AGT対応 - 入門から最近の進展まで -

2016.10/22 @千葉工大セミナー

Masato Taki RIKEN, iTHES group

duality (双対性)

Theory A

Theory B

duality (双対性): 例1

Theory A

Theory B

LA = ̄i(�µ@µ + m) �g

2( ̄�µ )2

LB =

1

2

(@µ�)2 � h cos�

2d massive Thirring model

2d sine-Gordon model

duality (双対性): 例1

Theory A

Theory B

LA = ̄i(�µ@µ + m) �g

2( ̄�µ )2

LB =

1

2

(@µ�)2 � h cos�

2d massive Thirring model

2d sine-Gordon model

duality (双対性): 例2 Montonen-Olive S-duality

Theory A

Theory B

4d YM

4d YM

Theory A

Theory B

4d YM

4d YM

duality (双対性): 例2 Montonen-Olive S-duality

Theory A

Theory B

4d YM

4d YM

duality (双対性): 例2 Montonen-Olive S-duality

Theory A

Theory B

4d YM

4d YM

duality (双対性): 例2 Montonen-Olive S-duality

1. Sketch of AGT correspondence

AGT: a kind of duality

Theory A

Theory B

4d N=2 susy gauge theory

2d conformal field theory (CFT)

AGT: a kind of duality

Theory A

Theory B

4d N=2 susy gauge theory

2d conformal field theory (CFT)

10d string theory

D-branes

hyper-plane in 10d spacetime

Open strings have their ends on it

higher

dimensional

SU(N)

Open string and gauge fields

hyper-plane in 10d spacetime

Open strings have their ends on it

Open string and gauge fields

Z(1)

Z(2)

W±

brane 1 brane 2

U(2) ! U(1)1 ⇥ U(1)2

Type I

Type IIB

Type IIAE8 x E8 Het.

SO(32) Het.

S

T T

Perturbative String Theories

M-theory

M5 on Cylinder 4D Gauge Theory

M5sNc SU(Nc)

R4 �

AGT via M-theory

M5 on Cylinder 4D Gauge Theory

M5sNc SU(Nc)

Quarks ?

R4 �

AGT via M-theory

flavors live on the edges

flavors flavors

R4 �

AGT via M-theory

� Out | | In �

Boundary Condition as a State

AGT via M-theory

Gauge Coupling is the Length

� Out | | In �� =

1g2

YM

AGT via M-theory

AGT via M-theory

Partition function is Matrix Element

� Out | | In �� =

1g2

YM

Z 4D = � Out |�2NcL0 | In ⇥

What’s the state?

state in CFT with Virasoro symmetry

AGT via M-theory

@ critical point

No typical scale

2d CFT describes critical phenomena

z ! f(z) ' z + ✏(z)

conformal symmetry

z ! f(z) ' z + ✏(z)

= z +X

n

✏n

✓zn+1 @

@z

◆z

conformal symmetry

z ! f(z) ' z + ✏(z)

= z +X

n

✏n

✓zn+1 @

@z

◆z

conformal symmetry

ln = �zn+1 @

@z

conformal symmetry

2. instanton &

partition function

minimum of Euclidean Yang-Mills action

Aµ

Fµ⌫ = @µA⌫ � @⌫Aµ + i[Aµ, A⌫ ]

S =

Zd

4x(Fµ⌫)

2

SU(2) instanton

self-dual equation

Fµ⇤ =12

�µ⇤⌅�F ⌅� Aaµ(x) =

3�

b=1

Mab(x � X)⇥�b

µ⇥

(x � X)2 + ⇥2

1-instanton solution

�

X

xµ

Xµ

�

M � SO(3){ center of the instanton

size

Mk

M1 = R4 � R+ � SO(3)

SU(2) instanton

: k-instanton parameter space

instanton partition function

�

R2

dX1dX2 ��

R2

dX1dX2e��1((X1)2+(X2)2)

