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

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. Hikami (Naruto) Around the Volume Conjecture 8 / 19 Quantum Dilogarithm Function

S2,3 S1,2 S1,2 S1,3 S2,3 = Geometry: pentagon identity in γ 0 is now → 1 π2 p′ p3 p′ p2 p3 exp Li2(e 3− ) Li2(e 2− − ) p2 (p′ p3) p1 (p′ p2 p3) 2iγ − 3 + + + 3 − + 2 − − · ¸ ³ 2 ´ 1 π z p2 p p3 z dz exp Li2(e − ) Li2(e 2′ − − ) ∼ 2iγ − 2 + + Z · ³ p p z Li2(e 3′ − 2′ + ) z p2 p′ p′ z p1 p2 p′ p3 (p1 p2) + + − + 3 − 2 + − + 2 − + ¸ ¡ ¢ ¡ ¢ ´

We apply saddle point method; saddle point condition is

1 p′ z p3 − p2 z p′ z p′ consistency 1 e 2− − 1 e − 1 e 2− − 3 1 − − − = ←→ (gluing around center axis) ³ ´ ¡ ¢ ³ ´

K. Hikami (Naruto) Around the Volume Conjecture 8 / 19 Quantum Dilogarithm Function

S2,3 S1,2 S1,2 S1,3 S2,3 = ideal hyperbolic tetrahedra

d a z[2] a b z[1] a,b S c,d z[3] z[2]

b 〈 | | 〉 = c = z[1] z[3] c z[3] z[2] z[1] 0 1

d b

b z[1] z e − a z[2] = = 1 a d z[1] [ ] 1 a,b S− c,d z[3] z 2 1 z− d = − = c = z[1] z[3]  1 D ¯ ¯ E z[3] (1 z)− ¯ ¯ c = − ¯ ¯ z[2]  Our Claim: saddle point condition consistency condition ←→

K. Hikami (Naruto) Around the Volume Conjecture 8 / 19 Partition Function

hyperbolic cusped 3-manifold is constructed from oriented ideal tetrahedra;

M ( ) ( ; ) ( ) ( ) , ( ) εi ( ) , ( ) Zγ Mu dpδC p u δG p p2−i 1 p2−i S p2+i 1 p2+i = i 1 − − Ï = D ¯ ¯ E Y ¯ ¯ ¯ ¯ δG: gluing condition u δC: completeness m e (u: deformation parameter) = cf. Dimofte–Gukov–Lenells–Zagier [arXiv:0903.2472]

K. Hikami (Naruto) Around the Volume Conjecture 9 / 19 Partition Function

hyperbolic cusped 3-manifold is constructed from oriented ideal tetrahedra;

M ( ) ( ; ) ( ) ( ) , ( ) εi ( ) , ( ) Zγ Mu dpδC p u δG p p2−i 1 p2−i S p2+i 1 p2+i = i 1 − − Ï = D ¯ ¯ E Y ¯ ¯ ¯ ¯ δG: gluing condition u δC: completeness m e (u: deformation parameter) =

In the classical limit γ 0, →

M (ε ) ( ε ) ( ε ) i i − i , − i (εi) ( εi) ( εi) 2γ i εi V p2i p2i p2i 1 Zγ (Mu) dpδC(p;u)δG(p) δ p p − p − e − − 2i 1 2i 1 2i ³ ´ ∼ "i 1 − − − − # P Z = ³ ´ 1 Y dx exp ΦM(x;u) = 2iγ Z µ ¶

K. Hikami (Naruto) Around the Volume Conjecture 9 / 19 Partition Function

In the classical limit γ 0, →

Neumann–Zagier function

∂ ΦM(x;u) 0 saddle point (consistency) ∂xi = ∂ ΦM(x;u) 2v 2logℓ u 0: completeness ∂u = ≡ =

