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瀧 雅人 RIKEN, Hashimoto Lab
2012. 7/27 @ YITP
Instantons&
Whittaker states of CFT
based on [H.Kanno, M.T., arXiv:1203.1427]
(simple version of) AGT correspondence
Instanton partition function
Whittaker state
4D
2D
(simple version of) AGT correspondence
Instanton partition function
Whittaker state
4D
2D
What is the Whittaker state !?
What is the Whittaker state !?
: coherent state of annihilation operators of 2D CFT
Today I will talk on
Instanton partition function
Whittaker state
4D
2D
Today I will talk on
Instanton partition function
Whittaker state
4D
2D
generalized Whittaker state
from M5 to N=2 gauge theories
6=4+2 :
! "! "! "
! "! "! "Gaiotto used
me to get N=2 theories
M5 on Cylinder 4D Gauge Theory
M5sNc SU(Nc)
R4 ×
M5 on Cylinder 4D Gauge Theory
M5sNc SU(Nc)
Quarks ?
R4 ×
Quarks ?flavors live on the edges
flavors flavors
R4 ×
Flavors via Boundary Conditions
R4 ×
Flavors via Boundary Conditions
quarks quarksNInNOut
R4 ×
Flavors via Boundary Conditions
susy QCD
quarks quarksNInNOut
NOut + NIn = Nf
R4 ×
How to describe BCs?
How to describe BCs?
not easy at all (in M5 language)
How to describe BCs?
not easy at all (in M5 language)
But, we have a nice description!
gauge theory via 2d CFT
2. AGT correspondence
� Out | | In �
Boundary Condition as a State
Gauge Coupling is the Length
� Out | | In �� =
1g2
YM
Partition function is Matrix Element
� Out | | In �� =
1g2
YM
Z 4D = � Out |Λ2NcL0 | In �
[Ln, Lm] = (n − m)Ln+m
What’s the state?
[Ln, Lm] = (n − m)Ln+m
L 1,2,3,···
L −1,−2,−3,···
L 0 = H
a
a†
: annihilation op.s ( )
: creation op.s ( )
. . . harmonic oscillators
What’s the state?
eigenstates
Whittaker states for gauge theory
3. flavorful states
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 �
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
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
4. Summary
“Generalized” is ubiquitous M5 configuration
generalized Whittaker states :
surface operators4D SCFTs
flavorful cases of colorful ABCDEFG{
GaugeTheory
Next step : feedback to M-theory
M5 branes
M5 branes GaugeTheory
FIN
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