- AGT 入門から最近の進展まで 対応 -

AGT

-

-

2016.10/22

@千葉工大セミナー

Masato Taki ^{RIKEN , iTHES group}

duality (

)

Theory A

Theory B

duality (

):

1

Theory A

Theory B

L

= ¯ i(

@

+ m) g

2 ( ¯

)

L

= 1

2 (@

)

h cos

2d massive Thirring model

2d sine-Gordon model

duality (

):

1

Theory A

Theory B

L

= ¯ i(

@

+ m) g

2 ( ¯

)

L

= 1

2 (@

)

h cos

2d massive Thirring model

2d sine-Gordon model

duality (

):

2 Montonen-Olive S-duality Theory A

Theory B

4d YM

4d YM

Theory A

Theory B

4d YM

4d YM

duality (

):

2 Montonen-Olive S-duality

Theory A

Theory B

4d YM

4d YM

duality (

):

2 Montonen-Olive S-duality

Theory A

Theory B

4d YM

4d YM

duality (

):

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

dimens ional

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

Z

W

brane 1 brane 2

U (2) ! U (1)

⇥ U (1)

Type I

Type IIB

Type IIA E⁸ x E⁸ Het.

SO(32) Het.

S

T T

Perturbative String Theories

M-theory

M5 on Cylinder 4D Gauge Theory

N _c M5s SU (N _c )

R ⁴

AGT via M-theory

M5 on Cylinder 4D Gauge Theory

N _c M5s SU (N _c )

Quarks ?

R ⁴

AGT via M-theory

flavors live on the edges

flavors flavors

R ⁴

AGT via M-theory

Out | | In

Boundary Condition as a State

AGT via M-theory

Gauge Coupling is the Length

Out | ⁼ | In

1 g ²

YM

AGT via M-theory

AGT via M-theory

Partition function is Matrix Element

Out | ⁼ | In

1 g ²

YM

Z _4D = Out | ^{2N c L 0} | 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

✓

z ⁿ⁺¹ @

@ z

◆

z

conformal symmetry

z ! f (z ) ' z + ✏(z )

= z + X

n

✏ _n

✓

z ⁿ⁺¹ @

@ z

◆

z

conformal symmetry

l _n = z ⁿ⁺¹ @

@ z

conformal symmetry

2 . instanton &

partition function

minimum of Euclidean Yang-Mills action

A

F

= @

A

@

A

+ i[A

, A

]

S = Z

d

x(F

)

SU(2) instanton

self-dual equation

F

= 1

2

F

3

b=1

M ^ab (x X)^⇥ _µ⇥^b (x X)² + ⇥²

1-instanton solution

X

x

X

M SO (3)

{

size

M

M¹ = R⁴ R⁺ SO(3)

SU(2) instanton

:  k-instanton parameter space

instanton partition function

R²

dX

dX

R²

dX

dX

e

= 1

1

instanton partition function

U (1)

three ’s : ✏ ₁ , ✏ ₂ , a

Z^Nek = 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)

.

.

.

L _n>0 | = 0 L ₀ | = |

Rep. of Virasoro algebra

analogy of

· · ·

weight basis

L ₁|

L

|

L

|

L ₂|

L ₂L ₁| L ₃|

L _Y | ⇤ ⇥ L _Y1L _Y2 · · · | ⇤ ^{n = |Y | = Yi}

|

+1 +2 +3 + n

L

Young diagram

Y = [Y

,Y

,Y

, …] = [4, 3, 1, 1, 0, 0, …]

· · ·

weight basis

L ₁|

L

|

L

|

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

L

| ⇥, = ⇥

| ⇥, L

| ⇥, = 0

Gaiotto state

L

| ⇥, = ⇥

| ⇥, L

| ⇥, = 0

ansatz

Gaiotto state

L

| ⇥, = ⇥

| ⇥, L

| ⇥, = 0

ansatz

Gaiotto state

L

| ⇥, = ⇥

| ⇥, L

| ⇥, = 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 )) + · · ·

Z^Nek = 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 )) + · · ·

Z^Nek = 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

=

^{c = 1 + 6 (}

⁺

⁾

1 2

= ( ₁ + ₂)² 4 _{1 2}

a²

1 2

AGT dictionary (universal)

AGT relation

4 . flavorful states

g / 2N _c N _f

beta function of N=2 gauge theories

many of them are asymptotically non-free

z_vector(⇥a, _1,2; Y⇥ )

=

N

a,b=1 (i,j) Y_a

(a_a a_{b 1}(Y_bj^T i + 1) + ₂(Y_ai j + 1)) ¹

(i,j) Y_b

(a_a a_b + ₁(Y_bi j + 1) ₂(Y_aj^T i + 1)) ¹ [Nekrasov, ’02]

generic partition function for SU(N) theory

AGT has a wide range of generalizations

N _c

N _f

0 1 2

3 4 2

| N

Landscape of

0 1 2

3 4 2

[AGT, ’09]

| N

Landscape of

0 1 2

3 4 2

[AGT, ’09]

