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embedding tensors (non)geometric fluxes exotic branes +
D U-duality G 0 R-symmetry H dim(g 0 /H) T-duality 11 1 1 0 1 IIA R + 1 1 1 IIB SL(2, R) SO(2) 2 1 9 GL(2, R) SO(2) 3 SO(1, 1) 8 SL(3, R) SL(2, R) SO(3) SO(2) 7 SL(2, R) SL(2, R) 7 SL(5, R) Sp(2) 14 SL(4, R) 6 SO(5, 5) Sp(2) Sp(2) 25 SO(4, 4) 5 E 6(6) USp(8) 42 SO(5, 5) 4 E 7(7) SU(8) 70 SO(6, 6) 3 E 8(8) SO(16) 128 SO(7, 7) 11
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+ (non)geometric fluxes
embedding tensors +
+ embedding tensors (non)geometric fluxes
+ exotic branes
+ (non)geometric fluxes exotic branes
+ embedding tensors exotic branes
+ embedding tensors (non)geometric fluxes exotic branes
Contents Q ab c (Non)geometric Fluxes Θ M α Embedding Tensors b c n Exotic Branes
Calabi-Yau Calabi-Yau 3-fold Ricci Kähler SU(3) SU(4) SO(6) ds 2 10D = η µν (x) dx µ dx ν 4D + g mn (x, y) dy m dy n CY Levi-Civita 2 (J) 3 (Ω) dj = [m J np] = 0 dω = [m Ω npq] = 0 THREE EXOTICA - 19 -
non-cy 3-fold Ricci 2-form (non-kähler) dj 0 and/or dω 0 CY dj = 3 2 Im(W 1Ω) + W 4 J + W 3, dω = W 1 J J + W 2 J + W 5 Ω THREE EXOTICA - 20 -
(dj, dω) D d H fl f Q R ( ) ( β I I eλ m ΛI D α I e ΛI m Λ I (H = H fl + db) )( ) ω Λ ω Λ e 0 I, e 0I : H-flux charges (H fl = e 0 I α I + e 0I β I ) e a I, e ai : m ΛI, m Λ I: geometric flux charges ( ) Non-geometric flux charges (e Λ I, e ΛI ) THREE EXOTICA - 21 -
Exotic feature Non-geometric structure = Diffeo (GL(d, R)) (O(d, d), U- ) }{{} GL(d, R) duality transf. Generalized Geometry Doubled Geometry THREE EXOTICA - 22 -
Generalized geometries g mn geometry associated with g mn Conventional geometry (manifold) O(6) global symmetry M 6 geometry associated with g mn, B mn Generalized geometry O(6, 6) T-duality symmetry geometry associated with g mn, B mn, C (p) Exceptional generalized geometry E 7(7) U-duality symmetry THREE EXOTICA - 23 -
4 ( ) µ q u = µ q u + g k u Λ A Λ µ + g k uλ A µλ k Λ = [ 2 e RΛ + e I ] Λ (C H ξ) I a e Λ I k Λ = [ 2 m Λ R + m ΛI (C H ξ) I ] a + mλi ξ I ξ I (RR fluxes m Λ R ) h uv µ q u µ q v M AB H A µνρ H µνρb THREE EXOTICA - 24 -
A M µ G 0 T M Θ M α t α t α Lie G 0 global T M Lie G gauge µ D µ µ ga M µ T M... THREE EXOTICA - 26 -
Exotic feature [T M, T N ] = T MN P T P T M = Θ M α t α T MN P T (MN) P Θ P α = 0 F 2 = da + A A THREE EXOTICA - 27 -
