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Around the Volume Conjecture Around the Volume Conjecture Kazuhiro Hikami Department of Mathematics, Naruto University of Education, Japan March 2009 K. Hikami (Naruto) Around the Volume Conjecture 1 / 19 Outline 1 Introduction: Volume Conjecture Volume Conjecture Colored Jones Polynomial Hyperbolic Geometry Hyperbolic Knot 2 Quantum Dilogarithm Function & Hyperbolic Geometry Partition Function Examples Figure-Eight Knot 52 Pretzel Knot 3 Mock Theta Functions Quantum Invariants for Torus Link Ramanujan Mock Theta Function A Walk Through Ramanujan’s Garden Superconformal Algebra 4 Conclusion K. Hikami (Naruto) Around the Volume Conjecture 2 / 19 Volume Conjecture 1995: Kashaev constructed knot invariant K N based on quantum dilogarithm 〈 〉 function, and proposed conjecture. 2π 3 lim log K N Vol S \ K N N |〈 〉 | = →∞ ³ ´ K : knot Vol : hyperbolic volume 2000: H.Murakami & J.Murakami proved that K N is a specific value of the 〈 〉 N-colored Jones polynomial 2πi K N JK N;q e N 〈 〉 = = ³ ´ K. Hikami (Naruto) Around the Volume Conjecture 3 / 19 Volume Conjecture 2π 2πi 3 lim log JK N;q e N Vol S \ K N N = = →∞ ¯ ³ ´¯ ³ ´ ¯ ¯ ¯ ¯ Key towards geometrical interpretation of quantum invariants. relation with classical topological invariants Outline 1 introduction to volume conjecture 2 origin of hyperbolic structure quantum dilogarithm function and ideal tetrahedron 3 torus link T(s,t) mock theta functions whose shadow is CFT character 4 related topics superconformal algebra and mock theta functions K. Hikami (Naruto) Around the Volume Conjecture 3 / 19 Colored Jones Polynomial Same knot diagrams are related by a sequence of Reidemeister Moves R I: ←→ ←→ R II: ←→ ←→ R III: ←→ Kauffman Bracket K ; 〈 〉 1 A A− = + D E ­ ® 2 2 ­ ® D A A− D 1 ⊔ = − − 〈 〉 = ­ ® ³ ´ ­ ® Jones polynomial V(K); (q A4) = w(K) V(K) A3 − K writhe : 1 : 1 = − 〈 〉 + − ³ ´ ³ ´ K. Hikami (Naruto) Around the Volume Conjecture 4 / 19 G SU(2), R N-dim rep N-colored Jones poly JK (N) = = → Colored Jones Polynomial Kauffman Bracket K ; 〈 〉 1 A A− = + D E ­ ® 2 2 ­ ® D A A− D 1 ⊔ = − − 〈 〉 = ­ ® ³ ´ ­ ® Jones polynomial V(K); (q A4) = w(K) V(K) A3 − K writhe : 1 : 1 = − 〈 〉 + − ³ ´ ³ ´ N-colored Jones polynomial (1) (2) (N 1) JK (N;q) linear combination of V ,V ,...,V − = (Jones–Wenzl projection) where V(n)(K) V n-parallel of K = K. Hikami (Naruto) ¡ Around the Volume Conjecture¢ 4 / 19 Colored Jones Polynomial Chern–Simons Theory (Witten 1989) The quantum invariant for link L aKa in 3-manifold M is =∪ L 1 Ka iS W ,..., (L) DA W (A) e R1 RL Ra = Z(M) Ãa 1 ! Z Y= K ( ) Chern–Simons action S; Wilson loop operator WR ; partition function Z M ; k 2 S Tr A dA A A A ; gauge group G = 4π ∧ + 3 ∧ ∧ ZM µ ¶ K W (A) TrR P exp A ; irreducible rep. R R = µ IK ¶ Z(M) DAeiS; quantum invariant for M = Z G SU(2), R N-dim rep N-colored Jones poly JK (N) = = → G SU(N), R fundamental