=1�1

instanton partition function

U(1)three ’s : ✏1, ✏2, a

ZNek = 1 +2�4

�1�2(�1 + �2 + 2a)(�1 + �2 � 2a)

+�8(8(�1 + �2) + �1�2 � 8a2)

�21�2

2((�1 + �2)2 � 4a2)((2�1 + �2)2 � 4a2)((�1 + 2�2)2 � 4a2)

+ · · ·

1-instanton

2-instanton

instanton partition function

Origin of instanton partition function

N=2 SUSY

Origin of instanton partition function

N=2 SUSY

Origin of instanton partition function

twisting N=2 SUSY

Origin of instanton partition function

twisting N=2 SUSY

( )

Origin of instanton partition function

twisting N=2 SUSY

3. conformal symmetry

Virasoro algebra

Rep. of SU(2)

Rep. of SU(2)

Rep. of SU(2)

weight shift

Rep. of SU(2)

Rep. of SU(2)

.

.

.

Ln>0|�� = 0L0|�� = �|��

Rep. of Virasoro algebra

analogy of

· · ·

weight basis

L�1|��

L�12|��

L�13|��

L�2|��

L�2L�1|��L�3|��

L�Y |�⇤ ⇥ L�Y1L�Y2 · · · |�⇤ n = |Y | =�

Yi

|���

� +1

� +2

� +3

� + n

L+n

Young diagram

Y = [Y1,Y2,Y3, …] = [4, 3, 1, 1, 0, 0, …]

· · ·

weight basis

L�1|��

L�12|��

L�13|��

L�2|��

L�2L�1|��L�3|��

L�Y |�⇤ ⇥ L�Y1L�Y2 · · · |�⇤ n = |Y | =�

Yi

|���

� +1

� +2

� +3

� + n

A building block of AGT relation

L1|⇥, �� = ⇥2|⇥, �� Ln>1|⇥, �� = 0

Gaiotto state

L1|⇥, �� = ⇥2|⇥, �� Ln>1|⇥, �� = 0

ansatz

Gaiotto state

L1|⇥, �� = ⇥2|⇥, �� Ln>1|⇥, �� = 0

ansatz

Gaiotto state

L1|⇥, �� = ⇥2|⇥, �� Ln>1|⇥, �� = 0

ansatz

Gaiotto state

An AGT relation for pure SU(2) YM

⇤�, ⇥|�, ⇥⌅ = 1 +⇥4

2�+

⇥8(c + 8�)4�(16�2 � 10� + c(1 + 2�))

+ · · ·

ZNek = 1 +2�4

�1�2(�1 + �2 + 2a)(�1 + �2 � 2a)

+�8(8(�1 + �2) + �1�2 � 8a2)

�21�2

2((�1 + �2)2 � 4a2)((2�1 + �2)2 � 4a2)((�1 + 2�2)2 � 4a2) + · · ·

�

�

� �� �

⇤�, ⇥|�, ⇥⌅ = 1 +⇥4

2�+

⇥8(c + 8�)4�(16�2 � 10� + c(1 + 2�))

+ · · ·

ZNek = 1 +2�4

�1�2(�1 + �2 + 2a)(�1 + �2 � 2a)

+�8(8(�1 + �2) + �1�2 � 8a2)

�21�2

2((�1 + �2)2 � 4a2)((2�1 + �2)2 � 4a2)((�1 + 2�2)2 � 4a2) + · · ·

�

�

� �� �

�1�2��2 = �2 c = 1 + 6(�1 + �2)2

�1�2� =

(�1 + �2)2

4�1�2�

a2

�1�2

AGT dictionary (universal)

AGT relation

4. flavorful states

�g / 2Nc � Nf

beta function of N=2 gauge theories

many of them are asymptotically non-free

zvector(⇥a, �1,2; ⇥Y )

