Exceptional Groups as Symmetries of Nature ’17 @ KEK July 18, 2017 Exotic Branes and Exotic Branes and Superconformal Field Theories Superconformal Field Theories Tetsuji KIMURA Tokyo Institute of Technology
Contents 1. Exotic branes from F-theory Exotic SL ( 2 , Z ) monodromy 2. 3. Applications a. 6D N = ( 2 , 0 ) b. G-theory 4. Summary and discussions Tetsuji KIMURA : Exotic Branes @ ExGraS17, KEK - 2 -
1. Exotic branes from F-theory
Exotic branes Consider charged particles in 3D maximal supergravity : They are D7-brane wrapped on 7-torus and its dualized objects T 7 → − M D7 = R 1 R 2 · · · R 7 g s ℓ 8 − s → S ℓ 2 ℓ s R y : compact radius of y -direction s T y : R y → g s → , g s R y R y g s : string coupling constant 1 ℓ 2 s → g s ℓ 2 S : g s → , ℓ s : string length s g s Tetsuji KIMURA : Exotic Branes @ ExGraS17, KEK - 4 -
Exotic branes Consider charged particles in 3D maximal supergravity : They are D7-brane wrapped on 7-torus and its dualized objects R 1 R 2 · · · R 6 T 7 = M D6 → − g s ℓ 7 s M D7 = R 1 R 2 · · · R 7 g s ℓ 8 − R 1 R 2 · · · R 7 s → ← exotic! g 3 s ℓ 8 S s ℓ 2 ℓ s R y : compact radius of y -direction s T y : R y → g s → , g s R y R y g s : string coupling constant 1 ℓ 2 s → g s ℓ 2 S : g s → , ℓ s : string length s g s Tetsuji KIMURA : Exotic Branes @ ExGraS17, KEK - 5 -
Exotic branes Consider charged particles in 3D maximal supergravity : They are D7-brane wrapped on 7-torus and its dualized objects T F1 P S S T T T T T T T D7 D6 D5 D4 D3 D2 D1 D0 S NS5 T KK5 S S 5 2 T 2 S 6 1 5 2 4 3 3 4 2 5 1 6 0 7 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 Branes @ ExGraS17, KEK - 6 -
Exotic branes Consider charged particles in 3D maximal supergravity : They are D7-brane wrapped on 7-torus and its dualized objects D -dim IIB IIA D7 (1) D5 (21) D3 (35) D1 (7) D6 (7) D4 (35) D2 (21) D0 (1) F1 (7) P (7) NS5 (21) F1 (7) P (7) NS5 (21) 3 KK5 (42) KK5 (42) 0 ( 1 , 6 ) 0 ( 1 , 6 ) (240) 1 6 5 2 1 6 5 2 4 (7) (7) 2 (21) 4 (7) (7) 2 (21) 4 4 5 2 3 4 1 6 6 1 4 3 2 5 0 7 7 3 (1) 3 (21) 3 (35) 3 (7) 3 (7) 3 (35) 3 (21) 3 (1) # of charged particles = # of U ( 1 ) gauge one-form potentials = # of scalar fields = dim( E 8 ( 8 ) /SO ( 16 )) = 128 < 240 ! Tetsuji KIMURA : Exotic Branes @ ExGraS17, KEK - 7 -
Exotic branes Consider charged particles in 3D maximal supergravity : They are D7-brane wrapped on 7-torus and its dualized objects D -dim IIB IIA D7 (1) D5 (21) D3 (35) D1 (7) D6 (7) D4 (35) D2 (21) D0 (1) F1 (7) P (7) NS5 (21) F1 (7) P (7) NS5 (21) 3 KK5 (42) KK5 (42) 0 ( 1 , 6 ) 0 ( 1 , 6 ) (240) 1 6 5 2 1 6 5 2 4 (7) (7) 2 (21) 4 (7) (7) 2 (21) 4 4 5 2 3 4 1 6 6 1 4 3 2 5 0 7 7 3 (1) 3 (21) 3 (35) 3 (7) 3 (7) 3 (35) 3 (21) 3 (1) n has mass (tension) = R 1 R 2 · · · R b ( R b + 1 · · · R b + c ) 2 b c g n s ℓ b + 2 c + 1 s Tetsuji KIMURA : Exotic Branes @ ExGraS17, KEK - 8 -