K. Hikami (Naruto) Around the Volume Conjecture 9 / 19 Figure-Eight Knot

1 P 2 P 3 P P 2 P P 4 Zγ S \  Zγ  4 3  = 3 P 1 P         1 dpδC(p;u) p1,p2 S p3,p4 p4,p3 S− p2,p1 = 〈 | | 〉 Z D ¯ ¯ E ¯ ¯ Completeness condition δC can be read from developing map ¯ ¯

w w w w p4 p2 z z w e − = p1 p3 z z z e − ½ =

w 2u e− , i.e. p4 p2 (p1 p3) 2u −→ z = − − − =−

K. Hikami (Naruto) Around the Volume Conjecture 10 / 19 Figure-Eight Knot

1 P 2 P 3 P P 2 P P 4 Zγ S \  Zγ  4 3  = 3 P 1 P         1 dpδC(p;u) p1,p2 S p3,p4 p4,p3 S− p2,p1 = 〈 | | 〉 Z D ¯ ¯ E ¯ ¯ γ 0 1 x x 2¯u ¯ → dx exp Li2(e ) Li2(e− − ) 4u(u x) ∼  2iγ  − − +  Z Φ   Neumann–Zagier (x;u)     Conditions | {z }

0 1 0 ∂ x Φ(x;u) 0 1 0 1 0 2 2 − ∂xi = − m (1 x)(1 m x) =  1 2 1  ∂ ⇔  1− − ⇔ − − −  Φ(x;u) 2logℓ  ℓ  0 1 0    m2 x (m2 x 1) =   ∂u =  −  −  0 1 0     

K. Hikami (Naruto) Around the Volume Conjecture 10 / 19 52

 3  Zγ S \           1 1 1 dpδC(p;u) p1,p5 S− p4,p3 p2,p4 S− p6,p5 p3,p6 S− p1,p2 = ZR D ¯ ¯ ED ¯ ¯ ED ¯ ¯ E ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯

z1 z3 z3 z1 p3 p5 z1 e − = z2 z1 p5 p4 z2 e − z z2 = 1  p2 p6 z2 z2 z3 z3 e − z3 = 

p5 p4 p6 p2 2u − + − = K. Hikami (Naruto) Around the Volume Conjecture 11 / 19 52

 3  1 x y u Zγ S \ dx dy exp V(e ,e ;e ) ∼ 2iγ   Ï µ ¶        2  π y 1 1 2 V(x,y;m) Li2 Li2 Li2 log y/x log m /y = 2 − 2 − 2 − y + µ x m ¶ µ y m ¶ µ ¶ ¡ ¢ ³ ´ 1 1 − 2  −  2 1  − −   1  3 A(ℓ,m) :   , Vol(S \ K) 2.82812  1  = ···    1 2   − −   2   −   1 1  −   

K. Hikami (Naruto) Around the Volume Conjecture 11 / 19 Pretzel Knot ( 2,3,7) −

  3 Zγ S \                dpδC(p;u) p4,p7 S p6,p1 p5,p8 S p7,p5 p1,p6 S p8,p2 p3,p2 S p4,p3 = 〈 | | 〉〈 | | 〉〈 | | 〉〈 | | 〉 Ñ

z 3 p1 p7 p5 p8 z1 e − z2 e − = = z ep2 p6 z ep3 p2 z4 ( 3 − 4 − z 2 z3 z = = 1 z2 z1 p p p p 2u z 2 2 6 1 7 z2 − − + =− z3 z3 z 1 z1 z4 z4

K. Hikami (Naruto) Around the Volume Conjecture 12 / 19 Pretzel Knot ( 2,3,7) −

  3 1 x y z u Zγ S \  dx dy dz exp V(e ,e ,e ;e )   ∼ 2iγ   Ñ µ ¶           2π2 1 1 1 m2 VM (x,y,z;m) Li2 Li2 Li2 Li2 =− 3 + z + 4 + 2 + x µ ¶ µ xyzm ¶ µ zm ¶ µ ¶ 2 y5 3 log m2 log log m2 logx 2 logy 2 + + 2 + + µ x ¶ ³ ³ ´´ ³ ´ ¡ ¢ ¡ ¢ 1 m16 2m18 m20 ℓ − + − + A(ℓ,m) : 2m36 m38 ℓ2 ℓ4 m72 2m74 Vol(S3 \ K) 2.82812 + ¡ + − ¢ + = ··· ℓ5 m90 2m92 m94 m110 ℓ6 −¡ − ¢ + ¡ + ¢ ¡ ¢

