Labastida-Marino-Ooguri-Vafa予想 と

その精密化

亀山 昌也 名古屋大学　多元数理科学研究科

結び目の数学X

Based on arXiv:1703.05408 with 縄田 聡 (Fudan University)

Introduction物 Chern-Simons theory 数 Quantum knot invariants

物 Topological string theory 数 Gromov-Witten theory

Large N duality

物 Refined Chern-Simons theory 数 Homological knot invariants

物 Refined Topological string 数 ?????????

Refined large N duality

・Labastida-Marino-Ooguri-Vafa (LMOV)予想はlarge N dualityの一種 ・LMOV予想(の一部分)を紹介する

・Refined Chern-Simons invariantsとknot homologyのreview ・LMOV予想の精密化とpositivity予想

LMOV予想(の一部)定義

次の式を通じてreformulated invariants 　　　　 を定義するfµ(K; a, q)

はSchur関数はYoung図、� s�(x)

注意

X

�

H�(K; a, q)s�(x) = exp

1X

d=0

X

�

1

d

f�(K; ad, qd)

qd2 � q

� d2

s�(xd)

!,

H�(K; a, q) はunreduced colored HOMFLY-PT polynomial

H�(K; a, q) = H�(K; a, q)H�(K; a, q)

H�(K; a, q) 2 Z[a± 12 , q

± 12 ]

LMOV予想(の一部)例 f

q12 � q

� 12

=H ,

f

q12 � q

� 12

=H ,�1

2H

2 � 1

2H

(2),

f

q12 � q

� 12

=H � 1

2H

2 +1

2H

(2),

f

q12 � q

� 12

=H �H H +1

3H

3 � 1

3H

(3),

f

q12 � q

� 12

=H �⇥H +H

⇤H +

2

3H

3+

1

3H

(3),

f

q12 � q

� 12

=H �H H +1

3H

3 � 1

3H

(3).

H(n)� := H�(K; an, qn)

LMOV予想(の一部)予想

f�(a, q) =X

�,⇢

1X

g=0

X

�2 12Z

C��⇢B�(q) bN⇢,g,�(q12 � q�

12 )2ga� ,

reformulated invariantsは次の形をもつ

C��⇢ は巡回群のClebsch-Gordon係数

B�(t) =

⇢(�t)dt�

|�|�12 � : hook rep for ^d V

0 � : otherwise.

は整数(LMOV invariantsと言う)bN⇢,g,�

bf⇢(K; a, q) :=X

g,�

bN�,g,�(q12 � q�

12 )2ga� .

LMOV予想(の一部)例

bf (K; a, q) = a12 � a

12 他のYoung図に対しては全て消えるUnknot

f = � (a � 1)�

a�a � g2 � 2

�,

f = � (a � 1)2a(g + 2)�a2 � a(g + 2) + 1

�,

f =(a � 1)2a(g + 2)2��a2 + a(g + 2) � 1

�,

f = � (a � 1)2a(g + 2)�a4

�g3 + 6g2 + 10g + 6

�� a3

�g4 + 8g3 + 23g2 + 31g + 17

�+ a2

�2g3 + 13g2 + 25g + 17

�� a

�g2 + 6g + 7

�+ 1

�,

f = � (a � 1)2a�a4

�g6 + 13g5 + 66g4 + 167g3 + 224g2 + 155g + 46

�� a3

�g7 + 15g6 + 93g5 + 311g4 + 613g3 + 724g2 + 486g + 147

�+ a2

�2g6 + 27g5 + 145g4 + 399g3 + 603g2 + 485g + 167

�� a

�g5 + 13g4 + 64g3 + 150g2 + 171g + 77

�+ g3 + 7g2 + 15g + 11

�,

f = � (a � 1)2a(g + 2)�a4

�g6 + 12g5 + 55g4 + 122g3 + 139g2 + 80g + 20

�� a3

�g7 + 14g6 + 80g5 + 243g4 + 428g3 + 444g2 + 260g + 70

�+ a2

�2g6 + 25g5 + 122g4 + 299g3 + 397g2 + 280g + 86

�� a

�g5 + 12g4 + 53g3 + 108g2 + 106g + 42

�+ g3 + 6g2 + 10g + 6

�.

g = (q12 � q� 1

2 )2^

Trefoil

^^^^^

Introduction物 Chern-Simons theory 数 Quantum knot invariants

物 Topological string theory 数 Gromov-Witten theory

Large N duality

物 Refined Chern-Simons theory 数 Homological knot invariants

物 Refined Topological string 数 ?????????