[Wyllard, ’09]

| N

Landscape of

0 1 2

3 4 2

[Gaiotto, ’09]

[AGT, ’09]

[Wyllard, ’09]

| N

Landscape of

0 1 2

3 4 2

[Gaiotto, ’09]

[AGT, ’09]

[M.T, ’09]

[Wyllard, ’09]

| N

Landscape of

| N

Landscape of

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

| N

Landscape of

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

[Kanno-M.T, ’12]

| N

Landscape of

SU(2)

0 1 2

3 4 2

[Gaiotto, ’09]

SU(2) with 0 Flavors

SU(2) with 0 Flavors

L ₂ | 0 = 0 N _f = 0

L ₁ | 0 = | 0

SU(2) with 0 Flavors

L ₂ | 0 = 0 N _f = 0

L ₁ | 0 = | 0

It means 0-flavor, not vacuum

*

SU(2) with 0 Flavors

L ₂ | 0 = 0 N _f = 0

L ₁ | 0 = | 0

Z

= 0 | 0 ⇥

L ₂ | 1 i = ⇤ ² | 1 i

L ₁ | 1 i = 2m⇤ | 1 i

N _f = 1

SU(2) with 1 Flavors

Z _SU ^N

⁼¹ ₍₂₎ = h 0 | 1 i = h 1 | 0 i

L ₂ | 1 i = ⇤ ² | 1 i

L ₁ | 1 i = 2m⇤ | 1 i

N _f = 1

SU(2) with 1 Flavors

4-flavors case is special

(original AGT)

SU(2)

N_f = 4

Out | | In

SU(2)

N_f = 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 (Y

; Y

) = | L

L

| ⇥

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 4

2

SU(3) without Flavor

SU(3) Whittaker state without Flavor

L _m ^W n

[L _m , W _n ] = (2m n)W _n+m

: theory with and

[W _m , W _n ] =

L _m ^W n

[L _m , W _n ] = (2m n)W _n+m

: theory with and

[W _m , W _n ] =

SU(3) Whittaker state without Flavor

[L_n, W_m] = (2n m)W_n+m

[W_n, W_m] = 9 2

c

3 · 5!(n² 1)(n² 4) _{n, m} + 16

22 + 5c(n m) _n+m +(n m) (n + m + 2)(n + m + 3)

15

(n + 2)(m + 2) 6

⇥

L_n+m⇤

[L_n, L_m] = (n m)L_n+m + c

12(n³ n) _{n, m}

n =

m⇥Z

: L_mL_{n m} : +x_n

5 L_n

x_2l = (1 l)(1 + l), x_2l+1 = (1 l)(12 + l)

non-linear algebra (Lie)

SU(3) Whittaker state without Flavor

L ₁ | 0 = 0 W ₁ | 0 = | 0

N _f = 0

L ₁ | 0 = 0 W ₁ | 0 = | 0 N _f = 0

Z

= 0 | 0 ⇥

SU(3) Whittaker state without Flavor

SU(3) Whittaker states with 0,1 Flavors

L ₁ | 0 = 0 W ₁ | 0 = | 0 N _f = 0

N _f = 1

L ₁ | 1 = | 1 W ₁ | 1 = m | 1

Z

= 0 | 1 ⇥ = 1 | 0 ⇥

Z

= 1 | 1 ⇥

SU(3) Whittaker states with 0,1 Flavors

Z

= 0 | 1 ⇥ = 1 | 0 ⇥

Z

= 1 | 1 ⇥

Z

= 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 must be eigenstate. L ₁ , L ₂ , W ₂ , W ₃

W

= [L

, W

] 3W

= [L

, W

] But

Qestion.

SU(3) with 2 Flavors Trouble !?

| 2 must be eigenstate. L ₁ , L ₂ , W ₂ , W ₃

W

= [L

, W

] 3W

= [L

, W

] But

Qestion.

Answer.

[L

, L

] = nL

(W ₁ + L ₀ ) | 2 ⇥ | 2 ⇥

generalized Whittaker states

{ L ₁ , L ₂ }

{ L ₁ , L ₂ } { L ₀ } ^{: 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 0

0 a

dz z

m =

C^z=0

F

C² F F

{

F + ⇤F = ⇥ (

)

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

| V

(1)V

(q )⇥

(z ) |

⇥

(z) = exp ¹

2 G(z) + · · ·

2,1

(L² ₁ b²L ₂) _2,1(z) = 0 insertion of degenerate field

surface 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

S

monodromy

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

W (a) =

1

j= 1

e^{4⇥ijb(a /2)}

W (a) =

1/2

j= 1/2

e

[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