K. Hikami (Naruto) Around the Volume Conjecture 12 / 19 Torus Link T(s,t)

non-hyperbolic #component gcd(s,t) = Trefoil T(2,3) Solomon’s Seal T(2,5) T(2,4)

K. Hikami (Naruto) Around the Volume Conjecture 13 / 19 Kashaev invariant T(s,t) for Torus Link 〈 〉N

n ∞ j 1 Kontsevich (1997): F(q) (q)n (x)n (x;q)n 1 xq − = n 0 = = j 1 − X= Y= ³ ´ n 2 i πi 1 π 3/2 N 3 ∞ bn 2πi F(e N ) N e− 12 − + N ; b0,1,... 1,1,3,19,207,... N ∼ ³ ´ + n! − N = →∞ n 0 µ ¶ X= Zagier (1999): nearly modularity of F(q) F(q) as period function (Eichler integral) of η(τ).

We know that T(2,3) N qF(q) 2πi 〈 〉 = |q e N =

T(s,t) N: a limiting value of Eichler integral of modular form (“shadow”). 〈 〉 “shadow” is related to character of CFT link CFT T(s,t) minimal model M(s,t) T(2,2P) SU(2)P 2 −

K. Hikami (Naruto) Around the Volume Conjecture 14 / 19 Kashaev invariant T(s,t) for Torus Link 〈 〉N

We show T(2,2P) for P 2 ≥ wt-3/2 vector modular form q e2πiτ; τ H = ∈ 1 2 1 for n a mod 2P Ψ(a)( ) (a)( ) n /4P; (a)( ) P τ nψ2P n q ψ2P n ± =± = 2 n Z = (0 otherwise X∈

3/2 P 1 2 (a) τ − 2 ab (b) (a) a πi (a) Ψ ( 1/τ) sin π Ψ (τ); Ψ (τ 1) e 2P Ψ (τ) P − = i P P P P + = P b 1s µ ¶ ³ ´ X=

K. Hikami (Naruto) Around the Volume Conjecture 14 / 19 Kashaev invariant T(s,t) for Torus Link 〈 〉N wt-3/2 vector modular form q e2πiτ; τ H = ∈ 1 2 1 for n a mod 2P Ψ(a)( ) (a)( ) n /4P; (a)( ) P τ nψ2P n q ψ2P n ± =± = 2 n Z = (0 otherwise X∈

3/2 P 1 2 (a) τ − 2 ab (b) (a) a πi (a) Ψ ( 1/τ) sin π Ψ (τ); Ψ (τ 1) e 2P Ψ (τ) P − = i P P P P + = P b 1s µ ¶ ³ ´ X= Eichler integral

2 Ψ(a)( ) ∞ (a)( ) n /4P P τ ψ2P n q = n 0 = 2PN X 2 (a) e (a) π k k 1 Limiting value is: Ψ (1/N) ψ (k)e 2PN B1 ; B1(x) x P =− 2P 2PN = − 2 k 0 µ ¶ X= e (P 1)2 − πi (P 1) Kashaev invariant: T(2,2P) PNe− 2PN Ψ − (1/N) 〈 〉N = P K. Hikami (Naruto) Around the Volume Conjecture e 14 / 19 Kashaev invariant T(s,t) for Torus Link 〈 〉N wt-3/2 vector modular form q e2πiτ; τ H = ∈ 1 2 1 for n a mod 2P Ψ(a)( ) (a)( ) n /4P; (a)( ) P τ nψ2P n q ψ2P n ± =± = 2 n Z = (0 otherwise X∈

3/2 P 1 2 (a) τ − 2 ab (b) (a) a πi (a) Ψ ( 1/τ) sin π Ψ (τ); Ψ (τ 1) e 2P Ψ (τ) P − = i P P P P + = P b 1s µ ¶ ³ ´ X= Eichler integral

