Tao Probing the End of the World 2 Futoshi Yagi (KIAS) Based on - PowerPoint PPT Presentation

Tao Probing the End of the World 2 Futoshi Yagi (KIAS) Based on the collaboration with Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Masato Taki arXiv:1504.03672, 1509.03300, 1505.04439

§1 Review of the previous talk + α

09’ Benini Benvenuti Tachikawa ’96 Seiberg 5 d N = 1 SU (2) N f flavor , 0 ≤ N f ≤ 7 N f = 0 N f = 1 N f = 2 N f = 3 N f = 4 N f = 5 N f = 6 N f = 7

Period 6d KK mode || 5d Instanton “Tao diagram” Infinite spiral rotation, Periodic structure 5 d N = 1 SU (2) N f = 8 flavor “period” ∝ 1 1 R ∝ g 2

No consistent 5-brane web diagram We cannot move all the 7-branes to infinity 5d N = 1, SU (2), N f = 9 flavor

Finite diagram “Tao diagram” No diagram 5D UV fixed point 6D UV fixed point No UV fixed point Observation For 5 d N = 1 SU (2) , N f flavor 0 ≤ N f ≤ 7 N f = 8 N f ≥ 9

Finite diagram: “Tao diagram”: No diagram: 5D UV fixed point 6D UV fixed point No UV fixed point Conjecture Tao diagrams for “class T ”

§1 Overview of the previous talk + Conjecture §2 Evidence for the conjecture §3 Generalization §4 Conclusion Plan of this talk

§2 Evidence for the conjecture

Finite diagram: “Tao diagram”: No diagram: 5D UV fixed point 6D UV fixed point No UV fixed point Conjecture Tao diagrams for “class T ”

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

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

Conjecture For 5d N = 1 SU ( N ), N f flavor, Chern-Simons level κ 6d fixed point for N f = 2 N + 4, κ = 0

Bergman, Zafrir ‘14 Via “Mass deformation” Conjecture For 5d N = 1 SU ( N ), N f flavor, Chern-Simons level κ 6d fixed point for N f = 2 N + 4, κ = 0 5d fixed point for N f < 2 N + 4, κ ≤ 2 N + 4 − N f No fixed point for others

Bergman, Zafrir ‘14 M5-brane probing D N+2 singularity “(D N+2 , D N+2 ) conformal matter” Del Zotto - Heckman - Tomasiello - Vafa ’14 Via “Mass deformation” Conjecture For 5d N = 1 SU ( N ), N f flavor, Chern-Simons level κ 6d fixed point for N f = 2 N + 4, κ = 0 5d fixed point for N f < 2 N + 4, κ ≤ 2 N + 4 − N f No fixed point for others

5d SU(N>3) theories [Intriligator-Morrison-Seiberg ’97] “All” UV complete theories were claimed to be classified. Comments on the previously known classification

5d SCFT 5d SU(N>3) theories [Intriligator-Morrison-Seiberg ’97] “dead” (Landau pole) Comments on the previously known classification N f = 0 , 1 , · · · , 2 N, 2 N + 1 , 2 N + 2 , 2 N + 3 , 2 N + 4 0 , 1 , · · · , 2 N { {

O verlooked for 20 years 5d SU(N>3) theories [Intriligator-Morrison-Seiberg ’97] [Bergman, Zafrir ’14] Comments on the previously known classification N f = 0 , 1 , · · · , 2 N, 2 N + 1 , 2 N + 2 , 2 N + 3 , 2 N + 4 0 , 1 , · · · , 2 N 2 N + 4 { { This talk { Previously known 5d SCFT

M5-brane probing D N+2 singularity Tensor branch (≒ Coulomb branch) O8 (2N+4) D8 NS5 (2N-4) D6 Brunner, Karch ’97, Hanany, Zaffaroni ’97 5 7,8,9 6d N = (1 , 0) Sp ( N − 2) gauge theory N f = 2 N + 4, w/tensor multiplet S 1

5 6 (1,-1) 7-brane (1,1) 7-brane transition Hanany-Witten + (1,-1) 7-brane = (1,1) 7-brane O7 - -plane T-duality (N=3) Diagramatic “Derivation” 7,8,9 5 (2N-4) D6 NS5 (2N+4) D8 O8 Sen ‘96 5d SU ( N ) N f = 2 N + 4

Tao Probing the End of the World 2

Tao Probing the End of the World 2 Tao probing the D-type singularity

§3 Generalization

What about still other types of Tao diagrams?

’15 Yonekura k 5d [ N + 2] − SU ( N ) − · · · − SU ( N ) − [ N + 2] k = 2 n + 1 6d Sp ( N 0 ) − SU (2 N 0 + 8) − SU (2 N 0 + 16) − · · · − SU (2 N 0 + 8( n − 1)) − [2 N 0 + 8 n ] N 0 = N − 2 n k = 2 n 6d [ A ] − SU ( N 0 ) − SU (2 N 0 + 8) − SU (2 N 0 + 16) − · · · − SU (2 N 0 + 8( n − 1)) − [2 N 0 + 8 n ] N 0 = N − 2 n − 1

’15 Zafrir ’15 Ohmori, Shimizu 5d [ N + 3] − SU ( N ) − SU ( N − 1) − SU ( N − 2) − · · · − SU (3) − SU (2) − [3] (“Tao-nization” of 5d T N ) N = 3 n : 6d SU (3) − SU (12) − · · · − SU (3 + 9( n − 1)) − [3 + 9 n ] N = 3 n + 1 : 6d SU (3) − SU (12) − · · · − SU (9 n − 6) − [9 n + 3] N = 3 n + 2 : 6d SU (0) − SU (9) − · · · − SU (9 n ) − [9 n + 9]

Finite diagram: “Tao diagram”: No diagram: 5D UV fixed point 6D UV fixed point No UV fixed point … §4 Conclusion Partially checked the conjecture Tao diagrams for “class T ”

Classification by Intriligator - Morrison - Seiberg Chern-Simons level VS Some quiver gauge theories have UV fixed point No UV fixed point for product gauge group Our conjecture flavor Im τ e ff ( a ) > 0 for ∀ a 5 d SU ( N ) ( N > 2) : N f ≤ 2 N κ ≤ 2 N − N f 5 d SU ( N ) : N f ≤ 2 N + 4 , κ ≤ 2 N + 4 − N f

Conflict between their classification and web diagram S-dual 5d UV fixed point No UV fixed point ?! Intriligator-Morrison-Seiberg D5 NS5 D5 NS5 SU (3) N f = 6 [2] − SU (2) − SU (2) − [2]