Exotic branes Consider charged particles in 3D maximal supergravity : They are D7-brane wrapped on 7-torus and its dualized objects D -dim IIB IIA D7 (1) D5 (21) D3 (35) D1 (7) D6 (7) D4 (35) D2 (21) D0 (1) F1 (7) P (7) NS5 (21) F1 (7) P (7) NS5 (21) 3 KK5 (42) KK5 (42) 0 ( 1 , 6 ) 0 ( 1 , 6 ) (240) 1 6 5 2 1 6 5 2 4 (7) (7) 2 (21) 4 (7) (7) 2 (21) 4 4 5 2 3 4 1 6 6 1 4 3 2 5 0 7 7 3 (1) 3 (21) 3 (35) 3 (7) 3 (7) 3 (35) 3 (21) 3 (1) eg.) 5 2 2 -particle in 3D is uplifted to 5 2 2 -brane in 8D( = 5 + 3) (as codim-2 object). When exotic 5 2 2 -brane in 8D is embedded into 10D, this does not depend on 2 = 10 − 8 transverse directions. (smeared / KK-reduced) necessary to keep aspects of codim-2 object Tetsuji KIMURA : Exotic Branes @ ExGraS17, KEK - 9 -
Exotic branes D U-duality # IIB IIA 10A 1 – – – 2 ⊂ 3 10B SL ( 2 , Z ) D7 (1) 7 3 (1) – SL ( 2 , Z ) × Z 2 2 ⊂ 3 6 1 9 D7 (1) 7 3 (1) D6 (1) 3 (1) 5 2 5 2 6 1 6 ⊂ ( 8 , 1 ) SL ( 3 , Z ) D7 (1) 7 3 (1) D5 (1) NS5 (1) D6 (2) KK5 (2) 3 (1) 2 (1) 3 (2) 8 × SL ( 2 , Z ) 2 ⊂ ( 1 , 3 ) 5 2 KK5 (2) NS5 (1) 2 (1) 5 2 5 2 6 1 4 3 D7 (1) 7 3 (1) D5 (3) NS5 (3) D6 (3) KK5 (6) D4 (1) 3 (3) 2 (3) 3 (3) 3 (1) 20 ⊂ 24 7 SL ( 5 , Z ) 5 2 KK5 (6) NS5 (3) 2 (3) 5 2 5 2 3 4 6 1 4 3 D7 (1) 7 3 (1) D5 (6) NS5 (6) D3 (1) D6 (4) KK5 (12) D4 (4) 3 (6) 2 (6) 3 (1) 3 (4) 3 (4) 40 ⊂ 45 6 SO ( 5 , 5 ; Z ) 5 2 KK5 (12) NS5 (6) 2 (6) 5 2 5 2 3 4 6 1 4 3 2 5 D7 (1) 7 3 (1) D5 (10) NS5 (10) D3 (5) D6 (5) KK5 (20) D4 (10) D2 (1) 3 (10) 2 (10) 3 (5) 3 (5) 3 (10) 3 (1) 72 ⊂ 78 5 E 6 ( 6 ) ( Z ) 5 2 KK5 (20) NS5 (10) 2 (10) 5 2 5 2 6 1 4 3 D7 (1) 7 3 (1) D5 (15) NS5 (15) D6 (6) KK5 (30) D4 (20) 3 (15) 2 (15) 3 (6) 3 (20) 126 ⊂ 133 3 4 1 6 1 6 2 5 1 6 4 E 7 ( 7 ) ( Z ) D3 (15) D1 (1) F1 (1) D2 (6) F1 (1) 3 (15) 3 (1) 4 (1) 3 (6) 4 (1) 5 2 KK5 (30) NS5 (15) 2 (15) 5 2 5 2 6 1 4 3 D7 (1) 7 3 (1) D5 (21) NS5 (21) D6 (7) KK5 (42) D4 (35) 3 (21) 2 (21) 3 (7) 3 (35) 0 ( 1 , 6 ) 0 ( 1 , 6 ) 240 ⊂ 248 3 4 1 6 1 6 2 5 1 6 0 7 3 E 8 ( 8 ) ( Z ) D3 (35) D1 (7) F1 (7) P (7) D2 (21) F1 (7) D0 (1) P (7) 3 (35) 3 (7) 4 (7) (7) 3 (21) 4 (7) 3 (1) (7) 4 4 5 2 KK5 (42) NS5 (21) 2 (21) For codim-2, all branes are (un)wrapped on torus along suitable directions. → Defect branes Tetsuji KIMURA : Exotic Branes @ ExGraS17, KEK - 10 -
Exotic branes Exotic b c n -brane : charged particle in 3D, codim-2 object in ( b + 3 ) -dim ◦ pair with standard b -brane of codim-2 in ( b + 3 ) -dim ◦ c smeared transverse directions from 10D viewpoint ◦ tension proportional to g − n ◦ s D7-brane (codim-2 object in 10D) has been studied for 20 years : F-theory Vafa: hep-th/9602022 Tetsuji KIMURA : Exotic Branes @ ExGraS17, KEK - 11 -
2. Exotic SL ( 2 , Z ) monodromy
SL ( 2 , Z ) duality in 10D F-string : couple to B ( 2 ) τ ( z ) = ϑ 2 π + i 2 π log Λ solution : r couple to C ( 2 ) D-string : ( z = x 8 + i x 9 = r e i ϑ ) couple to τ ( z ) = C + ie − φ D7(1234567) : ()F-string() ( p, q ) -string SL ( 2 , Z ) − − − − − − → S [] E [] E d 5 D7-brane[] d 5 [ p, q ] 7-brane[] ( 1 , 0 ) -string = F1 [ 1 , 0 ] 7-brane = D7(1234567) ( 0 , 1 ) -string = D1 [ 0 , 1 ] 7-brane = 7 3 (1234567) Open D-string is ending on 7 3 (1234567). Tetsuji KIMURA : Exotic Branes @ ExGraS17, KEK - 13 -