2 Ψ(a)( ) ∞ (a)( ) n /4P P τ ψ2P n q = n 0 X= e (a) P 1 L 2k,ψ k (a) N − 2 ab (b) ∞ 2k πi Ψ (1/N) sin π Ψ ( N) − P + i P P P − ≃ ³ k! ´ 2PN s b 1 s µ ¶ k 0 µ ¶ X= X= e e K. Hikami (Naruto) Around the Volume Conjecture 14 / 19 Mock Theta Function

Ramanujan’s last letter to Hardy (Jan. 12, 1920) 17 mock theta functions (4 3rd order / 10 5th order / 3 7th order) G.N. Watson: “The Final Problem” (1936) Transformation formulae for 3rd order “Lost Notebook” (1976, published in 1988)

3rd order mock theta function f(q)

3 1 1 3 2 2cosh βy cosh βy π 24 24β ∞ βy 2 q− 24 f( q) q − f( q1) e− 2 dy − + α 1 − = s α cosh 3βy r Z0 ¡ ¢ ¡ ¢ Mordell integral¡ ¢ 2 qn | {z } ( ) ∞ ; e α, 2, e β f q 2 q − αβ π q1 − = n 0 [( q)n] = = = X= −

K. Hikami (Naruto) Around the Volume Conjecture 15 / 19 Mock Modular Form

Zwegers (2002) harmonic Maas form (wt-1/2) [holomorphic part] [non-holomorphic part] = + £ “harmonic Maass form” with¤ weight k (cf. Bringmann–Ono) 2 2 2 ∆ f 0; ∆ y ∂ ∂ iky ∂x i∂y , τ x iy k = k = − x + y + + = + ³ ´ ¡ a b ¢ f(γ(τ)) ρ(γ) (cτ d)k f(τ); γ SL(2;Z) = + = c d ∈ µ ¶ “holomorphic part”: qλ mock theta function × “non-holomorphic part”: shadow is wt-3/2 modular form Φ(τ) £ ¤ i Φ 2 ∞ ( z) φn n − dz β1/ (4ny)q− s i p = p 2 Z τ z τ n 0 n − + X>

K. Hikami (Naruto) Around the Volume Conjecture 16 / 19 Mock Modular Form

Zwegers (2002) harmonic Maas form (wt-1/2) [holomorphic part] [non-holomorphic part] = + £ “holomorphic part”: qλ mock¤ theta function × “non-holomorphic part”: shadow is wt-3/2 modular form Φ(τ) £ ¤ i Φ 2 ∞ ( z) φn n − dz β1/2(4ny)q− s i p = pn Z τ z τ n 0 − + X> View from quantum topology

Kashaev invariants K N are limiting values of mock modular form whose shadow is character〈 of〉 CFT.

torus link shadow weight link shadow T(s,t) M(s,t) 1/2 ex. T(2,3) η(τ) 3 T(2,2P) SU(2)P 2 3/2 T(2,4) η(τ) − Witten–Reshetikhin–Turaev invariant for Seifert manifolds as£ a limiting¤ value of mock theta function. K. Hikami (Naruto) Around the Volume Conjecture 16 / 19 “A walk through Ramanujan’s Garden”

Freeman J. Dyson (June 2, 1987) The mock theta-functions give us tantalizing hints of a grand synthesis still to be discovered. Somehow it should be possible to build them into a coherent group-theoretical structure, analogous to the structure of modular forms which Hecke build around the old theta-functions of Jacobi. This remains a challenge for the future. My dream is that I will live to see the day when our young physicists, struggling to bring the predictions of superstring theory into correspondence with the facts of nature, will be led to enlarge their analytic machinery to include not only theta-functions but mock theta-functions.