Refined large N duality

Knot homology

8

Khovanov discovered categorification of knot invariants.

JK(q) =X

i,j

(�1)iqj dimHsl2i,j (K)

KhR2(K;q, t) =X

i,j

tiqj dimHsl2i,j (K)

(Jones poly)

He constructed a graded homology and showed Hsl2��

•Homological knot invariants have non-negative coef.

Remark

•The notations are different between physics and math.

This homology allowed to construct new invariant

Refined Chern-Simons invariants

定義

はMacdonald関数

の定義は次のスライド�(n,m)�|µ,⌫

rCS�(Tm,n; a, q, t) :=X

µ

Pµ(pn; q, t)�(n,m)�|µ; ,

Pµ(pn; q, t) (Pµ(pn; q, t = q) = s�(x))

pn(x) :=1X

i=1

xni =

a12 � a�

12

t12 � t�

12

1. The �-factors vanish when |µ|� |⌫|� n|�| 6= 0

2. The �-factors with last empty index are proportional to stabilized �-factors

�(n,m)�|µ; = const e�(n,m)

�|µ; , (1)

where const is an overall normalization factor which depends on �, n, m butis the same for all µ.

3. Stabilized �-factors satisfy

e�(1,0)�|µ⌫ = Nµ

�⌫ ,X

↵

eSµ↵e�(n,m)�|↵⌫ =

X

�

e�(m,�n)�|µ�

eS�⌫ ,

eTµe�(n,m)�|µ⌫ = e�(n,m+n)

�|↵⌫eT⌫ , (2)

where Nµ�⌫ is Littlewood-Richardson coe�cients for Macdonald functions.eS�µ

and eT� are

eS�µ =1

g�(q, t)M�(t

⇢; q, t)Mµ(t⇢q�; q, t),

eT� ⌘ eT�� = tk�T k2

2 q�k�k2

2 . (3)

rCS�(Tn,m; a, q, t = q) = H�(Tn,m; a, q)なぜならrefined Chern-Simons invariantsの定義式は　　　　で Rosso-Jones公式そのものに一致する

t = q

a = �a2t , q12 = �qt , t

12 = q ,変数変換 で

refined Chern-Simons invariantsはhomological HOMFLY-PT polynomialsと一致する

長方形でないYoung図に対するrefined Chern-Simons invariantsは 一般に負の整数係数を持ってしまう(即ちhomological invariantsではない)

①

Refined Chern-Simons invariantsの数学的定義はCherednikが DAHAを用いて行った

②

③

④

Remarks and Observations

ExamplesrCS�(K; a, q, t) = P�(pn; q, t) =

✓t

a

◆ |�|2 Y

x2�

tl0(x) � aqa

0(x)

1� qa(x)tl(x)+1

rCS (T2,3; a, q, t) =a(�aq + qt+ 1)

pqpt

= �a2

�a2q2t3 + q4t2 + 1

�

q2

rCS (T2,3; a, q, t) =a2

�a2q2 � aq(t+ 1)

�qt2 + 1

�+ t

�q2t4 + q(t+ 1)t+ 1

��

qt4

=a4

�a4q4t6 + a2q2

�q2 + 1

�t3

�q6t2 + 1

�+ q2

�q12t4 +

�q2 + 1

�q4t2 + 1

��

q10

rCS�(Tm,n; a, q, t) = rCS�(K; a, q, t) rCS�(Tm,n; a, q, t)