rep P.T.HOMFLY poly = = → K. Hikami (Naruto) Around the Volume Conjecture 4 / 19 Hyperbolic Geometry 3D hyperbolic geometry; H3 (x,y,z,t) R4 t2 x2 y2 z2 1, t 0 = ∈ − + + + =− > © ¯ ª ¯ Poincaré model t coshξ = 2 2 2 2 2 2 x sinhξ sinθ cosϕ ds dξ sinh ξ dθ sin θ dϕ = = + + y sinhξ sinθ sinϕ ³ ´ = z sinhξ cosθ = Beltrami model X Z sinhξ sinθ cosϕ dX2 dY2 dZ2 = ds2 + + Y Z sinhξ sinθ sinϕ = 2 = Z 1 Z− coshξ sinhξ cosθ = + K. Hikami (Naruto) Around the Volume Conjecture 5 / 19 Hyperbolic Geometry Beltrami half-space model dX2 dY2 dZ2 H3 (X,Y,Z) R3 Z 0 ; ds2 + + = ∈ > = Z2 n ¯ o ¯ ¯ ∂H3 C = ∪∞ geodesics are intersections of hemisphere with vertical plane, i.e., circles/straight lines orthogonal to XY-plane Orientation preserving isometry of H3, Isom (H3) PSL(2;C); + =∼ az b a b z + ; PSL(2;C) 7→ cz d c d ∈ + µ ¶ K. Hikami (Naruto) Around the Volume Conjecture 5 / 19 Hyperbolic Geometry ideal tetrahedron ∆α,β,γ 3 every vertices of ∆α,β,γ are on ∂H α β γ π + + = dihedral angles of opposite edges are equal hyperbolic volume of ideal tetrahedron; dX dY dZ Vol ∆ , , α β γ = Z3 Ñ ¡ ¢ Z p1 X2 Y2,(X,Y) ∆ ≥ − − ∈ L(α) L(β) L(γ) = + + θ Lobachevsky function: L(θ) log 2sint dt =− | | Z0 K. Hikami (Naruto) Around the Volume Conjecture 5 / 19 Hyperbolic Geometry ideal tetrahedron ∆α,β,γ dX dY dZ Vol ∆ , , α β γ = Z3 Ñ ¡ ¢ Z p1 X2 Y2,(X,Y) ∆ ≥ − − ∈ L(α) L(β) L(γ) D(z) = + + = z z[1] z = [ ] 1 L(v) z 2 z[2] 1 z− ≃ = − 1 z[1] z[3] z[3] (1 z)− 0 1 = − Bloch–Wigner function: D(z) Li2(z) arg(1 z) log z =z ℑn +z log(1− s)· | | ( ) ∞ − Euler dilogarithm: Li2 z 2 ds = n 1 n =− 0 s X= Z K. Hikami (Naruto) Around the Volume Conjecture 5 / 19 Hyperbolic Knot hyperbolic knot K is a knot whose complement S3 \ K can be given a metric of negative constant curvature 1. 3 − 3 H /Γ is hyperbolic manifold; Γ [repr of π1(S \ K)] is discrete subgrp of PSL(2;C) non-hyperbolic knots are either torus knots or satellite knots (Thurston). K. Hikami (Naruto) Around the Volume Conjecture 6 / 19 Hyperbolic Knot hyperbolic knot K is a knot whose complement S3 \ K can be given a metric of negative constant curvature 1. − non-hyperbolic knots are either torus knots or satellite knots (Thurston). almost all hyperbolic knots K are distinguished by hyperbolic volumes 3 S \ K is decomposed into ideal tetrahedra with zi (gluing faces together); 3 Vol S \ K D(zi) = i ³ ´ X dihedral angles around every edges sum to 2π (consistency) α [ ] , , ij βij zi eij 1 i.e. zi 1 zi 1 i around edge j = i − = ± Y Y eij {1,2,3} ¡ ¢ ∈ developing map of cusp is torus whose fundamental domain is quadrilateral (completeness) K. Hikami (Naruto) Around the Volume Conjecture 6 / 19 Hyperbolic Knot A-polynomial of K deformation parameter u from complete hyperbolic structure of S3 \ K; longitude λ & meridian µ of tubular neighbour of K 3 CCGLS (1994): A-polynomial AK (ℓ,m) is SL(2;C)-character variety of π1(S \ K). ℓ m ρ(λ) ∗ , ρ(µ) ∗ = 0 1/ℓ = 0 1/m