7-brane and monodromy When τ moves around D7-brane counterclockwise, τ → τ + 1 it receives a magnetic “charge” of D7-brane (monodromy) : D7 τ D7 + 1 τ D7 branch cut � � − 1 1 K [ 1 , 0 ] · ( τ + 1 ) = τ , ∈ SL ( 2 , Z ) K [ 1 , 0 ] = 0 1 Tetsuji KIMURA : Exotic Branes @ ExGraS17, KEK - 14 -
Exotic branes D7-brane : (localized in 89-plane) 2 π + i ϑ 2 π log Λ τ D7 ( z ) ≡ C + i e − φ = ( z = x 8 + i x 9 = r e i ϑ ) r When τ D7 moves around D7-brane counterclockwise ϑ → ϑ + 2 π , it receives a magnetic “charge” (monodromy) : τ D7 → τ D7 + 1 D7 T 67 , S , and T 67 − − − − − − − − − − → τ D7 + 1 τ D7 branch cut Tetsuji KIMURA : Exotic Branes @ ExGraS17, KEK - 15 -
Exotic branes Exotic 5 2 2 -brane : (localized in 89-plane, smeared along 67-directions) � ϑ � − 1 � 2 π + i 2 π log Λ ( z = x 8 + i x 9 = r e i ϑ ) det G mn = − ρ E ( z ) = B 67 + i r When ρ E moves around 5 2 2 -brane counterclockwise ϑ → ϑ + 2 π , it receives a magnetic “charge” (monodromy) : − 1 /ρ E → − 1 /ρ E + 1 5 2 D7 2 T 67 , S , and T 67 − − − − − − − − − − → − 1 − 1 τ D7 + 1 + 1 τ D7 ρ E ρ E branch cut branch cut T 67 S T 67 → 5 2 − − → D5 − → NS5 − − D7 2 Tetsuji KIMURA : Exotic Branes @ ExGraS17, KEK - 16 -
Monodromy of exotic 5 2 2 -brane Exotic 5 2 2 -brane : (localized in 89-plane, smeared along 67-directions) − 1 → − 1 − 1 SL ( 2 , Z ) : + 1 where = ρ NS5 ρ E ρ E ρ E 5 2 2 ✬ ✩ We cannot remove this shift − 1 − 1 + 1 B-field gauge transformation ρ E ρ E by branch cut coordinate transformations ✫ ✪ This property comes from × SL ( 2 , Z ) SL ( 2 , Z ) = SO ( 2 , 2 ; Z ) complex structure complexified K¨ ahler T 67 -duality Tetsuji KIMURA : Exotic Branes @ ExGraS17, KEK - 17 -
Exotic SL ( 2 , Z ) pairs Exotic SL ( 2 , Z ) pairs by monodromy matrix K [ p,q ] : K [ p,q ] K [ p,q ] [ p, q ] S [ p, q ] E − − − − − → − − − − − → D7 in 10-dim 7 -brane D b in ( b + 3 ) -dim db -brane K [ p,q ] K [ p,q ] [ p, q ] T [ p, q ] T − − − − − → − − − − − → NS5 in 8-dim s 5 -brane KK5 in 8-dim k 5 -brane T F1 P S S T T T T T T T D7 D6 D5 D4 D3 D2 D1 D0 S NS5 T KK5 S S 5 2 T 2 S 6 1 5 2 4 3 3 4 2 5 1 6 0 7 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 TK: arXiv:1602.08606 Tetsuji KIMURA : Exotic Branes @ ExGraS17, KEK - 18 -
3. Applications
U-duality and exotic branes D U-duality IIB IIA 10B SL ( 2 , Z ) D7 7 3 – 6 1 SL ( 2 , Z ) × Z 2 9 D7 7 3 D6 3 5 2 5 2 6 1 SL ( 3 , Z ) D7 7 3 D5 NS5 D6 KK5 3 2 3 8 × SL ( 2 , Z ) 5 2 KK5 NS5 2 5 2 5 2 6 1 4 3 D7 7 3 D5 NS5 D6 KK5 D4 3 2 3 3 7 SL ( 5 , Z ) 5 2 KK5 NS5 2 5 2 5 2 3 4 6 1 4 3 D7 7 3 D5 NS5 D3 D6 KK5 D4 3 2 3 3 3 6 SO ( 5 , 5 ; Z ) 5 2 KK5 NS5 2 . . . . . . . . . . . . Defect branes (codim-2 branes) in diverse dimensions Bergshoeff, Ort´ ın, Riccioni: arXiv:1109.4484 Tetsuji KIMURA : Exotic Branes @ ExGraS17, KEK - 20 -
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