five dimensional duality and e cft from m5...

Five dimensional duality

and E6 CFT

from M5 branes

Futoshi Yagi (SISSA & INFN, Trieste)

Based on

Ling Bao, Elli Pomoni, Masato Taki, FY arXiv:1112.5228 [hep-th] Ling Bao, Vladimir Mitev, Elli Pomoni, Masato Taki, FY Work in progress

This talk consists of two topics

Five dimensional duality from M5-brane

E6 CFT from M5-brane

Ling Bao, Elli Pomoni, Masato Taki, FY

arXiv:1112.5228 [hep-th]

Ling Bao, Vladimir Mitev, Elli Pomoni, Masato Taki, FY

Work in progress

Five dimensional duality and E6 CFT from M5 branes

=

+

Five dimensional duality

from M5-brane

§1 Introduction

…

…

5D N=1 SU(N)M-1 SYM compactified on S1

5D N=1 SU(M)N-1 SYM compactified on S1

SU(N) SU(N) SU(N)

M-1

N-1

SU(M) SU(M) SU(M)

SU(N) SU(N)

SU(M) SU(M)

Gauge

symmetry Global

symmetry

fundamental

Bi-fundamental

Five dimensional duality ’97 Katz, Mayer, Vafa

’97 Aharony, Hanany, Kol

Anti-fundamental

Claim

)(

f

af

)1,(

bif

2)( )(

i

N

ii

ii

a

m

m

m

eqi

SU(N)M-1 theory (original) SU(M)N-1 theory (dual)

1,1

1,,1

N

Mi

di

diN

di

d

d

a

m

m

m

q

)(

f

af

)1,(

bif

)(

Map UV gauge coupling constants:

Bi-fundamental masses:

Anti-fundamental masses:

Fundamental masses:

Coulomb moduli parameters:

d

i

dNddIRd

iN

iii

IR

ammmqS

ammmqS

)(faf)1,(

bif

)(

)(faf)1,(

bif

)(

,,,,

,,,,

11,,1 MNI

Both theory give the identical IR effective action under proper map

Coulomb branch: SU(N) M-1 →U(1)(M-1)(N-1) ← SU(M) N-1

§1 Introduction

§2 Understanding the duality from brane setup

§3 Finding the map

§4 Summary and discussion 1

§5 Implication of the duality map

Plan of the first half Five dimensional duality from M-theory

Work in progress

Gives motivation to the second half

4d N=2 pure

SU(N) gauge theory on R3,1 (x0,x1,x2,x3)

N D4

NS5 NS5

x4

x6

x5

FFxdLdxxdS Tr464

2

1

g

L

World volume theory of the D4-branes

§2 Understanding the duality

’96 Hanany, Witten

’97 Witten

…..

4d N=2 superconformal

SU(N) gauge theory (2N flavor) on R3,1 (x0,x1,x2,x3)

NS5 NS5

x4

x6

x5

N D4 (flavor brane)

Vector

multiplet Anti-fundamental

hypermultiplet

SU(N) SU(N) SU(N)

Gauge

symmetry Global

symmetry

Global

symmetry

N D4 (flavor brane)

N D4 (color brane)

4d N=2 superconformal

SU(N)M-1 gauge theory (2N flavor) on R3,1 (x0,x1,x2,x3)

N D4

NS5 NS5

x4

x6

x5

N D4 N D4

Vector


hypermultiplet

…

NS5

Bi-fundamental

hypermultiplet Fundamental

hypermultiplet

…

SU(N) SU(N) SU(N) SU(N) SU(N)

