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
multiplet Anti-fundamental
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
multiplet Anti-fundamental
hypermultiplet
…
NS5
Bi-fundamental
hypermultiplet Fundamental
hypermultiplet
M NS5-branes
N D4 N D4
x5 direction is periodic
52 R
Type IIA
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
multiplet Anti-fundamental
hypermultiplet
…
NS5
Bi-fundamental
hypermultiplet 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
5d N=1 SU(N)M-1 gauge theory
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 Fundamental
hypermultiplet
M NS5-branes
N D4
D4
Type IIA
Bi-fundamental
mass
D4
D4 Anti-fundamental
mass D4
Anti-fundamental
mass
D4
N
…
5d N=1 SU(N)M-1 gauge theory
with 2N flavor on S1 with mass deformation
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
5d N=1 SU(N)M-1 gauge theory
with 2N flavor on S1 with mass deformation
in the Coulomb phase
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
5d N=1 SU(N)M-1 gauge theory
with 2N flavor on S1 with mass deformation
in the Coulomb phase
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
Local structure of the intersection
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)
Local structure of the intersection
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
・Seen as a single M5-brane (IR theory is Abelian)
x4
x6
M5
Local structure of the intersection
11x
M5(D4)
M5(NS5)
M5
5x
D4 NS5
M5(D4)
11x
5x
NS5: M5 wrapping x5
D4: M5 wrapping x11
Comment
・The NS5-brane is pulled by the D4-branes
・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