K. Hikami (Naruto) Around the Volume Conjecture 17 / 19 Superconformal Algebra

N 4 SCFT (c 6k): string compactiﬁcation on K3 (k 1) = = = a b Ln: energy momentum tensor Gr , Gs (a,b 1,2): supercharge i = Tn (i 1,2,3): SU(2) current = 3 c 2πizT L0 character (Eguchi–Taormina, 1988): ch(z;τ) TrH e 0 q − 24 = ³ ´ 3 L0 Ω h Ω T Ω ℓ Ω | 〉= | 〉 0 | 〉= | 〉 1 unitarity condition: h k ≥ 4

K. Hikami (Naruto) Around the Volume Conjecture 18 / 19 Superconformal Algebra

N 4 SCFT (c 6k): string compactiﬁcation on K3 (k 1) = = =

massless character (BPS);

πiz n(n 1)/2 2πinz ie θ11(z;τ) q + e chR (z;τ) ( 1)n ℓ 0 3 n 2πiz = = η(τ) n Z − 1 q e e X∈ − 2 R £ R¤ 1 [θ11(z;τ)] ( ; ) ( ; ) 8 2 chℓ 0 z τ ch 1 z τ q− 3 = + ℓ 2 = ( ) e e= η τ massive character (non-BPS); £ ¤

2 n 1 [θ11(z;τ)] 8 ; ( 0) q − 3 n η(τ) > £ ¤

K. Hikami (Naruto) Around the Volume Conjecture 18 / 19 Superconformal Algebra

R massless character chℓ 0(z;τ) is a mock theta function, i.e., a holomorphic part of = 3 harmonic Maass forme whose shadow is η(τ) (Eguchi, KH) £ ¤ We have harmonic Maass form 1 µ(z;τ) µ(z;τ) R(0;τ) = − 2 3 [θ (z;τ)]2 1 i η(z) R ( ; ) 11 b ( ; ); ( ; ) ∞ chℓ 0 z τ 3 µ z τ R 0 τ dz = = η(τ) = pi τ £pz ¤τ e Z− + which satisifes £ ¤

i z 1 1 πi µ(z;τ) µ ; ; µ(z;τ 1) e− 4 µ(z;τ); µ(z 1;τ) µ(z τ;τ) µ(z;τ) =− τ τ − τ + = + = + = s µ ¶ 2 2 2 1 b by ∂ ∂ b i y ∂x i∂y µb(z;τ) 0;b (τ x biy) b − x + y + 2 + = = + · ¸ ³ ´ ¡ ¢ b K. Hikami (Naruto) Around the Volume Conjecture 18 / 19 Superconformal Algebra

Character Decomposition of Elliptic Genus (Eguchi, KH)

3 3 3 c c πi(T T ) 2πiT z L0 L0 ZK3(z;τ) TrH R H R e 0 − 0 e 0 q − 24 q − 24 = ⊗ µ ¶ θ (z;τ) 2 θ (z;τ) 2 θ (z;τ) 2 8 10 00 01 = θ (0;τ) + θ (0;τ) + θ (0;τ) "µ 10 ¶ µ 00 ¶ µ 01 ¶ # 2 R R ∞ n 1 [θ11(z;τ)] 20ch (z;τ) 2ch (z;τ) An q − 8 ℓ 0 ℓ 1 3 = = − = 2 + n 1 η(τ) e e X= £ ¤ where the number of non-BPS reps An is given

4π ∞ 1 π 3πis(d,c) 2πind/c A I1/ p8n 1 e− + n 1/4 2 = (8n 1) c 1 c 2c − d mod c − X= ³ ´ (c,Xd) 1 =

n 12345 6 7 8 · An 90 462 1540 4554 11592 27830 61686 131100 91.50 462.03 1538.81 4551.03 11594.14 27831.87 61686.17 131099.32

K. Hikami (Naruto) Around the Volume Conjecture 18 / 19 Concluding Remarks/Quantum Topology

Generalized Volume Conjecture hyperbolic geometry & quantum dilog function originates from discrete integralbe systems torus link & limiting value of Eichler integral mock modular form

logarithmic CFT AJ conjecture superconformal algebras & hyper-Kähler

applications to topological order experimental observation of fractional charge of non-abelian fQHE ν 5/2 = topological q-computing (Kitaev, Freedman) topological entanglement entropy

K. Hikami (Naruto) Around the Volume Conjecture 19 / 19