6d Superconformal Field
Theories
Kimyeong Lee
KEK Theory Workshop, January 2015
Jungmin Kim, Seok Kim, KL, Little strings and T-duality, to appear
Joonho Kim, Seok Kim, KL, Cumrun Vafa [1411.2324] Elliptic genus of E-strings
Hee-Cheol Kim, Seok Kim, Sung-Soo Kim, KL [arXiv:1307.7660] The general M5-brane superconformal Index
Hee-Cheol Kim, KM [arXiv:1210.0853] M5 brane theories on R x CP2
Hee-Cheol Kim, Seok Kim, Eunkyung Ko, KL [arXiv:1110.2175] On instantons as KK modes of M5 branes
Stefano Bolognesi, KL [arXiv:1105.5073] 1/4 BPS string junctions and N3 problem in 6-dim conformal field theories
Outline
•
M5 Branes
•
N=2 SCFT
•
N=1 SCFT
•
6d Little String Theories
•
Summary
M5 Branes
•
M Theory: 11-dim, only one parameter !P
•
Low energy mechanics: GMN, CMNP, ΨM
•
M2 & M5 branes are electric and magnetic objects
of tension 1/ !P3, 1/!P6
•
Strongly interacting: electric and magnetic strength
of order one
•
6d (0,2) AN-1 superconformal field theory
•
AdS7 X S4 geometry
M5 Branes
•
Single M5 brane: abelian
•
low energy mechanics: Bμν,Ψα,ΦI , H=dB=*H
•
•
3+5=8, self-dual => quantum, chiral fermion (2,0)
2 M5 branes in Coulomb phase
•
M2 brane connecting M5 branes: 1/2 BPS selfdual
string
•
tensionless string in the symmetric phase
•
Lagrangian for nonabelian symmetry is not known
•
N3 degrees of freedom, anomaly, AdS/CFT
6d (2,0) SCFTs
•
ADE classification, type IIB on R1+5xC2/ZK
•
DN: OM5+ N M5 branes
•
Coulomb Phase:
•
•
1/2 BPS objects:
•
massless tensor multiplets : O(N)
•
selfdual strings : Order (N2)
1/4 BPS objects: later….
5d N=2 SYM on R1+4
Douglas (10),Lambert-Papageorgakis-Schmidt-Sommerfeld (10)
•
x5~x5+ 2πR,
•
Instantons= Kaluza-Klein modes, 8π2/gYM2 = 1/R
•
duality between KK modes and instantons
•
threshold bound state of k instantons
•
strong coupling limit = 6d (2,0) SCFT theory
•
Perturbative Approach: 6-loop divergence
•
UV incomplete
Bern et.al. (12)
YΜ couplings and off-shell 1/4 BPS objects
•
•
•
[ta,tb]=ifabc tc
i
•
structure constant fabc
j
•
fα-αh, [H,E±α]=±αΕ±α,
wave on selfdual strings
self-dual string junctions
•
i
j
k
[Ε+α,Ε-α]=α⋅Η
•
fαβγ, [Εα,Εβ]=fαβγ Εγ
•
|fabc|=0 or 1
3
N
•
Anomaly polynomial:
•
•
•
dimension of group*dual Coxeter number
Counting 1/4BPS object
# of root (fα-αh)+ # of junction (fαβγ) = hd/3
•
sum of |fabc|2 (two-loop) = hd/6
•
Weyl vector ρ= 1/2 sum of positive roots, |ρ|2=hd/6
N (N
2
1)
+
N (N
1)(N
6
2)
N (N 2
=
6
1)
= hG dG
High temperature Phase
8
1
8
7 4
4
5 1
2 8
8
3
1
6
3
1
2
7
5
8
6
Counting Instantons on R1+4
Index for BPS states with k instantons
h
i
F
Q2
Ik (µ , 1 , 2 , 3 ) = Trk ( 1) e
e
•
•
Q = Q+̇
+̇
i
µ ⇧i
e
i
1 (2J1L )
i
2 (2J2L )
i
) SU (2)R
R (2JR )
i
adjoint hyper flavor
U(1)N ⊂"U(N)color
•
µi : chemical potential for
•
γ1, γ2, γR : chemical potential for SU(2)1L, SU(2)2L , SU(2)R
calculate the index by the localization:
SU (2)2R
SU (2)1R
I(q, µi ,
1,2,3 )
=
1
X
q k Ik
k=0
•
5d N=2*% instanton partition function on R4 x S1:
•
In β → 0 and small chemical potential limit, the index becomes 4d Nekrasov
instanton partition function :
µi
1
R
1+ R
2
2⇡i⌧
ai =
✏1 = i
✏2 = i
, m=i
q
=
e
2
2
2
2
t ～ t+ β
instanton fugacity
Scalar Vev
Omega deformation parameter
Adj hypermultiplet mass
6d (2,0) Index Function
H.C. Kim, S. Kim(12); H.-C. Kim,J. Kim, S.Kim(12)
Minahan, Nedelin, Zabzine,(12)
•
Partition function on S1xS5
•
Radius of S1= β
•
small beta: S5 partition function of 5d N=2 SYM
•
large beta: S5=S1 fiber over CP2
•
Zk modding of S1 fiber with twist
•
5d supersymmetric Yang-Mills Chern-Simons
theory
5d SYMCS on
2
RxCP
✴
Lagrangian on R x CP2 with 2 supersymmetries for any p:
✴
Supersymmetry Transformation
✴
p/2=-1/2 : k = j1+j2+j3+ R1+ 2R2
✴
Q = Q++ , S = Q+++
Jmn: Kahler 2-form of CP2
additional supersymmetries: Total 8 supersymmetries
Q+++ , Q+
+
+
,
Q
++
+
conjugates
2
Localization of 5d theory on RxCP
•
Quantization of the coupling constant: K/4π2
•
’t Hooft coupling: λ = Ν/Κ
Expected supersymmetries
•
•
K ≧ 4: 8 supersymmetries
•
K=3: 10 supersymmetries
•
K=2: 16 supersymmetries
•
K=1: 32 supersymmetries
•
three fixed points of CP2
•
Ground State for U(N): uniform anti-instanton background
F=2(N-1,N-3,N-1,…,-(N-3),-(N-1)) J
•
Vacuum energy E0=-N(N2-1)/6
•
higher fluxes + localized instantons
•
Field theory calculation matches AdS/CFT calculation.
