Brane Tilings: NSVZ Beta Function

Transcription

1 Brane Tilings: NSVZ Beta Function Amihay Hanany Imperial College & KITP UCSB 1

2 NSVZ Beta Function 2

3 NSVZ Beta Function β 1 g 2 = 1 8π 2 3N M µ[r M ](1 γ M (g)) 1 g 2 N/8π 2 2

4 NSVZ Beta Function β 1 g 2 = 1 8π 2 3N M µ[r M ](1 γ M (g)) 1 g 2 N/8π 2 D = 1 + γ 2 = 3 2 r 2

5 Look at the numerator 3

6 Look at the numerator β = N M µ[r M ](1 r M ) 3

7 Look at the numerator β = N M µ[r M ](1 r M ) µ(fund) = 1 2 3

8 Look at the numerator β = N M µ[r M ](1 r M ) µ(fund) = 1 2 µ(adj) =N 3

9 Matter in Adj & Bi-fund 4

10 Matter in Adj & Bi-fund β a = N a N a (1 r A,a ) 1 2 N b (1 r B,ab ) A adj[a] B bif und[a,b] 4

11 Vanishing Beta Function 5

12 Vanishing Beta Function B bif und[a,b] N b (1 r B,ab )=2N a 5

13 All ranks equal 6

14 All ranks equal B bif und (1 r B )=2 6

15 A superpotential term 7

16 A superpotential term B monomial r B =2 7

17 Conditions for conformal invariance (1 r B )=2 B bif und r B =2 B monomial 8

18 Look for a graphical representation 9

19 Look for a graphical representation 3 objects in N=1 supersymmetry in 3+1d: 9

20 Look for a graphical representation 3 objects in N=1 supersymmetry in 3+1d: gauge fields - vector multiplets 9

21 Look for a graphical representation 3 objects in N=1 supersymmetry in 3+1d: gauge fields - vector multiplets matter fields - chiral multiplets 9

22 Look for a graphical representation 3 objects in N=1 supersymmetry in 3+1d: gauge fields - vector multiplets matter fields - chiral multiplets interactions - superpotential 9

23 A Typical Quiver: Star of David 10

24 Quiver 11

25 Quiver Encodes the gauge groups and matter fields 11

26 Quiver Encodes the gauge groups and matter fields Good for gauge theories on D branes 11

27 Quiver Encodes the gauge groups and matter fields Good for gauge theories on D branes A string stretched between 2 D branes 11

28 Quiver Encodes the gauge groups and matter fields Good for gauge theories on D branes A string stretched between 2 D branes Always bi-fundamentals 11

29 Quiver Encodes the gauge groups and matter fields Good for gauge theories on D branes A string stretched between 2 D branes Always bi-fundamentals no information on W 11

30 Improve by introducing Brane Tilings 12

31 Introduce arrows in alternating fashion 13

32 Brane Tilings Dictionary 14

33 Brane Tilings Dictionary Face (tile) - U(N) Gauge group; U(N) V-plet 14

34 Brane Tilings Dictionary Face (tile) - U(N) Gauge group; U(N) V-plet Edge - A bi-fundamental chiral multiplet 14

35 Brane Tilings Dictionary Face (tile) - U(N) Gauge group; U(N) V-plet Edge - A bi-fundamental chiral multiplet Node - Interaction term in W 14

36 Brane Tilings Dictionary Face (tile) - U(N) Gauge group; U(N) V-plet Edge - A bi-fundamental chiral multiplet Node - Interaction term in W +(-) sign for a white (black) node 14

37 Brane Tilings Dictionary Face (tile) - U(N) Gauge group; U(N) V-plet Edge - A bi-fundamental chiral multiplet Node - Interaction term in W +(-) sign for a white (black) node 2+1d: Each Face - integer CS level 14

38 An Infinite class of SCFT s in 3+1d & 2+1d 15

39 Conditions for conformal invariance 16

40 Conditions for conformal invariance (1 r B )=2 B bif und 16

41 Conditions for conformal invariance (1 r B )=2 B bif und r B =2 B monomial 16

42 Conditions for conformal invariance (1 r B )π =2π B around face r B =2 B monomial 16

43 Conditions for conformal invariance (1 r B )π =2π B around face r B π =2π B around node 16

44 Conditions for conformal invariance 17

45 Conditions for conformal invariance Locally flat tiles (NSVZ) 17

46 Conditions for conformal invariance Locally flat tiles (NSVZ) Locally flat nodes (W has R charge 2) 17

47 Conditions for conformal invariance Locally flat tiles (NSVZ) Locally flat nodes (W has R charge 2) Periodic, bi-partite, 2d tilings 17

48 3 Hexagon tiling 18

49 Ex: Chessboard Tiling 19

50 Isoradial Embedding; dp1 20

51 Seiberg Duality 21

52 Brane Tilings Moduli space of Vacua 22

53 Brane Tilings Moduli space of Vacua In 3+1d the moduli space is a non-compact singular toric CY3 cone 22

54 Brane Tilings Moduli space of Vacua In 3+1d the moduli space is a non-compact singular toric CY3 cone In 2+1d with a choice of CS levels it is a noncompact singular toric CY4 cone 22

55 The 2+1d Lagrangian d 4 θ X ab X ab e V a X ab e V b G 1 + i d 4 θ k a dtv a Dα (e tv a D α e tv a ) a=1 0 + d 2 θw (X ab )+c.c. 23

