Strings and Fields 2016 @ Yukawa Institute, Kyoto University August 09, 2016 Exotic Brane Junctions Exotic Brane Junctions from F-theory from F-theory JHEP 05 (2016) 060 JHEP 05 (2016) 060 arXiv:1602.08606 arXiv:1602.08606 Tetsuji KIMURA Tetsuji KIMURA Keio University revised version
• • What is exotic brane? • • How can we use exotic branes? Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 2 -
What is exotic brane?
Exotic branes Exotic brane is ✔ from standard brane via string dualities in lower dim ✔ vortex-like (codim 2) ✔ ✔ Obers and Pioline: hep-th/9809039 Eyras and Lozano: hep-th/9908094 de Boer and Shigemori: arXiv:1209.6056 etc.. Exotic b c n -brane has a tension R 1 R 2 · · · R b ( R b + 1 · · · R b + c ) 2 g n s ℓ b + 2 c + 1 s Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 4 -
Exotic branes Performing string dualities, the tension is transformed : ℓ 2 ℓ s T y : s R y → g s → R y : compact radius of y -direction , g s R y R y ℓ s : string length 1 S : ℓ 2 s → g s ℓ 2 g s → , g s : string coupling constant s g s Example duality chain of 5-branes : T 9 T 8 S S 5 2 5 2 − − → − − − → − − − → − − → D5(12345) NS5(12345) KK5(12345,9) 2 (12345,89) 3 (12345,89) 5 1 5 1 5 2 2 Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 5 -
Exotic branes Exotic brane is ✔ from standard brane via string dualities in lower dim ✔ vortex-like (codim 2) ✔ ✔ Obers and Pioline: hep-th/9809039 Eyras and Lozano: hep-th/9908094 de Boer and Shigemori: arXiv:1209.6056 etc.. NOTE D7-brane is an object of codim 2 in 10D. D7-brane physics has been studied for 20 years : F-theory Vafa: hep-th/9602022 Sen: hep-th/9605150 etc.. Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 6 -
D7-brane D7-brane : 2 π + i θ � Λ ρ ( z ) ≡ C ( 0 ) + ie − φ = � ( z = x 8 + i x 9 = r e i θ ) 2 π log r When ρ moves around D7-brane counterclockwise θ → θ + 2 π , it receives a magnetic “charge” of D7-brane (monodromy) : ρ → ρ + 1 D7 ρ + 1 ρ branch cut There exists a branch cut in z -plane. Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 7 -
defect NS5-brane defect NS5-brane : (transverse a, b directions are smeared) 2 π + i θ � Λ ab + ie + 2 φ = � ρ NS5 ( z ) ≡ B ( 2 ) ( z = x 8 + i x 9 = r e i θ ) 2 π log r When ρ moves around NS5-brane counterclockwise θ → θ + 2 π , it receives a magnetic “charge” of NS5-brane (monodromy) : ρ NS5 → ρ NS5 + 1 dNS5 ρ NS5 + 1 ρ NS5 branch cut ρ NS5 + 1 ≃ ρ NS5 In this case, we can identify by B -field gauge transformation. Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 8 -
Exotic 5-brane Exotic 5 2 2 -brane : ( T ab -dualized from defect NS5-brane) monodromy : 1 1 ρ E = − 1 → − 1 where ρ E ρ E ρ NS5 5 2 2 ✬ ✩ We cannot identify this monodromy change ρ E ρ E − ρ E + 1 B -field gauge transformation by branch cut coordinate transformations ✫ ✪ 5 2 2 -brane’s property is from that of defect NS5-brane via SL ( 2 , Z ) ∈ SO ( 2 , 2 ; Z ) T ab -duality Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 9 -
How can we use exotic branes?
Brane junctions Consider an NS5-brane crossing the branch cut of D7-brane from the left. If D7-brane goes across NS5-brane, a new 5-brane and a junction appear (Hanany-Witten effect). − D 5 + NS 5 NS 5 D 7 crossing! D 5 − − − − − − − − → Hanany-Witten NS 5 − D 5 + NS 5 D 7 X, Y ∈ z -plane Note: D7-brane(1234567), D5(1234X), NS5(1234Y), This is a brane junction in F-theory. Gaberdiel and Zwiebach: hep-th/9709013 DeWolfe and Zwiebach: hep-th/9804210 etc.. Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 11 -
Exotic brane junctions Perform the T 34 - and S -duality of the system of NS5-brane with D7-brane. We obtain a config. that a new “3-brane” with exotic 5 2 2 -brane : D 3 + wD 5 D 3 5 2 2 crossing! wD 5 − − − − − − − − → Hanany-Witten D 3 D 3 + wD 5 5 2 2 We find that D5(1234Y)-brane wrapped on T 2 34 ( ≡ wD5) is ending on 5 2 2 -brane. Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 12 -