6d (1,0) SCFT
Witten(95), Ganor and Hanany (96),
Seiberg and Witten (96),Barshadsky and Johansen (96),
Morrison ad Vafa (96), Witten (96)…
Heckman, Morrison, Vafa(13),Del Zotto et.al (14),
Gaiotto and Tomasiello (14), Morrison and Taylor (12)
•
M5 brane near M9 E8 Wall
•
M5 on RxC2/ΓADE
•
F-theory on elliptically fired CY 3-fold with base B,
•
•
D3 brane wrapping collapsed cycle in B=tensionless string
F-theory construction of minimal model (single tensor multiplet)
•
elliptic fibration over Hirzebruch surfaces Fn (n=0,1,2,…,12)
•
small (12+n,12-n) instantons of E8xE8 string theory
•
F theory on simple orbifold of C2xT2 with (x,y,λ)->( ζx,ζy,ζλ)
n=3,4,5,8,12
E8 (1,0) Theory
•
Joonho Kim, Seok Kim, KL, Jaemo Park, Cumrun Vafa (14)
M2 branes between 2 M5 branes =
M-string elliptic genus
•
M2 branes between M5 and M9
branes = E-string elliptic genus
•
wrap x11 to a circle with E8 Wilson line
248 →120+128 with SO(16) symmetry
•
D8+O8,NS5,D2,
•
D6(un-compactify S1 in IR)
Hwang, JKim,SKim,Park(14), Haghighat,Lockhart,Vafa(14),
Cai,Huang,Sun(14),Haghighat,Klemm,Lockhart,Vafa (14),..
M9
M2
M5
x0x9 2d SQFT on D2
branes
UV theory on D2 branes
•
The theory on D4 (wrap instead NS5)=Sp(k) theory
•
Symmetries SO(4)1234xSO(3)567
=SU(2)LxSU(2)RxSU(2)I
, , · · · = 1, 2 ˙ , ˙ , · · · = 1, 2 A, B, · · · = 1, 2,
•
•
boundary condition + boundary degrees of freedom
2d field content:
vector : O(n) antisymmetric (Aµ ,
hyper : O(n) symmetric (
Fermi : O(n)
, ˙,
˙A
+ )
A
)
SO(16) bifundamental
•
2d N=(0,4) SUSY Q ˙ A dictates the interaction
•
SO(16)→E8 symmetry enhancement in IR
l
Elliptic Genus
•
•
Gadde and Gukov (13),
Benini,Eager,Hori,Tachikawa I,II(13)
Take (0,2) subset of (0,4) SUSY,
Define partition function for n-strings
8
Zn (q,
1,2 , ml )
= TrRR ( 1)q HL q̄ HR e2
i
1 (J1 +J2 )
e2
2 (J2 +JI )
e2
iml Fl
l=1
J1 , J2 , JI are the Cartans ofSU (2)L SU (2)R SU (2)I
Fl are the Cartans of SO(16)
•
All string sum: Z = Σn=0 Zn
•
Path integral representation of Zn
Holonomy
•
•
gauge zero mode= O(n) flat connection =O(2p) and O(2p+1) cases
eigenvalues ui=u1i+τu2i, of the holonomy exp(u1i σ2), exp(u2i σ2)
O(2p)
O(2p+1)
Determinant
•
hyper, fermi, vector
•
Integration: Jeffery-Kirwan Residues
Benini-Eager-Hori-Tachikawa
Calculations..
•
single string: Ganor and Hanany, Klemm, Mayr and Vafa
•
two E-strings: Haghighat, Lockhart,Vafa
•
3,4 E-strings, any E-strings…
•
5d YM theory on D4 with Nf=8: Hwang, Kim ,Park
•
E8 symmetry is manifest for lower number of strings
2
K. Mohri (02), K. Sakai (14)
Cai, Huang, Sun(14)
Little String Theories
•
Low energy dynamics of NS5 branes + fundamental strings in
the limit where gravity decouples
•
type IIA, compactify one of R5 transverse to 6d (2,0) SCFT
•
type IIB, S-dual of D5-D1 system and decouple gravity
•
UV completion of 6d N=2 SYM theory (ADE)
•
Two theories on R1+4xS1 with momentum p and winding w are
T-dual to each other with exchange of p and w.
•
elliptic genus of instanton strings and M-strings are needed to
show this.
•
to appear soon Jungmin Kim, Seok Kim, KL
Conclusion
•
Very rich structures on 6d (2,0) and (1,0) SCFTs
•
A lot to explore and calculate
•
the 4d reduction of (1,0) theories on Riemann surface is
interesting
•
There are more of 6d little string theories to be discovered.