=N�

a,b=1

�

(i,j)�Ya

(aa � ab � �1(Y Tbj � i + 1) + �2(Yai � j + 1))�1

��

(i,j)�Yb

(aa � ab + �1(Ybi � j + 1) � �2(Y Taj � i + 1))�1

[Nekrasov, ’02]

generic partition function for SU(N) theory

AGT has a wide range of generalizations

Nc

Nf

0

1

2

3 42

| Nf �Landscape of

0

1

2

3 42

[AGT, ’09]

| Nf �Landscape of

0

1

2

3 42

[AGT, ’09]

[Wyllard, ’09]

| Nf �Landscape of

0

1

2

3 42

[Gaiotto, ’09]

[AGT, ’09]

[Wyllard, ’09]

| Nf �Landscape of

0

1

2

3 42

[Gaiotto, ’09]

[AGT, ’09]

[M.T, ’09]

[Wyllard, ’09]

| Nf �Landscape of

| Nf �Landscape of

[Keller-Mekareeya-Song-Tachikawa, ’12]

| Nf �Landscape of

[Keller-Mekareeya-Song-Tachikawa, ’12]

[Kanno-M.T, ’12]

| Nf �Landscape of

SU(2)

0

1

2

3 42

[Gaiotto, ’09]

SU(2) with 0 Flavors

SU(2) with 0 Flavors

L2| 0 � = 0

Nf = 0

L1| 0 � = | 0 �

SU(2) with 0 Flavors

L2| 0 � = 0

Nf = 0

L1| 0 � = | 0 �

It means 0-flavor, not vacuum*

SU(2) with 0 Flavors

L2| 0 � = 0

Nf = 0

L1| 0 � = | 0 �

ZNf =0

SU(2) = � 0 | 0 ⇥

L2|1i = ⇤2|1iL1|1i = 2m⇤|1i

Nf = 1

SU(2) with 1 Flavors

ZNf=1SU(2) = h0|1i = h1|0i

L2|1i = ⇤2|1iL1|1i = 2m⇤|1i

Nf = 1

SU(2) with 1 Flavors

4-flavors case is special (original AGT)

SU(2)

Nf = 4

� Out | | In �

SU(2)

Nf = 4

� Out | | In �

��1|V2(1)V3(q)|�4⇥ = ��1|V2(1)�

Y,Y �

|�, Y ⇥Q�1� (Y, Y ⇥)��, Y ⇥|V3(q)|�4⇥

the basis of the Verma module is not orthonormal

Q�(Y1 ; Y2) = ��|LY2L�Y1 |�⇥Shapovalov matrix

[1] [2] [12]

[1]

[2]

[12]

0

0

0

0

[3]

[3]

Q� =

n = 1

n = 2

n = 3

��1|V2(1)V3(q)|�4⇥ = ��1|V2(1)�

Y,Y �

|�, Y ⇥Q�1� (Y, Y ⇥)��, Y ⇥|V3(q)|�4⇥

the basis of the Verma module is not orthonormal

�

�1

�2 �3

�4

=⇤

�

q�(�)��1��2 b�(�)

��1 �4

�2 �3

⇥B�(�)

��1 �4

�2 �3

⇥(q)

SU(3)

0

1

2

3 42 [M.T, ’09]

SU(3) without Flavor

SU(3) Whittaker state without Flavor

Lm Wn

[Lm, Wn] = (2m � n)Wn+m

: theory with and

[Wm, Wn] =

Lm Wn

[Lm, Wn] = (2m � n)Wn+m

: theory with and

[Wm, Wn] =

SU(3) Whittaker state without Flavor

[Ln, Wm] = (2n � m)Wn+m

[Wn, Wm] =92

� c

3 · 5!(n2 � 1)(n2 � 4)�n,�m +

1622 + 5c

(n � m)�n+m

+(n � m)�

(n + m + 2)(n + m + 3)15

�(n + 2)(m + 2)

6

⇥Ln+m

⇤

[Ln, Lm] = (n � m)Ln+m +c

12(n3 � n)�n,�m

�n =�

m⇥Z: LmLn�m : +

xn

5Ln

x2l = (1 � l)(1 + l), x2l+1 = (1 � l)(12 + l)