注意

ExamplesrCS (T2,3; a, q, t) = a3q�3/2t�9/2

��a3q6 + a2

�q6t3 + q6t2 + q5t2 + q5t� q4t2 + q4t+ q4 + q3

�

+a��q6t5 � 2q5t4 � q5t3 + q4t4 � 3q4t3 � q4t2 + q3t3 � 3q3t2 � q3t+ q2t2 � 2q2t� q

�

+q5t6 + 2q4t5 � q3t5 + 2q3t4 + q3t3 � q2t4 + 2q2t3 � qt3 + 2qt2 + t�

=� a6q�10�q20

�a2t13 + t10

�+ q16t8

�a4t6 + 2a2t3 + 2

�

+q14t6�a4t8 + a2

�t2 � 1

�t3 � 1

�+ q12t6

�a4t6 + 3a2t3 + 2

�

+q10t4�a6t11 + a4t8 � a4t6 + a2t5 � a2t3 + t2 � 1

�

+q8t4�a4t6 + 3a2t3 + 2

�

+q6t2�a4t8 + a2

�t2 � 1

�t3 � 1

�+ q4t2

�a4t6 + 2a2t3 + 2

�+ a2t3 + 1

�

Introduction物 Chern-Simons theory 数 Quantum knot invariants

物 Topological string theory 数 Gromov-Witten theory

Large N duality

物 Refined Chern-Simons theory 数 Homological knot invariants

物 Refined Topological string 数 ?????????

Refined large N duality

X

�

H�(K; a, q)s�(x) = exp

1X

d=1

X

µ

1

d

fqµ(K; ad, qd)

qd2 � q

� d2

sµ(xd)

!

Refined large N duality

?

X

�

H�(K; a, q)s�(x) = exp

1X

d=1

X

µ

1

d

fqµ(K; ad, qd)

qd2 � q

� d2

sµ(xd)

!Large N duality

Idea

rCS�(K; a, q, t)

Cauchy formulas

�

�

g�(q, t)P�(x; q, t)P�(y; q, t) = exp

��

d>0

1

d

td2 � t�

d2

qd2 � q� d

2

pd(x)pd(y)

�

�

�

P�(x; q, t)P�T (y; t, q) = exp

��

d>0

(�1)d�1

dpd(x)pd(y)

�

Macdonald 変えない！

X

�

rCS�(Tm,n; a, q, t) g�(q, t)P�(x; q, t) = exp

1X

d=1

X

µ

1

d

fqµ(Tm,n; ad, qd, td)

qd2 � q�

d2

sµ(xd)

!

X

�

rCS�(Tm,n; a, q, t) P�T (�x; t, q) = exp

1X

d=1

X

µ

1

d

f t̄µ(Tm,n; ad, qd, td)

t�d2 � t

d2

sµ(xd)

!

X

�

H�(K; a, q)s�(x) = exp

1X

d=1

X

µ

1

d

fqµ(K; ad, qd)

qd2 � q

� d2

sµ(xd)

!

Refined large N dualityReformulated invariantsの定義

Refined reformulated invariantsの定義

Examples

18

fq

t12 � t�

12

=rCS ,

t12

q12

fq

t12 � t�

12

=qt � 1

q2 � 1rCS � t � 1

2(q � 1)(rCS )2 � t + 1

2(q + 1)rCS

(2),

t12

q12

fq

t12 � t�

12

=t � q

q2 � 1rCS +

t2 � 1

qt � 1rCS � t � 1

2(q � 1)(rCS )2 +

t + 1

2(q + 1)rCS

(2),

t

q

fq

t12 � t�

12

=(qt � 1)

�q2t � 1

�

(q2 � 1) (q3 � 1)rCS � (t � 1)(qt � 1)

(q � 1)2(q + 1)rCS rCS

+(t � 1)2

3(q � 1)2(rCS )3 � t2 + t + 1

3 (q2 + q + 1)rCS

(3),

t

q

fq

t12 � t�

12

= � (q � t)(qt � 1)