Gauge

symmetry Global

symmetry

fundamental

Bi-fundamental

M NS5-branes

SU(N)M-1

Anti-fundamental

N D4 N D4

5d N=1 SU(N)M-1 gauge theory

with 2N flavor on S1

N D4

NS5 NS5

x4

x6

x5

N D4 N D4

Vector


hypermultiplet

…

NS5

Bi-fundamental


hypermultiplet

M NS5-branes

N D4 N D4

x5 direction is periodic

52 R

Type IIA


with 2N flavor on S1

N D4

NS5 NS5

x4

x6

x5

N D4 N D4

Vector


hypermultiplet

…

NS5

Bi-fundamental


hypermultiplet

M NS5-branes

N D4 N D4

x5 direction is periodic

T-dual

N D5

NS5 NS5 NS5

… N D5 N D5 N D5 N D5

52 R

5

2

R

Type IIA

Type IIB


with 2N flavor on S1 with mass deformation

N D4

NS5 NS5

x4

x6

x5

D4 N D4

Anti-fundamental

hypermultiplet

…

NS5

Bi-fundamental


hypermultiplet

M NS5-branes

N D4

D4

Type IIA

Bi-fundamental

mass

D4

D4 Anti-fundamental

mass D4

Anti-fundamental

mass

D4

N

…



in the Coulomb phase

NS5 NS5

x4

x6

x5

D4

…

NS5

M NS5-branes

D4

Type IIA

D4

D4 Anti-fundamental

mass D4

Anti-fundamental

mass

D4

Coulomb

moduli

parameter

D4

D4

D4

Bi-fundamental

mass

D4

D4

D4

N Center of mass

Center of mass

Center of mass

…

D4

D4

D4

NS5 NS5

…

NS5

M NS5-branes

D4

x4

x6

5x




D4 D4

D4

D4

D4

D4

D4 D4 D4

D4

D4

D4

D4

D4 N

…

NS5 NS5

…

NS5

M NS5-branes

D4

x4

x6

5x




D4 D4

D4

D4

D4

D4

D4 D4 D4

D4

D4

D4

D4

D4 N

M-theory uplift

M5 M5

…

M5

M5

M5

M5

M5

M5

M5

11x M5

M5

M5

M5

M5

M5

M5

M5

M5

…

…

Local structure of the intersection

M5(D4)

M5(NS5)

M5(D4)

11x

5x

NS5: M5 wrapping x5

D4: M5 wrapping x11


M5(D4)

M5(NS5)

M5(D4)

11x

5x

NS5: M5 wrapping x5

D4: M5 wrapping x11

Comment

・The NS5-brane is pulled by the D4-branes

x4

x6

M5(D4)

M5(NS5)

M5(D4)


M5(D4)

M5(NS5)

M5(D4)

11x

5x

NS5: M5 wrapping x5

D4: M5 wrapping x11

Comment


・Seen as a single M5-brane (IR theory is Abelian)

x4

x6

M5


11x

M5(D4)

M5(NS5)

M5

5x

D4 NS5

M5(D4)

11x

5x

NS5: M5 wrapping x5

D4: M5 wrapping x11

Comment


・Seen as a single M5-brane (IR theory is Abelian)

・If we regard x5 direction instead of x11 direction as M-theory circle,

D4-brane looks like compactified NS5-brane and vice-versa

x4

x6

11x

5x5x

11 x

11x

5x

(NS5)

5D N=1 SU(N)M-1 SYM

compactified on S1

N

M

5x11x

…

…

…

…

…

… …

x4

x6

5D N=1 SU(N)M-1 SYM

compactified on S1

5D N=1 SU(M)N-1 SYM

compactified on S1

M

N

Dual!!

11x

5x

…

…

…

…

…

…

…

N

M

5x11x

…

…

…

…

…

… …

x4

x6

x4

x6

115

64

xx

xx

5D N=1 SU(N)M-1 SYM

compactified on S1

5D N=1 SU(M)N-1 SYM

compactified on S1

M

N

Dual!!

11x

5x

…

…

…

…

…

…

…

N

M

5x11x

…

…

…

…

…

… …

x4

x6

x4

x6

115

64

xx

xx

11

116

5

54

,R

ixx

R

ixx

etew

00 0

N

i

M

j

ji

ij twC wt 00 0

M

j

N

i

ij

ij twC

’97 Brandhuber Sonnenschein, Itzhaki

Theisen, Yankielowicz

M5-brane configuration = (R3,1 ×) Seiberg-Witten curve

SW curve: 0,,0 0

N

i

M

j

ji

ij twamqC

moduliCoulomb:

mass:

couplinggauge:2

a

m

eq i

SW 1-form: tdwwdt loglogloglogSW

SW solution: ,, SWSW

ii Bi

Ai

a

Fa

ji

ijaa

F

eff

11 )()(

NM MSUNSU

wt

Gauge group: SU(N)M-1

’94 Seiberg, Wittem

…

’97 Witten

Boundary condition

5

2

2~ R

m

em

5

54

R

ixx

ew

11

116

R

ixx

et

5

1

1~ R

m

em

5

4

4~ R

m

em

5

3

3~ R

m

em

0~,~

,~,~

43

21

tmmw

tmmw

0??,

,??,

wt

wt

§3 Finding the map

2 MNConcentrate on Self duality of SU(2) 4 flavor

Asymptotic behavior of the NS5-brane

NS5

D4

x4

x6

x5 54 ixxv

av

Minimize the volume

)(62 avx

11

116

R

ixx

et

1

211

116

21

6

)(

log

log

act

cavRixx

vcavcx

if x5 is not compactified

Under the periodic boundary condition

NS5

D4

x4

x6

x5

54 ixxv

av

Minimize the volume

)(sinh

)sinh(log

)2log(

1

5

11

116

avt

cav

vcnRavR

ixx

n

av

52 Rav

52 Rav 54 Rav

n

nRavx )2( 5

62

when x5 is compactified

NS5

x4

x6

x5

4,3,2,1~,,, 55

116

554

IemetewixxvR

m

I

R

ixx

R

v I

av av

2mv

1mv

5

2

1

21

2

1

21

21)1(

0~~

~~

)sinh()sinh(

)sinh()sinh(

RC

wmm

wmmC

avav

mvmvCt

4mv

3mv

5

2

1

43

2

1

43

43

)2(

0~~

~~

)sinh()sinh(

)sinh()sinh(

RC

wmm

wmmC

mvmv

avavCt

NS5

C

Ceq i

2

2~mw

1~mw

4~mw

3~mw

2

1

21~~ mmCt

2

1

21~~

mmCt

2

1

43~~ mmCqt

2

1

43~~

mmCqtw

t

0~~~~~~~~

0~~~~

2

1

432

1

21

22

1

432

1

21

43

2

21

mmCqtmmCtDwwmmCqtmmCt

mwmwdttmwmw

M5-brane configuration 2

1

32

4121

4

1

~~

~~

q

mm

mmCaaa ddd

Consistent boundary

condition is restricted!

0~~~~

~~

~~~~~~~~~~

~~

43

2

1

43

21

2

1

432

1

2121

22

1

432

1

21

2

21

mwmwmm

mmq

C

tmmqmmmmUwwmmqmm

C

tmwmw

1q

0~~~~

~~

~~~~~~

43

2

1

43

21

2

1

21

22

1

21

2

21

mwmwmm

mmq

tmmUwwmmtmwmwC

tt

5R

Relation to known curve

5D SU(2) curve

0

)1(

43

4

1

22

21

mvmvq

tuvmqvqtmvmvI

I

’96 Nekrasov

’97 Brandhuber, Sonnenschein, Itzhaki

Theisen, Yankielowicz

’00 Eguchi, Kanno

(’94 Seiberg, Witten)

’09 Eguchi, Maruyoshi

5~ R

m

I

I

em

(’97 Minahan, Nemeschansky, Warner)