�non-linear algebra (Lie)

SU(3) Whittaker state without Flavor

L1| 0 � = 0 W1| 0 � = | 0 �

Nf = 0

L1| 0 � = 0 W1| 0 � = | 0 �

Nf = 0

ZNf =0

SU(3) = � 0 | 0 ⇥

SU(3) Whittaker state without Flavor

SU(3) Whittaker states with 0,1 Flavors

L1| 0 � = 0 W1| 0 � = | 0 �

Nf = 0

Nf = 1

L1| 1 � = | 1 � W1| 1 � = m| 1 �

ZNf =1

SU(3) = � 0 | 1 ⇥ = � 1 | 0 ⇥

ZNf =2

SU(3) = � 1 | 1 ⇥

SU(3) Whittaker states with 0,1 Flavors

ZNf =1

SU(3) = � 0 | 1 ⇥ = � 1 | 0 ⇥

ZNf =2

SU(3) = � 1 | 1 ⇥

ZNf =2

SU(3) = � 0 | 2 ⇥ = � 2 | 0 ⇥ ?

SU(3) Whittaker states with 0,1 Flavors

[Kanno-M.T, ’12]

SU(3) with 2 Flavors

SU(3) with 2 Flavors Trouble !?

SU(3) with 2 Flavors Trouble !?

| 2 � L1, L2, W2, W3must be eigenstate.

W2 = [L1, W1]

3W3 = [L2, W1]

But

Qestion.

SU(3) with 2 Flavors Trouble !?

| 2 � L1, L2, W2, W3must be eigenstate.

W2 = [L1, W1]

3W3 = [L2, W1]

But

Qestion.

Answer.

[Ln, L0] = nLn

(W1 + L0)| 2 ⇥ � | 2 ⇥

generalized Whittaker states

{L1, L2}

{L�1, L�2}

{L0} : Cartan

eigenstate of linear combi.

of them

This is actually very ubiquitous B.C. for M5s !

}

Landscape of flavorful AGT

Generalized

G

G

G

G

�1

�2

z

A ��

a 00 �a

�dz

z

m =�

C z=0

F

k =�

C2

F � F{

F + ⇤F = ⇥(�C)+

As a defect, a surface operator creates the singularity near the locus of it’s insertion:

This is a solution of the following modified SD eq.

Instantons in the presence of such a are operator labelled by these two topological numbers.

Surface defect

�2,1

��1|V2(1)V3(q)⇥2,1(z)|�4⇥�(z) = exp�

�1

�2G(z) + · · ·

�2,1

(L2�1 � b2L�2)�2,1(z) = 0

insertion of degenerate fieldsurface operator

Surface defect

Loop operators

Wilson loop (electric)

Loop operators

Wilson loop (electric)

t’Hoft loop (magnetic)

path integral with

Loop operators

t’Hoft loop (magnetic)

path integral with

monopole

S2

monodromyWilson-‘tHooft loop operator for the ele.-mag. charge (p,q)

W��(a) =1�

j=�1

e4⇥ijb(a��/2)

W�(a) =1/2�

j=�1/2

e4⇥ijb(a��/2)

[Alday-Gaoitto-Gukov-Tachikawa-Verlinde, ‘09]

[Drukker-Gomiz-Okuda-Techner, ‘09]

Loop operators

5d and 6d gauge theories

CFT

integrable system

integrable mass deformation

5d and 6d gauge theories

CFT

integrable system

integrable mass deformation

deformed Virasoro (W) algebras

Virasoro (W) algebras

q-deformed, elliptic, …

mathematically, AGT is proved (for special cases)

proof of AGT

geometric representation theory

5. Conclusion

Conclusion

AGT has a stringy origin: 6d -> 4d vs 2d

Many variations (gauge group, matters, operators, dimensions ..)

Math proof, but no intuitive understandings