Exotic feature p-form potentials Hodge-dual in D-dim (D p 2)-form potentials ( ) D = 4 { Θ M α ( ) ( ) THREE EXOTICA - 28 -
Embedding tensor formalism : Θ M α dim G dim G 0 ( Dµ = µ ga M µ Θ M α t α ) D U-duality G 0 constraints on R(M) R(α) 9 GL(2) (2 1) (3 1) = 1 2 2 3 4 8 SL(3) SL(2) (3, 2) [(8, 1) (1, 3)] = (3, 2) (3, 2) (3, 4) (6, 2) (15, 2) 7 SL(5) 10 24 = 10 15 40 175 6 SO(5, 5) 16 45 = 16 144 560 5 E 6(6) 27 78 = 27 351 1728 4 E 7(7) 56 133 = 56 912 6480 3 E 8(8) 248 248 = 1 248 3875 27000 30380 F.Riccioni, D.Steele and P.West, arxiv:0906.1177 THREE EXOTICA - 29 -
Embedding tensors µ ϕ A = µ ϕ A g K A Σ A Σ µ g K AΣ A µσ K Σ = Θ Σ m (t m ) α β B a β (U 1 ) Aa α K Σ = Θ Σm (t m ) α β B a β (U 1 ) Aa α ϕ A ϕ A (Θ M m ) G AB µ ϕ A µ ϕ B M mn H m µνρ H µνρn THREE EXOTICA - 30 -
Embedding tensors Θ Σm : nongeometric flux charges (?) Nongeometric flux compactifications work in progress... THREE EXOTICA - 31 -
Embedding tensors D 32-SUSY 16-SUSY 8-SUSY 9 arxiv:1105.1760 (unknown) 8 arxiv:1203.6562 (unknown) 7 hep-th/0506237 (unknown) 6 arxiv:0712.4277 (unknown) arxiv:1012.1818 5 hep-th/0412173 hep-th/0702084 (unknown) 4 arxiv:0705.2101 hep-th/0602024 arxiv:1107.3305 3 hep-th/0103032 arxiv:0806.2584 arxiv:0807.2841 THREE EXOTICA - 32 -
: M-theory on S 1 (R s ) mass/tension (l s 1) type IIA longitudinal M2 1 F1 transverse M2 longitudinal M5 transverse M5 longitudinal KK6 RTN 2 gs 2 KK6 with R TN = R s 1 1 g s 1 g s 1 g 2 s g s D2 D4 NS5 KK5 D6 0 1 2 3 4 5 6 7 8 9 M S 1 R 3 KK6 6 1 3 Taub-NUT b c n : M = (R 1 R c ) 2 g n s transverse KK6 R 2 TN g 3 s 6 1 3 for review: N. Obers and B. Pioline, hep-th/9809039 THREE EXOTICA - 34 -
Exotic feature 5 2 2-brane M = (R 8R 9 ) 2 g 2 s NS5 0 1 2 3 4 5 6 7 8 9 T-dual along x 9 KK5 0 1 2 3 4 5 6 7 8 9 T-dual along x 8 5 2 2 0 1 2 3 4 5 6 7 8 9 THREE EXOTICA - 35 -
Exotic feature ds 2 = dt 2 + dx 2 12345 + H(dr 2 + r 2 dθ 2 ) + H K dx2 89 B 89 = θ σ K, e2φ = H K, K H2 + σ 2 θ 2 ( µ ) H(r) = h + σ log, σ R 8R 9 r 2πα θ = 0 : G 88 = G 99 = H 1 θ = 2π : G 88 = G 99 = H H 2 + (2πσ) 2 Globally nongeometric : θ- fiber T 89 single-valued Locally geometric : (non)geometric flux Q ab c T-fold THREE EXOTICA - 36 -
Exotic feature exotic branes D- co-dim. 2, 1 co-dim. 2 : Defect Branes (D 2)-form potentials co-dim. 1 : Domain Walls (D 1)-form potentials D- THREE EXOTICA - 37 -
(D 1)-form potentials D-form field strengths Domain Walls