56 Vacuum Equations Xab W = 0 µ a (X) := G X ab X ab G X cax ca +[X aa,x aa] = 4k a σ a b=1 c=1 σ a X ab X ab σ b = 0 24

57 Fibration of a CY3 over a complex line 25

58 Example: Chessboard Tiling; CS levels (1,-1) 26

59 Example: Chessboard Tiling; CS levels (1,-1) 26

60 Example: Chessboard Tiling; CS levels (1,-1) W = Tr(X 1 12X 1 21X 2 12X 2 21 X 1 12X 2 21X 2 12X 1 21) 26

61 4 fields in the quiver 27

62 1 hexagon; 1 double edge, G=2 28

63 Toric Duality 29

64 2 hexagon tiling; (1-,1) Conifold (C ) x C II W = φ 1 (X 1 12X 2 21 X 2 12X 1 21)+φ 2 (X 1 21X 2 12 X 2 21X 1 12) 30

65 Ex: 2 hexagon tiling Conifold x C II 31

66 Ex: 2 hexagon tiling Conifold x C II In 3+1d this is C 2 /Z2 x C 31

67 Ex: 2 hexagon tiling Conifold x C II In 3+1d this is C 2 /Z2 x C Master space - 2+1d mesonic moduli space 31

68 Ex: 2 hexagon tiling Conifold x C II In 3+1d this is C 2 /Z2 x C Master space - 2+1d mesonic moduli space Non trivial scaling dimensions 31

69 Ex: 2 hexagon tiling Conifold x C II In 3+1d this is C 2 /Z2 x C Master space - 2+1d mesonic moduli space Non trivial scaling dimensions 1/2 for ϕ s, 3/4 for X s 31

70 Ex: 2 hexagon tiling Conifold x C II In 3+1d this is C 2 /Z2 x C Master space - 2+1d mesonic moduli space Non trivial scaling dimensions 1/2 for ϕ s, 3/4 for X s Non-trivial SCFT in the IR 31

71 Ex: 2 hexagon tiling Conifold x C II In 3+1d this is C 2 /Z2 x C Master space - 2+1d mesonic moduli space Non trivial scaling dimensions 1/2 for ϕ s, 3/4 for X s Non-trivial SCFT in the IR a test of AdS/CFT 31

72 Toric Diagram C x C 32

73 5 fields in the Quiver Master space - C 5 33

74 Chessboard tiling; 1 double edge; (1,-1,0) 34

75 Chessboard tiling; 1 double edge; (1,-1,0) 35

76 Chessboard tiling; 1 double edge; (1,-1,0) mesonic moduli space is conifold x C 35

77 Chessboard tiling; 1 double edge; (1,-1,0) mesonic moduli space is conifold x C 1 dimensional baryonic moduli space 35

78 Chessboard tiling; 1 double edge; (1,-1,0) mesonic moduli space is conifold x C 1 dimensional baryonic moduli space Combined mesonic baryonic space - C5 35

79 Chessboard tiling; 1 double edge; (1,-1,0) mesonic moduli space is conifold x C 1 dimensional baryonic moduli space Combined mesonic baryonic space - C5 Scaling dimensions 1/2 for X12, 3/8 other 35

80 Conifold x C Phase III (0,1,-1); (-2,1,1) 36

81 Global symmetry conifold x C 37

82 Global symmetry conifold x C SU(2) x SU(2) x U(1)q x U(1)R x U(1)B 37

83 Conifold x C Table of charges 38

84 Toric Duality conifold x C 39

85 Toric Duality conifold x C Three phases 39

86 Toric Duality conifold x C Three phases 2 tiles 3 tiles 3 tiles 39

87 Toric Duality conifold x C Three phases 2 tiles 3 tiles 3 tiles Master space: mesonic mesonic baryonic 39

88 Toric Duality conifold x C Three phases 2 tiles 3 tiles 3 tiles Master space: mesonic mesonic baryonic mesonic generators: linear bi-linear 39

89 Hilbert Series conifold x C 40

90 Hilbert Series conifold x C (1 t 1 x 1 b) ( 1 t 1x 2 b 1 ) ( )( ) 1 t 1b 1 t 1 x 1 x 2 b (1 t 2 ) 40

91 Hilbert Series conifold x C (1 t 1 x 1 b) ( 1 t 1x 2 b 1 ) ( )( ) 1 t 1b 1 t 1 x 1 x 2 b (1 t 2 ) t 1 = t 3 q t 2 = t 4 /q 4 40

92 Lattice of generators conifold x C 41

93 Summary 42

94 Summary Brane Tilings - An infinite class of SCFT s in 3+1d and in 2+1d 42

95 Summary Brane Tilings - An infinite class of SCFT s in 3+1d and in 2+1d mesonic moduli space 42

96 Summary Brane Tilings - An infinite class of SCFT s in 3+1d and in 2+1d mesonic moduli space mesonic baryonic moduli space 42

97 Summary Brane Tilings - An infinite class of SCFT s in 3+1d and in 2+1d mesonic moduli space mesonic baryonic moduli space Complete BPS spectrum of Scaling dimensions 42

98 Summary Brane Tilings - An infinite class of SCFT s in 3+1d and in 2+1d mesonic moduli space mesonic baryonic moduli space Complete BPS spectrum of Scaling dimensions Toric Duality 42

99 Happy Birthday Misha! 43