(q � 1)(q3 � 1)rCS +

(t � 1)�qt2 � 1

�

(q � 1) (q2t � 1)rCS

��

(t � 1)2

(q � 1)2rCS +

(t � 1)2(t + 1)

(q � 1)(qt � 1)rCS

�rCS

+2(t � 1)2

3(q � 1)2(rCS )3 +

t2 + t + 1

3 (q2 + q + 1)rCS

(3),

t

q

fq

t12 � t�

12

=(q � t)

�q2 � t

�

(q2 � 1)(q3 � 1)rCS � (t2 � 1)(q � t)

(q � 1) (q2t � 1)rCS +

(t2 � 1)(t3 � 1)

(qt � 1) (qt2 � 1)rCS

+

�(t � 1)(q � t)

(q � 1)2(q + 1)rCS � (t � 1)2(t + 1)

(q � 1)(qt � 1)rCS

�rCS

+(t � 1)2

3(q � 1)2(rCS )3 � t2 + t + 1

3 (q2 + q + 1)rCS

(3)

It looks like that refined reformulated invariants have poles but all pole vanish for concrete knot.

rCS(n)� = rCS�(K; an, qn, tn)

f t̄

t12 � t�

12

=rCS ,

�f t̄

t12 � t�

12

=rCS +1

2rCS

(2) � 1

2(rCS )2 ,

�f t̄

t12 � t�

12

=rCS +q � t

qt � 1rCS � 1

2rCS

2 � 1

2rCS

(2),

f t̄

t12 � t�

12

=rCS � rCS rCS +1

3(rCS )3 � 1

3rCS

(3),

f t̄

t12 � t�

12

=rCS +(t + 1)(q � t)

qt2 � 1rCS �

�rCS +

(q � 1)(t + 1)

qt � 1rCS

�rCS

+2

3(rCS )3 +

1

3rCS

(3),

f t̄

t12 � t�

12

=rCS +(q + 1)(q � t)

q2t � 1rCS +

(q � t)�q � t2

�

(qt � 1) (qt2 � 1)rCS

+

��rCS +

(q � t)

qt � 1rCS

�rCS +

1

3(rCS )3 � 1

3rCS

(3).

Examples

19

It looks like that refined reformulated invariants have poles but all pole vanish for concrete knot.

rCS(n)� = rCS�(K; an, qn, tn)

Positivity conjecture

�f�(Tm,n; a, q, t) =�

charges

(�1)2Jr �N�,g,�,Jr (q12 � q� 1

2 )g(t�12 � t

12 )g

�q

t

�Jr� �2

a�

予想

Refined reformulated invariantsは次の形をもつ

fqµ(a, q, t) =

X

�,⇢,g,�,Jr

(�1)2JrCµ�⇢B�(t) bN⇢,g,�,Jr (q12 � q�

12 )g(t�

12 � t

12 )g

⇣qt

⌘Jr� �2a�

f t̄µ(a, q, t) =

X

�,⇢,g,�,Jr

(�1)2JrCµ�⇢B�(q�1) bN⇢,g,�,Jr (q

12 � q�

12 )g(t�

12 � t

12 )g

⇣qt

⌘Jr� �2a�

Positivity conjecture�f�(Tm,n; a, q, t) =

�

charges

(�1)2Jr �N�,g,�,Jr (q12 � q� 1

2 )g(t�12 � t

12 )g

�q

t

�Jr� �2

a�

change of variables

non-negative integers!

a = �a2t , q12 = �qt , t

12 = q ,

�f�(Tm,n;a,q, t) =�

charges

�N�,g,�,Jr (qt � q�1t�1)g(q � q�1)gt2Jr��a�

=�

i,j,k

�N�,i,j,ka2iq2jtk up to shift

Example

+

+

+

+

+...

continue

g = (qt� q�1t�1)(q� q�1)

