 2 0 1 6 0 2 1 2 j=0 ··· . Hiroyuki FUJI Volume Conjecture and Topological Recursion . Computational Results top1 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] √ 1 − 2m2 − 2m4 − m6 + m8 + (1 − m4) 1 − 2m2 + m4 − 2m6 + m8 ℓ(m) = , 2m4 S′ (u) = log ℓ(m), 0 [ √ ] 1 σ0(m) −4 −2 2 4 S1(u) = − log , σ0(m) := m − 2m + 1 − 2m + m , 2 2 −1 S (u) = (1 − m2 − 2m4 + 15m6 − 2m8 − m10 + m12), 2 3/2 6 12σ0(m) m 2 S (u) = (1 − m2 − 2m4 + 5m6 − 2m8 − m10 + m12). 3 3 6 σ0(m) m S1(u) coincides with the Reidemeiser torsion. [Porti] " # X3 1 0 T(M; u) = exp − n(−1)n log det ∆Eρ : 2 n n=0 Eρ Eρ: flat line bdle, ∆n : Laplacian on n-forms. Hiroyuki FUJI Volume Conjecture and Topological Recursion . 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) 2 C∗ × C∗jH(y; x) = 0 : • Free energies for closed world-sheet: Symplectic invariants F (g,0) • Free energies for world-sheet with boundaries: (g,h) Spectral invariants F (u1; ··· ; uh), ui: open string moduli (g) These free energies are integrals of the meromorphic forms Wh . Z Z eu1 euh [Bouchard-Klemm-Marino-Pasquetti] F (g,h) ··· ··· (g) ··· (u1; ; uh) = ∗ dx1 ∗ dxh W (x1; ; xh): u u h e 1 e h Hiroyuki FUJI Volume Conjecture and Topological Recursion D-brane Partition Function The D-brane partition function is defined by X ZD(ξ1; ··· ; ξn) = ZRTrRV R ∞ ∞ X X X 1 2g−2+h (g) w1 wh log ZD = g F ··· TrV ··· TrV ; h! s w1, ,wh g=0 h=1 w1,··· ,wh V = diag(ξi; ··· ξn) ξi (i = 1; ··· n) are location of non-compact D-brane in X. ξ non-compact ξ D-branes ξ Hiroyuki FUJI Volume Conjecture and Topological Recursion BKMP's Free Energy and D-brane Partition Function D-brane partition function is related with BKMP's free energies (g,h) F (u1; ··· ; uh). [Marino] Dictionary w1 ··· wh $ w1 ··· wh ui TrV TrV x1 xh ; xi = e : Identification of the free energy: X 1 (g,h) (g) w1 wh F (u1; ··· uh) = F ··· x ··· x : h! w1, wh 1 h w1,··· ,wh Hiroyuki FUJI Volume Conjecture and Topological Recursion Topological Recursion Eynard-Orantin's topological recursion (2g + h ≥ 3): W(g) (x; x ; ··· ; x ) h+1 1 h " X dE (x) q (g−1) ··· = Res Wh+2 (q; ¯q; x1; ; xh) q=qi y(q) − y(¯q) xi # Xg X (g−`) (`) + WjJj+1 (q; pJ)WjH|−|Jj+1(¯q; pHnJ) : `=0 J⊂H q, ¯q: points x = q on the 1st sheet and 2nd sheet of the spectral curve q : End points of cuts in the double covering of C.
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