6d SCFTs and Tao diagrams

Transcription

1 6d SCFTs and Tao diagrams Kim, Sung-Soo Korea Institute for Advanced Study APCTP Focus Research Program (FRP)

2 This work is based on various collaborations with Hirotaka Hayashi (Instituto de Fisica Teorica), Kimyeong Lee (KIAS), Masato Taki (RIKEN), Futoshi Yagi (KIAS) arxiv: arxiv: arxiv:

3 We report a new understanding of 5d SCFTs in connection with 6d SCFTs through Tao web diagrams

4 6d SCFTs (2,0) theory: worldvolume theory of M5 branes but mysterious (1,0) theory: F-theory classification. [Del Zotto-Heckman-Tomasiello-Vafa 14] [Heckman-Morrison-Vafa 13] [many more ]

5 5d/6d relation: N=(2,0) A circle compactification of 6d (2,0) theory = 5d Maximally supersymmetric Yang-Mills theory g 2 YM = R 6 circle radius

6 E-string theory 5d/6d relation: N=(1,0) One of the famous examples of (1,0) theory: M2 M5 F1 D4 D0 M9 E-string O8 8 D8 s 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 ]

7 5d/6d relation: N=(1,0) E-string theory M2 M5 F1 D4 D0 M9 E-string O8 8 D8 s E-string theory on a circle = 5d SU(2) theory with Nf=8 KK modes = Instantons UV completion is a 6d SCFT!

8 Q1: How do we determine what 5d theories have UV completion as 6d SCFTs?

9 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

10 (p,q) web diagram and 5d SU(2) theory [Aharony-Hanany, 97] Type IIB configuration with charge conservation, tension balance:(p,q) web diagram NS5 D5 (1,0) D5-brane (1,1) (0,1) NS5-brane (1,1) (1,-1) 6

11 Flavors = semi infinite D NS5 D5 F1 D1 (1,1) (1,1) (1,-1) 6

12 5d SU(2) theory via 5-branes and 7-branes [1,-1] 7-brane [1,1] 7-brane NS5. D5. D7..

13 Flavors = D7 brane Nf= 0 Nf= 1 Nf= 2 Nf= 3 Nf= 4

14 Five, six, seven flavors seem problematic, N f =5 N f =6 N f =7 but possible to make sense using 7-brane monodromies e.g., Nf=5 [Benini-Benvenuti-Tachikawa, 09] finite configuration diagram

15 5d SCFTs Flavor decoupling of 5d SU(2) theory of Nf=8 SU(2) gauge theory with Nf = 0,1,,7 flavors has non-trivial 5d UV fixed point (Superconformal theory) [Seiberg 96]

16 Brane configuration for 5d SU(2) theory with Nf=8 flavors

17 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

18 The shape looks like or [p,q] [-p,-q] or

19 Laozi We call it Tao diagram Tao ( ) probes the end of the world (M9) arxiv:

20 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

21 spiral with a constant period

22 spiral with a constant period KK spectrum

23 Period period = instanton ~ R -1

24 Tao diagrams give new perspective on 6d SCFTs Tao diagram: infinite spirals (KK spectrum) constant period (compactified radius) - Naturally identified as a 6d theory on a circle (compactification radius emerges ) - Computational tool: Partition function

25 Partition function from Tao diagram h X 1 Z E-string =PE m=0 i h F m (y, A, q)q m 1 =PE (1 q)(1 q 1 ) 1X n=0 1 f n A n i, f 1 = (1) + c q (q) (1) + (3) + (1) q 2 (1) + s +2 2 (q) c q (q)+4 2 (q)+2) (1) +2 2 (q) (3) + (5) + (1) (2) q 4 + O(q 5 ), (4.54) f 2 = 2 2 s q 2 (4) +(3 2 (q)+2) (2) +4( 3 (q)+ 2 (q)+1) q 2 2 (2) s +3 2 (q) (1) c +4( 3 (q)+ 2 (q)+1) s + (5 4 (q)+6 3 (q)+11 2 (q)+8) (2) +(4 3 (q)+4 2 (q)) (4) +(3 2 (q) 2) (6) q 3 +(4 3 (q)+3 2 (q)+2)( (1) ) (q) (1) (3) +2 (1) (5) +2( (2) ) 2 +2( s ) 2 +(6 5 (q)+8 4 (q)+16 3 (q)+20 2 (q)+10) q 4 + O(q 5 ). (4.55) arxiv: reproduces the E-string partition function (elliptic genus) by (up to 4 instantons) [ 14 Kim, Kim, Lee, Park, Vafa ] Tao diagram indeed sees the E-string theory on a circle

26 Claim: Tao web diagrams imply that a 5d theory has UV completion as a 6d SCFT

27 Many more Tao web diagrams 5d SU(N) Nf=2N+4 arxiv:

28 { { 0, N f =0, 1,, 2N, 2N +1, 2N +2, 2N +3, 2N +4

29 { { { 5d SU(N>3) theories N f =0, 1,, 2N, 2N +1, 2N +2, 2N +3, 2N +4 5d SCFT [Intriligator-Morrison-Seiberg 97] [Bergman-Zafrir 14] 6d SCFT Tao diagram

30 What 6d SCFT is for this Tao? 5d SU(N) Nf=2N+4 arxiv:

31 5d N = 1, SU(N), N f =2N +4 N =2 N =3 N =4 SU(4),N f = 12

32 5d N = 1, SU(N), N f =2N +4

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34 6d N =(1, 0) Sp(N 2) gauge theory N f =2N + 4, w/tensor multiplet S 1

35 5d SU(N) N f =2N +4

36 5d SU(N) N f =2N +4

37 Quiver type?

38 5d [N + 2] SU(N) SU(N) [N + 2] k k =2n +1 6d Sp(N 0 ) SU(2N 0 + 8) SU(2N ) SU(2N 0 + 8(n 1)) [2N 0 +8n] k =2n N 0 = N 2n 6d [A] SU(N 0 ) SU(N 0 + 8) SU(N ) SU(N 0 + 8(n 1) [N 0 +8n + 8] N 0 = 2(N 2n + 1)

39 5d [N + 3] SU(N) SU(N 1) SU(N 2) SU(3) SU(2) [3] ( Tao-nization of 5d T N ) N =3n : 6d SU(0) SU(9) SU(9n) [9n + 9] N =3n + 1 : 6d SU(3) SU(12) SU(3 + 9(n 1)) [3 + 9n] N =3n + 2 : 6d 1 2 SU(6) SU(15) SU(6 + 9(n 1)) [6 + 9n] 3

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45 I II III 6d SU(6) theory with Nf=14, Na=1 IV V VI VII VIII IX X various 5d quivers (depending on D5, D7 distributions) XI XII

46 New understandings of 5d/6d SCFTs : Summary 6d SCFTs arxiv: d SCFTs Various dualities Tao diagram 5d SU(N) w/ Nf=2N+4 5d UV dualities