5D theory 5D theory on 5-branes with 5 D7-branes : [ 1 , 1 ] 7 [ 1 , 1 ] 7 IIB 0 1 2 3 4 5 6 7 8 9 D5 D 7 D 5 − − − − − − − − 5 D7 5 D 7 ( 1 , 1 ) 5 HW − − − → NS5 − − − − − − D5 − − − − − − NS5 NS 5 [ 1 , − 1 ] 7 − − − − − ( 1 , 1 ) 5 angle 7 3 [ 1 , 1 ] 7 D 7 5D T 3 theory 7 3 Aharony and Hanany: hep-th/9704170 DeWolfe, Hanany, Iqbal and Katz: hep-th/9902179 Gaiotto and Witten: arXiv:0804.2902, 0807.3720 Benini, Benvenuti and Tachikawa: arXiv:0906.0359 Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 13 -
3D theory 3D theory on “3-branes” with 5 exotic 5 2 2 -branes : [ 1 , 1 ] T s 5 [ 1 , 1 ] T 3 4 IIB 0 1 2 � � 5 6 7 8 9 wD5 s 5 5 2 5 5 2 2 5 5 2 • 2 • 2 wD 5 − − − − − − ( 1 , 1 ) 3 2 2 HW − − − → D3 − − − − − − wD5 − − − − D3 D 3 [ 1 , − 1 ] T − − − ( 1 , 1 ) 3 angle s 5 dNS 5 [ 1 , 1 ] T 1 s 5 2 5 2 2 = 1 2 3 2 1 dNS 5 mirror of 3D T 3 theory star-shaped quiver 5 2 2 -brane yields mirror of 3D T 3 theory. Benini, Tachikawa and Xie: arXiv:1007.0992 Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 14 -
3D theory 3D theory on “3-branes” with 5 exotic 5 2 3 -branes : [ 1 , 1 ] E d 5 [ 1 , 1 ] E 3 4 IIB 0 1 2 � � 5 6 7 8 9 wNS5 d 5 5 2 5 5 2 3 5 5 2 • 2 • 2 − − − − − − wNS 5 ( 1 , 1 ) 3 3 3 HW − − − → D3 − − − − − − wNS5 − − − − D3 D 3 [ 1 , − 1 ] E − − − ( 1 , 1 ) 3 angle d 5 dD 5 [ 1 , 1 ] E [ 1 , 1 ] T 1 d 5 s 5 2 5 2 5 2 3 2 mirror ← − − − − → = 1 2 3 S -duality 2 1 dD 5 dNS 5 3D T 3 theory mirror of 3D T 3 theory star-shaped quiver 5 2 3 -brane gives 3D T 3 theory. Exotic branes also play a role in generating (non-)Lagrangian theories. Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 15 -
Summary and Discussions
Summary • Exotic brane has a monodromy (charge) with branch cut. • • When D3-brane cross the branch cut of 5 2 • 2 -brane, it jumps to D3 + “D5 wrapped on two-torus”. • Exotic 5-branes are the building blocks of 3D T 3 -theory and its mirror theory. • • They correctly provide 3D theory even when spacetime is compactified. • • Non - Lagrangian theory from non - geometric background • Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 17 -
Discussions T P F1 S S T T T T T T T D0 D1 D2 D3 D4 D5 D6 D7 S NS5 T KK5 KK5 S S 5 2 T 2 S 0 7 1 6 2 5 3 4 4 3 5 2 6 1 7 3 3 3 3 3 3 3 3 T T T T T T T S 0 (1,6) 1 6 S 4 4 T Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 18 -
Discussions 4D theory on “4-branes” with 5 exotic 6 1 3 -branes : [ 1 , 1 ] E d 6 [ 1 , 1 ] E IIA 0 1 2 3 4 5 6 7 8 9 wNS5 � d 6 6 1 3 5 6 1 5 6 1 • 2 wNS 5 − − − − − − − ( 1 , 1 ) 4 3 3 HW − − − → D4 − − − − − − wNS5 − − − − − D4 D 4 [ 1 , − 1 ] E − − − − ( 1 , 1 ) 4 angle d 6 dD 6 [ 1 , 1 ] E d 6 6 1 3 4D T 3 theory can be realized. 4D T 3 theory can be realized. dD 6 dD 6 → KK 6, 6 1 3 → KK 6; D 4 → M 5, wNS 5 → M 5 Uplift to M-theory : g s ℓ 3 s = ℓ 3 g s ℓ s = R ♮ , P Gaiotto: arXiv:0904.2715 Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 19 -
Thanks
Appendix
3D theory 3D theory on “3-branes” with 5 defect D5-branes : [ 1 , 1 ] E d 5 [ 1 , 1 ] E 3 4 IIB 0 1 2 � � 5 6 7 8 9 D3 d 5 dD 5 D 3 − − − − − − 5 dD5 5 dD 5 ( 1 , 1 ) 3 HW − − − → wNS5 − − − − D3 − − − − − − wNS5 wNS 5 [ 1 , − 1 ] E − − − ( 1 , 1 ) 3 angle d 5 5 2 3 [ 1 , 1 ] E 1 d 5 2 dD 5 mirror ← − − − − → 1 2 3 S -duality 2 5 2 1 3 3D T 3 theory star-shaped quiver 3D T 3 theory is realized. Benini, Tachikawa and Xie: arXiv:1007.0992 Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 22 -
3D theory 3D theory on “3-branes” with 5 defect NS5-branes : [ 1 , 1 ] T s 5 [ 1 , 1 ] T 3 4 IIB 0 1 2 � � 5 6 7 8 9 D3 s 5 dNS 5 D 3 − − − − − − 5 dNS5 5 dNS 5 ( 1 , 1 ) 3 HW − − − → wD5 − − − − D3 − − − − − − wD5 wD 5 [ 1 , − 1 ] T − − − ( 1 , 1 ) 3 angle s 5 5 2 2 [ 1 , 1 ] T 1 s 5 2 dNS 5 = 1 2 3 2 5 2 1 2 mirror of 3D T 3 theory star-shaped quiver Mirror of 3D T 3 theory is realized. Benini, Tachikawa and Xie: arXiv:1007.0992 Tetsuji KIMURA : Exotic Brane Junctions from F-theory - 23 -
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