Non-torus knotscontinue

�f (41) =

�t31 + 3gt30 + 3g2t29 + 2t29 + g3t28 + 6gt28 + 4g2t27 + 4t27 + 6gt26 + 4t25

�a10

t15

+

�2t30 + 7gt29 + 9g2t28 + 6t28 + 5g3t27 + 20gt27 + g4t26 + 20g2t26 + 14t26 + 6g3t25 + 31gt25 + t25 + 15g2t24 + gt24 + 19t24 + 18gt23 + 2t23 + 8t22

�a8

t15

+

�t29 + 4gt28 + 6g2t27 + 6t27 + 4g3t26 + 21gt26 + g4t25 + 25g2t25 + 17t25 + 11g3t24 + 47gt24 + 2t24 + g4t23 + 36g2t23 + 3gt23 + 33t23 + 6g3t22 + g2t22 + 53gt22 + 7t22 + 15g2t21 + 5gt21 + 29t21 + 17gt20 + 5t20 + 7t19

�a6

t15

+

�2t26 + 7gt25 + 9g2t24 + 8t24 + 5g3t23 + 25gt23 + t23 + g4t22 + 24g2t22 + 2gt22 + 25t22 + 7g3t21 + g2t21 + 52gt21 + 8t21 + 27g2t20 + 10gt20 + 40t20 + 2g3t19 + 2g2t19 + 46gt19 + 14t19 + 9g2t18 + 7gt18 + 28t18 + 13gt17 + 5t17 + 6t16

�a4

t15

+

�t23 + 3gt22 + 3g2t21 + 7t21 + g3t20 + 17gt20 + 3t20 + 12g2t19 + 5gt19 + 25t19 + 2g3t18 + 2g2t18 + 42gt18 + 13t18 + 18g2t17 + 13gt17 + 41t17 + 2g3t16 + 2g2t16 + 41gt16 + 15t16 + 9g2t15 + 6gt15 + 26t15 + 13gt14 + 4t14 + 6t13

�a2

t15

+6t18 + 13gt17 + 4t17 + 9g2t16 + 6gt16 + 26t16 + 2g3t15 + 2g2t15 + 41gt15 + 15t15 + 18g2t14 + 13gt14 + 41t14 + 2g3t13 + 2g2t13 + 42gt13 + 13t13 + 12g2t12 + 5gt12 + 25t12 + g3t11 + 17gt11 + 3t11 + 3g2t10 + 7t10 + 3gt9 + t8

t15

+6t15 + 13gt14 + 5t14 + 9g2t13 + 7gt13 + 28t13 + 2g3t12 + 2g2t12 + 46gt12 + 14t12 + 27g2t11 + 10gt11 + 40t11 + 7g3t10 + g2t10 + 52gt10 + 8t10 + g4t9 + 24g2t9 + 2gt9 + 25t9 + 5g3t8 + 25gt8 + t8 + 9g2t7 + 8t7 + 7gt6 + 2t5

t15a2

+7t12 + 17gt11 + 5t11 + 15g2t10 + 5gt10 + 29t10 + 6g3t9 + g2t9 + 53gt9 + 7t9 + g4t8 + 36g2t8 + 3gt8 + 33t8 + 11g3t7 + 47gt7 + 2t7 + g4t6 + 25g2t6 + 17t6 + 4g3t5 + 21gt5 + 6g2t4 + 6t4 + 4gt3 + t2

t15a4

+8t9 + 18gt8 + 2t8 + 15g2t7 + gt7 + 19t7 + 6g3t6 + 31gt6 + t6 + g4t5 + 20g2t5 + 14t5 + 5g3t4 + 20gt4 + 9g2t3 + 6t3 + 7gt2 + 2t

t15a6+

4t6 + 6gt5 + 4g2t4 + 4t4 + g3t3 + 6gt3 + 3g2t2 + 2t2 + 3gt + 1

t15a8

• LMOV予想(の一部)、つまりreformulated invariantsの定義と整数係数予想を紹介した

• Refined LMOV予想(の一部)としてrefined reformulated invariantsを定義した

• Positivity予想、つまりRefined Chern-Simons invariantsは負整数係数を持つ場合があるが、refined reformulated invariantへ行くと非負整数係数性が見える

Summary