agt対応yasutake/matter/taki.pdfsketch of agt correspondence agt: a kind of duality theory a theory...
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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