(D 1)-form potentials D-form field strengths Domain Walls 0-form field strengths = Deformation Parameters
Domain Walls D8-brane in 10-dim. RR potential C 9 10 dc 9 = m ( ) IIA Romans massive IIA SUGRA (D 2)-branes in D-dim. SUSY Domain Walls Domain Walls Domain Walls THREE EXOTICA - 40 -
Domain Walls (D 1)-form potentials DWs + Θ M α D (D 1)-form potentials THREE EXOTICA - 41 -
U-duality G 0 form potentials D U-duality G 0 1-forms 2-forms 3-forms 4-forms 5-forms 6-forms 7-forms 8-forms 9-forms 10-forms IIA R + 1 1 1 1 1 1 1 1 1 1 IIB SL(2, R) 2 1 2 3 4 2 9 GL(2, R) 2 1 2 1 1 2 2 1 3 1 3 2 4 2 2 8 SL(3, R) SL(2, R) (3, 2) (3, 1) (1, 2) (3, 1) (3, 2) (8, 1) (1, 3) 7 SL(5, R) 10 5 5 10 24 40 15 6 SO(5, 5) 16 10 16 45 144 5 E 6(6) 27 27 78 351 4 E 7(7) 56 133 912 3 E 8(8) 248 3875 1 147250 3875 248 8645 133 1728 27 320 126 10 (6, 2) (3, 2) 70 45 5 (15, 1) (3, 3) (3, 1) (3, 1) (D 1)-forms Embedding Tensors Θ M α F.Riccioni, D.Steele and P.West, arxiv:0906.1177 THREE EXOTICA - 42 -
# of (Elementary SUSY) DWs fundamental Dirichlet solitonic D n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 # IIA 1 1 9 1 1 2 8 (1, 2) T 4 (3, 2) T 6 7 4 T 4 10 T 12 20 T 25 4 10 T 1 T 6 8 S T 32 56 C T 32 56 S T 8 C T 80 5 16 T 80 120 T 80 144 T 40 45 T 216 4 32 T 160 220 T 192 352 T 160 220 T 32 T 576 3 1 T 64 T 280 364 T 448 832 T 560 1001 T 14 104 T 448 832 T 2160 (α 6) 280 364 T, 6 64 T, 7 1 T, 8 brane s tension g +n s E.A. Bergshoeff et al, arxiv:1108.5067, arxiv:1210.1422 THREE EXOTICA - 43 -
String theory origin of EDWs in D-dim. D n = 0 n = 1 n = 2 n = 3 n = 4 n 5 IIA D8 [C 9 ] 9 D7 [C 8 ] 7 (0,1) 3 [E 9,1,1 ] 8 D6 [C 7 ] 6 (1,1) 3 [E 9,2,1 ] 7 D5 [C 6 ] NS5 [D 6 ] KKM [D 7,1 ] 5 2 2 [D 8,2 ] 5 (2,1) 3 [E 9,3,1 ] 5 3 4 [F 9,3 ] 6 D4 [C 5 ] 4 (3,1) 3 [E 9,4,1 ] 4 (3,1) 4 [F 9,4,1 ] 5 D3 [C 4 ] 3 (4,1) 3 [E 9,5,1 ] 3 (3,2) 4 [F 9,5,2 ] 4 D2 [C 3 ] 2 (5,1) 3 [E 9,6,1 ] 2 (3,3) 4 [F 9,6,3 ] 3 F1 [B 2 ] D1 [C 2 ] 1 (6,1) 3 [E 9,7,1 ] 1 (3,4) 4 [F 9,7,4 ] 1 (6,0,1) 4 [F 9,7,1,1 ] A D T,I1 +I 2,I 2 -forms : mixed-symmetry fields b (I 1,I 2 ) n -branes T + b + i I i = D 1 with T = 1 : transverse, b : spatial, I i : isometry directions THREE EXOTICA - 44 -
Exotic branes Exotic branes ( ) J. de Boer and M. Shigemori, arxiv:1004.2521 and arxiv:1209.6056 THREE EXOTICA - 45 -
embedding tensors (non)geometric fluxes exotic branes
