Slide

5d/6d SCFTs
on
Tao diagrams I
Kim, Sung-Soo
Korea Institute for Advanced Study
2015-11-17
This work is based on various collaborations with
Hirotaka Hayashi (Instituto de Fisica Teorica),
Kimyeong Lee (KIAS),
Masato Taki (RIKEN),
Futoshi Yagi (KIAS)
arXiv:1504.03672
arXiv:1505.04439
arXiv:1509.03300
We report
a new understanding of 5d SCFTs
in connection with 6d SCFTs
through “Tao web diagrams”
(An introduction to Tao diagrams)
Results of the work:
the next talk by Futoshi Yagi
6d SCFTs
(2,0) theory: worldvolume theory of M5 branes
but mysterious…
(1,0) theory: F-theory classification.
[Heckman’s lectures]
[Heckman-Morrison-Vafa ’13]
[Del Zotto-Heckman-Tomasiello-Vafa ’14]
[many more … ]
5d/6d relation: N=(2,0)
A circle compactification of 6d (2,0) theory
= 5d Maximally supersymmetric Yang-Mills theory
2
gY M
= R6
circle radius
5d/6d relation: N=(1,0)
One of the famous examples of (1,0) theory:
E-string theory
M2
M5
D4
F1
D0
M9
O8 8 D8’s
E-string
E-string theory on a circle = 5d SU(2) theory with Nf=8
KK modes = Instantons
E-string partition function
(elliptic genus)
[’14 Seok Kim, Joonho Kim, Kimyeong Lee, Jaemo Park, Vafa ]
5d/6d relation: N=(1,0)
E-string theory
M2
M5
D4
F1
D0
M9
O8 8 D8’s
E-string
E-string theory on a circle = 5d SU(2) theory with Nf=8
KK modes = Instantons
UV completion is a 6d SCFT !
Q1: How do we determine what 5d theories
have UV completion as 6d SCFTs?
Q1: How do we determine what 5d theories
have UV completion as 6d SCFTs?
A1: We developed a diagrammatic way
that distinguishes 6d UV completions
from 5d UV completions —> Tao diagram
(p,q) web diagram and 5d SU(2) theory
[Aharony-Hanany, ’97]
Configuration with charge conservation, tension balance
: the (p,q) web diagram
(1,0)
D5-brane
0 1 2 3 4 5 6 7 8 9
NS5
— —
— —
— —
D5
— —
— —
—
(1,1)
— —
— —
—
(0,1)
NS5-brane
—
-
(1,1)
-
5
(1,-1)
6
Brane configuration for 5d SU(2) theory with Nf=8 flavors
By pulling out 7-branes to infinity
(2,1)
(2,1)
(-2,-1)
(-2,-1)
Spirally rotating!
One revolution, charges remain the same.
Infinitely rotating spiral diagram
The shape looks like
or 陽
[p,q]
[-p,-q]
or 陰
We call it Tao diagram…
Tao (道) probes
the end of the
world (M9)…
arXiv:1504.03672
Laozi 老子
There are various equivalent forms of Tao diagram:
Another Tao diagram for SU(2) gauge theory with 8 flavors
This is more practical and useful for computations
spiral with a
constant period
spiral with a
constant period
KK spectrum
Period
period = instanton
-1
~R
Tao diagrams give new perspective
on 6d SCFTs
Tao diagram: infinite spirals (KK spectrum)
constant period (cptf. radius)
- Naturally identified as a 6d theory on a circle
(compactification radius emerges…)
-
Computational tool: Partition function
To compare to the partition function computed in [32, 39], we expand our
ms of the Coulomb
modulus
A (which
corresponds
to w in [39]) and obtain
Partition
function
from
Tao diagram
ZE-string
where
f˜1 =
1
hX
i
h
= PE
Fm (y, A, q)qm = PE
m=0
(1)
+
⇣
+ 3
f˜2 =
2
⇣
2
c
⇣
q+ 2
3 (q)
2 (q)
(1)
+ 4 2 (q) + 2)
⇣
2 sq
2 (4) + (3
(2)
⇣
+ (5
+
(1)
(3)
+
(1)
⌘
2
q +
(1)
s
+3
2 (q)
(1)
4 (q)
+6
3 (q)
+ 11
2 (q)
+ (4
3 (q)
+3
2 (q)
+ 2)(
+ (6
5 (q)
+8
4 (q)
+ 16
2 (q)
(2)
+ 8)
+ (4
(1) 2
) +3
3 (q)
+
+ 20
(5)
s
+2
(1
2 (q) c
+
3 (q)
2 (q)
2 (q)
(1) (2)
⌘
⌘
q 1)
+4
26
(1) (3)
⌘
2 (q))
+2
(4)
n=01
i
f˜n An ,
q3
q4 + O(q5 ),
⌘
(2)
+ 4( 3 (q) + 2 (q) + 1) q2
2 (q) + 2)
⌘
3
c + 4( 3 (q) + 2 (q) + 1) s q
+2
(3)
⇣
1
q)(1
1
X
+ (3
(1) (5)
+ 10) q4 + O(q5 ).
+ 2(
(4.54)
2 (q)
2)
(6)
(2) 2
) + 2( s )2
(4.55)
arXiv:1504.03672
Here f˜1 and f˜2 are precisely the ones that are used in [39]. We further checked fiveinstanton contribution by inputing yi with some specific values and found that F5 is also
reproduces
the E-string partition function (elliptic genus)
consistent . Hence, the result obtained from topological vertex amplitude based on the
agrees with the result obtained from the instanton calculus as well as elliptic
by (up Tao
to 4diagram
instantons)
[’14 Kim, Kim, Lee, Park, Vafa ]
6
genus of E-strings in [32, 39]7 .
Tao diagram
indeed
sees
the
E-string
theory
on
a
circle
5 Discussion
Claim:
Tao web diagrams imply
that a 5d theory has UV
completion as a 6d SCFT
Many more Tao web diagrams
5d SU(N) Nf=2N+4
For the corresponding 6d SCFT,
See Futoshi Yagi’s talk (Tao Talk II)
arXiv:1505.04439
Quiver type
New understandings of 5d/6d SCFTs : Summary
6d SCFTs
arXiv:1509.03300
Various dualities
5d SCFTs
Tao diagram
Tao Talk II by Futoshi Yagi
•
5d SU(N) w/ Nf=2N+4
•
5d UV dualities