Brane Tilings: NSVZ Beta Function
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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
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