6d Superconformal Field Theories

Transcription

1 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 [ ] Elliptic genus of E-strings Hee-Cheol Kim, Seok Kim, Sung-Soo Kim, KL [arxiv: ] The general M5-brane superconformal Index Hee-Cheol Kim, KM [arxiv: ] M5 brane theories on R x CP2 Hee-Cheol Kim, Seok Kim, Eunkyung Ko, KL [arxiv: ] On instantons as KK modes of M5 branes Stefano Bolognesi, KL [arxiv: ] 1/4 BPS string junctions and N3 problem in 6-dim conformal field theories

2 Outline M5 Branes N=2 SCFT N=1 SCFT 6d Little String Theories Summary

3 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/!P 3, 1/!P 6 Strongly interacting: electric and magnetic strength of order one 6d (0,2) AN-1 superconformal field theory AdS7 X S 4 geometry

4 M5 Branes Single M5 brane: abelian low energy mechanics: B μν,ψ α,φ I, 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 N 3 degrees of freedom, anomaly, AdS/CFT

5 6d (2,0) SCFTs ADE classification, type IIB on R 1+5 xc 2 /ZK DN: OM5+ N M5 branes Coulomb Phase: 1/2 BPS objects: massless tensor multiplets : O(N) selfdual strings : Order (N 2 ) 1/4 BPS objects: later.

6 5d N=2 SYM on R 1+4 x 5 ~x 5 + 2πR, Douglas (10),Lambert-Papageorgakis-Schmidt-Sommerfeld (10) Instantons= Kaluza-Klein modes, 8π 2 /gym 2 = 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)

7 YΜ couplings and off-shell 1/4 BPS objects wave on selfdual strings i j self-dual string junctions [ta,tb]=if abc tc structure constant f abc f α-αh, [H,E±α]=±αΕ±α, [Ε+α,Ε-α]=α Η i f αβγ, [Εα,Εβ]=f αβγ Εγ j f abc =0 or 1 k

8 N 3 Anomaly polynomial: dimension of group*dual Coxeter number Counting 1/4BPS object # of root (f α-αh )+ # of junction (f αβγ ) = 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 1) 2 + N(N 1)(N 2) 6 = N(N 2 1) 6 = h G d G

9 High temperature Phase

10 Counting Instantons on R 1+4 Index for BPS states with k instantons I k (µ i, 1, 2, 3) =Tr k h( 1) F e Q = Q + + SU(2) 2R SU(2) 1R Q 2 e µi i e i 1(2J 1L ) i 2 (2J 2L ) i R (2J R ) i ) SU(2) R µi : chemical potential for U(1) N "U(N)color adjoint hyper flavor γ1, γ2, γr : chemical potential for SU(2)1L, SU(2)2L, SU(2)R calculate the index by the localization: I(q, µ i, 1,2,3) = 1X q k I k k=0 5d N=2*% instanton partition function on R 4 x S 1 : t ~ t+ β In β 0 and small chemical potential limit, the index becomes 4d Nekrasov instanton partition function : a i = µ i 2 Scalar Vev 1 = i 1 2 R 2 = i 1 + R 2 Omega deformation parameter, m = i 2 2 q = e 2 i Adj hypermultiplet mass instanton fugacity

11 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 S 1 = β small beta: S 5 partition function of 5d N=2 SYM large beta: S 5 =S 1 fiber over CP 2 Zk modding of S 1 fiber with twist 5d supersymmetric Yang-Mills Chern-Simons theory

12 5d SYMCS on RxCP 2 Q = Q ++,S = Q +++ Lagrangian on R x CP 2 with 2 supersymmetries for any p: Supersymmetry Transformation Jmn: Kahler 2-form of CP 2 p/2=-1/2 : k = j1+j2+j3+ R1+ 2R2 additional supersymmetries: Total 8 supersymmetries Q + ++, Q + + +, Q + ++ conjugates

13 Localization of 5d theory on RxCP 2 Quantization of the coupling constant: K/4π 2 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 CP 2 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(N 2-1)/6 higher fluxes + localized instantons Field theory calculation matches AdS/CFT calculation.

14 6d (1,0) SCFT M5 brane near M9 E 8 Wall 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 on RxC 2 /Γ ADE 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 F n (n=0,1,2,,12) small (12+n,12-n) instantons of E 8 xe 8 string theory F theory on simple orbifold of C 2 xt 2 with (x,y,λ)->( ζx,ζy,ζλ) n=3,4,5,8,12

15 E8 (1,0) Theory Joonho Kim, Seok Kim, KL, Jaemo Park, Cumrun Vafa (14) Hwang, JKim,SKim,Park(14), Haghighat,Lockhart,Vafa(14), Cai,Huang,Sun(14),Haghighat,Klemm,Lockhart,Vafa (14),.. M2 branes between 2 M5 branes = M-string elliptic genus M9 M2 M5 M2 branes between M5 and M9 branes = E-string elliptic genus wrap x 11 to a circle with E 8 Wilson line with SO(16) symmetry D8+O8,NS5,D2, D6(un-compactify S1 in IR) x 0 x 9 2d SQFT on D2 branes

16 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 (A µ, hyper : O(n) symmetric(,, A + ) A ) Fermi : O(n) SO(16) bifundamental l 2d N=(0,4) SUSY Q A dictates the interaction SO(16) E8 symmetry enhancement in IR

17 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 Z n (q, 1,2,m l )=Tr RR ( 1)q HL q H R e 2 i 1(J 1 +J 2 ) e 2 2(J 2 +J I ) 8 l=1 e 2 im lf l J 1,J 2,J I are the Cartans ofsu(2) L SU(2) R SU(2) I F l are the Cartans of SO(16) All string sum: Z = Σn=0 Zn Path integral representation of Zn

18 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)

19 Determinant hyper, fermi, vector Integration: Jeffery-Kirwan Residues Benini-Eager-Hori-Tachikawa

20 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 N f =8: Hwang, Kim 2,Park E 8 symmetry is manifest for lower number of strings K. Mohri (02), K. Sakai (14) Cai, Huang, Sun(14)

21 Little String Theories Low energy dynamics of NS5 branes + fundamental strings in the limit where gravity decouples type IIA, compactify one of R 5 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 R 1+4 xs 1 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

22 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.