2~mw

1~mw

4~mw

3~mw

2

1

21~~ mmCt

2

1

21~~

mmCt

2

1

43~~ mmCqt

2

1

43~~

mmCqtw

t 1A

d

A1

1M

1T

011

SWSWSWSW MAAT d

dA

A

M

T

a

a

m

mmC

d

SW

SW

1SW

2

1

21SW

1

1

1

~log

~~log

2

1

32

4121

4

1

~~

~~

q

mm

mmCaaa ddd

tdwwdt loglogloglogSW

2

1

4

1

4

1

4

1

4

1

4

1

31

422

1

31

42

2

1

2

3

4314

2

1

3

3

21

43

2

1

4

3

32

12

2

1

3

42

3

11

~~

~~,~

~~

~~~

~

~~~~,~~~

~~

~~~

~~,~

~~~~

mm

mmqaq

mm

mma

qm

mmmmq

mmm

mm

qmmm

mmq

m

mmmm

dd

dd

dd

Self-duality map for SU(2) with four flavor

4,3,2,1~,~

4,3,2,1~,~

55

55

Iemea

Iemea

R

m

I

R

a

d

R

m

I

R

a

dId

I

Duality map for SU(N)M-1 ⇔ SU(M)N-1

Naa

aaq

aaqaC

N

Miq

aa

aa

Ca

M

M

d

N

d

NM

i

M

iMi

N

MM

i

k

k

Nii

Nii

di

,,1,~~

~~

1~,1~~

,,0

,,1,

~~

~~

~

2

1

)()0(

)(

1

)0(

1)(

1

)0(

1

)0(1

1

)(

1

2

1)(

1

1

)(

2

1

1 1

)()1(

1

)1(

1

)(

)(

Niii

Ni

Nii,i

NMM

N

N

aN

aa

aN

aN

m

aN

am

aaN

m

1

)()()(

1

)1(

1

)()1(

bif

1

)1()(f

)0(

1

)1(af

1

11

1

1

5

)(

)(~ R

a

i

i

ea

§4 Summary and discussion 1

The 5D SU(N)M-1 gauge theory compactified on S1 is dual to

the SU(M)N-1 theory.

We find the explicit map for the gauge theory parameters,

under which the identical Seiberg-Witten curve is obtained

from the dual theories.

Summary

・Translate the duality into 2D CFT

Via five dimensional extension of AGT correspondence,

N+2 point function of “q-deformed AM-1 Toda theory”

= M+2 point function of “q-deformed AN-1 Toda theory”

・Translate the duality into Matrix model

(q-deformed) New Dijkgraaf-Vafa matrix model for SU(N)M-1

(’09 Dikgraaf, Vafa, ’09 ITEP group, 09’ Itoyama .et.al ,09’ Eguchi Maruyoshi …)

⇔ (q-deformed) Eguchi-Yang type matrix model (’09 Sulkowski et.al)

for SU(M)N-1 (or old Dijkgraaf-Vafa matrix model (’02 Dijkgraaf-Vafa) ) ?

・Extension to the generic omega background

(especially in the Nekrasov Schatashvili limit):

→ duality for “quantum Seiberg-Witten curve”

(= Hamiltonian of the integrable system)