Q ab c vs Θ M α : G. Dall Agata et al, arxiv:0712.1026 Θ M α vs b c n : E. Bergshoeff et al, arxiv:1206.5697 4D N = 2 Q ab c vs Θ M α ( ) THREE EXOTICA - 49 -
Flux Compactifications on SU(3) SU(3) generalized geometry vs Embedding Tensor Formalism in 4D N = 2 theory gauged supergravity Q ab c vs Θ M α embedding tensor formalism rigid special Kähler vs local special Kähler hyper-kähler cone vs quaternonic Kähler etc. THREE EXOTICA - 50 -
Appendix
Calabi-Yau compactification in type IIA NS-NS ϕ(x, y) = φ(x) ( ) (χȷ g mn (x, y) = iv a (x) (ω a ) mn (y), g mn (x, y) = i z ȷ ) mpq Ω pq n (x) Ω 2 (y) B 2 (x, y) = B 2 (x) + b a (x)ω a (y) t a (x) b a (x) + iv a (x) R-R C 1 (x, y) = A 0 1(x) C 3 (x, y) = A a 1(x) ω a (y) + ξ I (x)α I (y) ξ I (x)β I (y) H (1,1) ω a a = 1,..., h (1,1) H (0) H (1,1) ω Λ = (1, ω a ) Λ = 0, 1,..., h (1,1) H (2,2) H (6) ω Λ = ( ω a, vol. vol. ) H (2,1) χ i i = 1,..., h (2,1) H (3) (α I, β I ) I = 0, 1,..., h (2,1) dω Λ dα I = 0 = d ω Λ = 0 = dβ I APPENDIX - 53 -
4D N = 2 SUGRA from type IIA on Calabi-Yau 10D type IIA action S (10D) IIA = S NS + S R + S CS : S NS = 1 e 2ϕ{ 1 } R 1 + 4dϕ dϕ 2 2Ĥ3 Ĥ3 S R + S CS = 1 { F2 4 F 2 + ( F4 Ĉ1 F ) ( 3 F4 Ĉ1 F ) } 3 1 4 B 2 F 4 F 4 4D N = 2 ungauged SUGRA: Neither gauge couplings, Nor scalar potential S (4D) = {1 2 R 1 G ab dta dt b h uv dq u dq v + 1 2 ImN ΛΣF Λ 2 F Σ 2 + 1 2 ReN ΛΣF Λ 2 F Σ 2 } gravitational multiplet g µν, A 0 1 vector multiplet (VM) A a 1, t a, t b t a SKG V hypermultiplet (HM) z i, z ȷ, ξ i, ξj z i SKG H universal hypermultiplet (UHM) φ, a, ξ 0, ξ0 a B 2 (Hodge dual) HM = Special QG App.top {q u } 4n H + 4 = {z i, z ȷ } + {ξ i, ξ j } + {φ, a, ξ 0, ξ 0 } 2n H (SKG H ) 2n H 4 (UHM) = {z i, z ȷ } SKG H + {φ} + {a, ξ I, ξ J } Heisenberg APPENDIX - 54 -
Intrinsic torsion classes of SU(3)-structure manifolds dj = 3 2 Im(W 1Ω) + W 4 J + W 3, dω = W 1 J J + W 2 J + W 5 Ω hermitian W 1 = W 2 = 0 balanced W 1 = W 2 = W 4 = 0 complex special hermitian W 1 = W 2 = W 4 = W 5 = 0 Kähler W 1 = W 2 = W 3 = W 4 = 0 Calabi-Yau W 1 = W 2 = W 3 = W 4 = W 5 = 0 conformally CY W 1 = W 2 = W 3 = 3W 4 + 2W 5 = 0 symplectic W 1 = W 3 = W 4 = 0 nearly Kähler W 2 = W 3 = W 4 = W 5 = 0 almost complex almost Kähler W 1 = W 3 = W 4 = W 5 = 0 quasi Kähler W 3 = W 4 = W 5 = 0 semi Kähler W 4 = W 5 = 0 half-flat ImW 1 = ImW 2 = W 4 = W 5 = 0 APPENDIX - 55 -