µ ¶ µ ¶ Neumann–Zagier (1985): There exists potential function ΦK (u) s.t. 1 ∂Φ v K = 2 ∂u u logm, v logℓ = = K. Hikami (Naruto) Around the Volume Conjecture 6 / 19 Hyperbolic Knot 2π 2πi 3 Volume Conjecture: lim log JK N;q e N Vol S \ K N N = = →∞ ¯ ³ ´¯ ³ ´ Generalized Volume Conjecture (Gukov,¯ H.Murakami,¯ ...) ¯ ¯ pair eb, eia is a zero locus of the A-polynomial for knot K. − ³ ´ d 1 2πi/k b lim logJK N;e = − da N,k k N/k→∞a ³ ´ = K. Hikami (Naruto) Around the Volume Conjecture 6 / 19 Figure-Eight Knot S3\ = Glue faces together to match orientation Developing map of cusp consistency & completeness 1 1 1 2 w2 1 1 w w w w 1 w − z 1 z = − µ ¶ µ − ¶ z z w z z z = ⋆Vol 2D(eπi/3) 2.02988 = = ··· K. Hikami (Naruto) Around the Volume Conjecture 7 / 19 Figure-Eight Knot 1 2 D0 D 1 4 2 D1 D1 : ⇐⇒ 3 D2 D 3 3 D 5 1 2 4 D4 : ⇐⇒ octahedron@crossing K. Hikami (Naruto) Around the Volume Conjecture 7 / 19 Quantum Dilogarithm Function Question::::::::: origin of hyperbolic geometry? Kashaev used quantum dilog at root of unity to construct K N 〈 〉 we treat quantum dilog at generic value, in integral form rather easy to study asymptotics relation with hyperbolic ideal tetrahedron & Neumann-Zagier function Quantum Dilog (Faddeev 1999) iϕx e− dx Φγ(ϕ) exp = 4sinh(γx)sinh(πx) x RZi0 + quantum discrete KdV, Volterra (Volkov) quantum Liouville theory (Ponsot–Teschner) Baxter Q-operator quantum Teichmüller space (Kashaev, Chekhov–Fock) circle packing (Bazhanov–Mangazeev–Stroganov) K. Hikami (Naruto) Around the Volume Conjecture 8 / 19 Quantum Dilogarithm Function Quantum Dilog (Faddeev 1999) iϕx e− dx Φγ(ϕ) exp = 4sinh(γx)sinh(πx) x RZi0 + Φ Φ γ duality : π2 (ϕ) γ ϕ γ = π ³ ´ 1 ϕ classical limit : Φγ(ϕ) exp Li2( e ) γ∼0 2iγ − → µ ¶ Φγ(ϕ iγ) 1 Φγ(ϕ iπ) 1 difference eq : + , + Φ ϕ Φ π ϕ γ(ϕ iγ) = 1 e γ(ϕ iπ) = 1 e γ − + − + 1 ϕ2 π2 γ2 inversion : Φγ(ϕ) Φγ( ϕ) exp + · − = − 2iγ 2 + 6 µ ³ ´¶ K. Hikami (Naruto) Around the Volume Conjecture 8 / 19 Quantum Dilogarithm Function pentagon identity: with canonical variables [p , q] 2iγ =− Φ (p)Φ (q) Φ (q)Φ (p q)Φ (p) γ γ = γ bγ b+ γ 1 qˆ1 pˆ2 Symmetric expression: By useb of Sb1,2 e 2iγb Φb (pˆb1 qˆ 2b pˆ 2) = γ + − S2,3 S1,2 S1,2 S1,3 S2,3 = momentum space; p p p p . In γ 0, | 〉 = | 〉 → 1 V(p′ p2,p1) p1,p2 S1,2 p′ ,p′ δ(p1 p2 p′ ) e− 2iγ 2− 〈 b| | 1 2〉 ∼ + − 1 · 1 1 V(p2 p′ ,p′ ) p1,p2 S− p′ ,p′ δ(p1 p′ p′ ) e 2iγ − 2 1 〈 | 1,2 | 1 2〉 ∼ − 1 − 2 · 2 π x V(x,y) Li2(e ) xy = 6 − − saddle point: ∂ ∂ V(x,y) D(1 ex) log ex V(x,y) log ey V(x,y) ℑ = − + | |·ℑ ∂x + | |·ℑ ∂y ³ ´ ³ ´ K. Hikami (Naruto) Around the Volume Conjecture 8 / 19 Quantum Dilogarithm Function S2,3 S1,2 S1,2 S1,3 S2,3 = Topology: usual interpretation of pentagon is Pachner move; a b e cf. c d f ½ ¾ 6j-symbol (Penrose, Ponzano–Regge) quantum 6j-symbol (Turaev–Viro) d b a b 1 a d a,b S c,d a,b S− c,d 〈 | | 〉 = c = c D ¯ ¯ E ¯ ¯ ¯ ¯ K.
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