Discussion

32

414

31

423

43

21243211 ~~

~~~,~~

~~~,~~

~~~,~~~~~

mm

mmM

mm

mmM

mm

mmMmmmmM

aqaq

MM

MqqM

MMMM

d

d

d

dd

dd

~~

~~

,~

,~

,~~

,~~

2

1

2

1

22

33

1441

Duality map for SU(2) theory can be nicely rewritten as

§5 Implication of the duality map

Weyl reflection symmetry of the global symmetry SO(8)

1~~

~~

ji

ji

mm

mm

1~~

~~

ji

ji

MM

MMor equivalently

+ Duality map

220341

~~,

~~,

~~MMMqMMM

=

Weyl reflection symmetry of

the enhanced global symmetry SO(10) .

1~~,

~~ JIJI MMMM

ji

ji

4,,1,

JI

JI

4,,1,0,

’96 Seiberg

Deformation of the 5D N=1 SU(2) four flavor

Two NS5

Two D4

UV fixed point

(CFT) IR

NS5 NS5

D4

D4

SO(10) global symmetry Weyl reflection symmetry

of SO(10)

Mass deformation

Finite coupling deformation

Go to Coulomb branch

(S1 compactification)

Seiberg’s claim 1. UV fixed points of five dimensional N=1 SYM with SU(2) Nf flavor exist for

Nf ≦ 7 (massless, infinite coupling)

2. They have the global symmetry (stringy argument)

8E

7E

6E

)10(5 SOE

)5(4 SUE

)2()3(3 SUSUE

)2()2(2 SUSUE

)2(1 SUE

1fNE

We have studied SU(2) with four flavor

and partially checked

the enhanced global E5=SO(10) symmetry.

How about SU(2) with five flavor?

Can we check the enhanced

global E6 symmetry?

E6 CFT (UV fixed point)

E6 CFT from M5-brane

Until recently, the brane setup for

E6 CFT ( 5D SU(2) five flavor ) was not known

D4

D4

D4

D4

D4

D4

D4

NS5 NS5

NS5

NS5

§6 Introduction 2

Recently, Benini ,Benvenutti, and Tachikawa

proposed the corresponding brane set-up

(in the type IIB picture)

Three NS-5 branes

Three D5 branes

Three (1,1)-5 branes

(bound state of D5 and NS5)

6)3()3()3( ESUSUSU

Recently, Benini Benvenutti Tachikawa

proposed the corresponding brane set-up

(in the type IIB picture)

Three NS-5 branes

Three D5 branes

Three (1,1)-5 branes

(bound state of D5 and NS5)

M5 wrapping x11

M5 wrapping x5

M5 wrapping

both x5 and x11

11x

5x

11x

5x

11x

5x

Calculate

the M5-brane configuration

= (R3,1 ×) Seiberg-Witten curve

of this setup

and check the Weyl reflection

invariance of the E6 symmetry

(not only SU(3) x SU(3) x SU(3) )

Purpose

§6 Introduction 2

§7 M5-brane configuration and E6 symmetry

§8 Summary and discussion

Plan of the second half E6 CFT from M-theory

§7 M5-brane configuration

and E6 symmetry

Boundary condition

M5 cf

11x

5x

11

116

5

54

,R

ixx

R

ixx

etew

x4

x6

We take into

account the brane tension

§7 M5-brane configuration

and E6 symmetry

Boundary condition Choose the convention

1~~~1~~~

321

321

nnn

mmm

11

116

5

54

,R

ixx

R

ixx

etew

From the consistency

1~~~321 lll

Seiberg-Witten curve

0~~1

~~

~~

321

211

22

3

ttntn

wtlwtUwm

twlwm

w

i ii i

i ii i

i ii i

By construction Weyl reflection symmetry of SU(3)xSU(3)xSU(3) is manifest

baJIji llnnmm~~

,~~,~~

3,2,1,3,2,1,3,2,1, baJIji

Coordinate transformation

)~~~)(

~~~)(~~~(

~,)~(~~

33

4

33

1

223

4

33

1

213

4

33

1

2

33

3

4

33

1

2

lnnWlnnWlnnW

TntWntnnw

Seiberg-Witten curve in the new coordinate

0)~

(~

1~

26

1

6

1

26

1

32

k

k

k

k

k

k MWTMWUWMWT

33

1

33

1

2623

1

33

1

2513

1

33

1

24

33

1

33

1

1323

1

33

1

1213

1

33

1

11

~~~~,~~~~

,~~~~

~~~~,

~~~~,

~~~~

mnnMmnnMmnnM

lnnMlnnMlnnM

Weyl reflection symmetry of SU(6) is manifest !

lk MM~~

6,,1, lk

ba ll~~

Before coordinate transformation After coordinate transformation

lk MM~~

6)3()3()3( ESUSUSU

ji mm ~~

JI nn ~~

6)6( ESU

Whole Weyl reflection symmetry of E6 is reproduced!

§8 Summary and discussion

The 5D SU(N)M-1 gauge theory compactified on S1 is dual

to the SU(M)N-1 theory.

We find the explicit map for the gauge theory parameters,

under which the identical Seiberg-Witten curve is

obtained from the dual theories.

The explicit map for SU(2) with four flavor is available to

check the enhanced SO(10) global symmetry.

The similar method is available to check the global

symmetry of the E6 CFT

Summary

Discussion

Seiberg Witten curve after the coordinate transformation

0)~

(~

1~

26

1

6

1

26

1

32

k

k

k

k

k

k MWTMUWWMWT

looks like special case of 5D SU(3) with six flavor with constraint

6

1

2 ~

k

kMq

together with the tuning of the Coulomb moduli parameter

It should be related to the discussion by Argyres and Seiberg that

4D E6 CFT is realized by the strong coupling limit of SU(3) 6 flavor

Mlnm eMelenem ~~~~

Generators of the Weyl symmetry

ijijji Cr

216325214

433322211

:::

:::

nnrmmrmmr

MMrllrllr

3r

five dimensional duality and e cft from m5...

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