Geometric flux compactification in type IIA 10D type IIA action S (10D) S NS = 1 2 IIA = S NS + S R = S NS + S R + S CS : (democratic form) e 2ϕ{ 1 } R 1 + 4dϕ dϕ 2Ĥ3 Ĥ3, SR = 1 8 [ F F] 10 with constraint F = λ( F) and EoM (Bianchi) (d + Ĥ ) F = 0 (d Ĥ ) F = 0 non-cy with SU(3)-structure with m Λ R = 0 non-cy with SU(3)-structure with mλ R = 0 4D N = 2 abelian gauged SUGRA (with ξ I (ξ I, ξ I ) T ): S (4D) = d 4 x g [ 1 2 R + 1 4 ImN ΛΣF Λ µνf Σµν ϵµνρσ 8 g ReN ΛΣF Λ µνf Σ ρσ g ab µ t a µ t b g iȷ µ z i µ z ȷ µ φ µ φ + e2φ 2 (M H) IJ D µ ξ I D µ ξ J e2φ 4 ( Dµ a ξ I (C H ) IJ D µ ξ J) 2 V (t, t, q) ] (e Λ I, e ΛI ) : geometric flux charges & e RΛ : RR-flux charges (with constraints e Λ I e ΣI e ΛI e Σ I = 0) t a SKG V and z i SKG H HM are ungauged (in general) non-cy data D µ ξ I = µ ξ I e Λ I A Λ µ & D µ ξi = µ ξi e ΛI A Λ µ D µ a = µ a (2e RΛ ξ I e ΛI + ξ I e Λ I )A Λ µ V (t, t, q): scalar potential D. Cassani, arxiv:0804.0595 APPENDIX - 56 -
Generic form of 4D N = 2 gauged SUGRA with B-field Non-vanishing m Λ R dualizes the axion field a in standard SUGRA to B-field. 4D gauged action is different from the standard one: S (4D) = [ 1 2 R( 1) + 1 2 ImN ΛΣF Λ 2 F Σ 2 + 1 2 ReN ΛΣF Λ 2 F Σ 2 g ab dt a dt b g iȷ dz i dz ȷ dφ dφ e 4φ 4 H 3 H 3 e2φ 2 (M H) IJ Dξ I Dξ J V ( 1) + 1 [ξ 2 db I (C H ) IJ Dξ J + ( 2e RΛ ξ I e ΛI + ξ I I e ) ] Λ A Λ 1 1 ] 2 mλ Re RΛ B 2 B 2 Constraints among flux charges: e Λ I e ΣI e ΛI e Σ I = 0, m Λ R e Λ I = 0 = m Λ R e ΛI APPENDIX - 57 -
Scalar potential Scalar potential from (non)geometric flux compactifications: V = g 2[ 4h uv k u k v + 3 x=1 V NS = g ab D a P + D b P + + g iȷ D i P + D ȷ P + 2 P + 2 ( g ab D a P x D b P x 3 P x 2)] =... V NS + V R (abelian: k a Λ = 0) = 2 g 2 e 2φ[ Π T Q H T M V Q ΠH + Π T V Q M H Q T Π V + 4Π T H C T H Q T( Π V Π T V + Π V Π T ) ] V Q CH Π H V R = g ab D a P 3 D b P 3 + P 3 2 = 1 2 g2 e 4φ( e RΛ e ΛI ξ I + e Λ I ξi ) (ImN ) 1 ΛΣ ( e RΣ e ΣI ξ I + e Σ I ξi ) Π V = e KV/2 (X Λ, F Λ ) T t a = X a /X 0 a = 1,..., n V SKG V of vector-moduli ( 0 1 C V,H = 1 0 ) P + P 1 + ip 2 = 2e φ Π T V Q C H Π H P P 1 ip 2 = 2e φ Π T V Q C H Π H ; Q = P 3 m ΛI m Λ I = e 2φ Π T V C V (c R + Qξ) ( ) I eλ e ΛI, Q = C T H Q C V c R = Π H = e KH/2 (Z I, G I ) T z i = Z i /Z 0 i = 1,..., n H SKG H of hyper-moduli ( ) m Λ R e RΛ Cassani et.al., arxiv:0804.0595, arxiv:0911.2708 APPENDIX - 58 -