Flux compactifications and SUSY-breaking

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1 Luca Martucci (INFN & University of Rome Tor Vergata ) Flux compactifications and SUSY-breaking Based on: ariv: ariv: in collaboration with J. Held, D. Lüst, F. Marchesano, D. Tsimpis November 2011, Saclay

2 The advantages of supersymmetry flux space-filling

3 The advantages of supersymmetry flux It trades second-order field equations for first-order ones space-filling

4 The advantages of supersymmetry flux It trades second-order field equations for first-order ones Provides automatically calibration structures: supersymmetric calibrated brane stability automatic space-filling

5 The advantages of supersymmetry flux It trades second-order field equations for first-order ones Provides automatically calibration structures: supersymmetric calibrated brane stability automatic space-filling Calibrated simplify bulk-brane coupled equations too! Koerber & Tsimpis `07 Lüst, Marchesano, L.M. & Tsimpis `08

6 The advantages of supersymmetry flux It trades second-order field equations for first-order ones Provides automatically calibration structures: space-filling supersymmetric calibrated brane stability automatic Calibrated simplify bulk-brane coupled equations too! Koerber & Tsimpis `07 Lüst, Marchesano, L.M. & Tsimpis `08 Can we break SUSY preserving (some of) these nice properties?

7 Prototypical example G3 -flux Graña & Polchinski `00 Giddings, Kachru & Polchinski `01 ds210 = e2a dxµ dxµ e2a ds2f-th. G3 = ig3 G 3 = F3 τ H F5 = de4a space-filling D7- and D3- tree-level (no-scale) SUSY-breaking Mink4 All bulk coupled EoM s satisfied! G0,3!= 0, Wtree =! Ω G3 "= 0

8 Prototypical example G3 -flux Graña & Polchinski `00 Giddings, Kachru & Polchinski `01 ds210 = e2a dxµ dxµ e2a ds2f-th. G3 = ig3 G 3 = F3 τ H F5 = de4a space-filling D7- and D3- tree-level (no-scale) SUSY-breaking Mink4 All bulk coupled EoM s satisfied! G0,3!= 0, Wtree = What is the mechanism behind? Can we extend it to more general settings?! Ω G3 "= 0

9 Strategy for general case flux 4D approach to 10D physics fields specifying internal configuration find functional δv I δφ V (φi ) such that: 10D SUGRA EoM space-filling

10 Strategy for general case flux 4D approach to 10D physics fields specifying internal configuration find functional δv I δφ V (φi ) such that: space-filling 10D SUGRA EoM V as 4D potential depending on all KK-modes more direct 4D interpretation on 10D equations SUSY should impose a structure on V

11 Potential and for bosonic fields flux Internal NS-NS fields: metric: dilaton: ds210 = e2a dxµ dxµ ds2 φ 3-form: H = db Σ,F space-filling

12 Potential and for bosonic fields flux Internal NS-NS fields: metric: dilaton: ds210 = e2a dxµ dxµ ds2 φ 3-form: H = db Σ,F space-filling

13 Potential and for bosonic fields flux Internal NS-NS fields: metric: dilaton: ds210 = e2a dxµ dxµ ds2 φ 3-form: H = db Internal R-R fields:! k even/odd FRR = Fk in IIA/IIB k Σ,F space-filling

14 Potential and for bosonic fields flux Internal NS-NS fields: metric: ds210 = e2a dxµ dxµ ds2 φ dilaton: 3-form: H = db Σ,F Internal R-R fields:! k even/odd FRR = Fk in IIA/IIB space-filling k Full set of 10D EoM can be obtained by extremizing V =! " # $ 1 % 1 e4a e2φ R H 2 4(dφ)2 8 2 A 20(dA)2 Fel2 2 2 # #! i τi " 4Aφ e Σi $ det(g Σi Fi ) Fi Σi C e %

15 Potential and for bosonic fields flux Internal NS-NS fields: metric: ds210 = e2a dxµ dxµ ds2 φ dilaton: 3-form: H = db Σ,F Internal R-R fields:! k even/odd FRR = Fk in IIA/IIB space-filling k Full set of 10D EoM can be obtained by extremizing V =! " # $ 1 % 1 e4a e2φ R H 2 4(dφ)2 8 2 A 20(dA)2 Fel2 2 2 # #! i τi " 4Aφ e Σi $ However, no information on supersymmetry det(g Σi Fi ) Fi Σi C e % Underlying supersymmetric structure invisible!

16 Bosonic fields and spinors flux Supersymmetry generators space-filling

17 Bosonic fields and spinors flux Supersymmetry generators space-filling

18 Bosonic fields and spinors flux Supersymmetry generators space-filling Full information on NS-NS sector, η1 and η2 can be packed into two pure spinors on T T Z e3aφ η1 η2t T eφ η1 η2

19 Bosonic fields and spinors flux Supersymmetry generators space-filling Full information on NS-NS sector, η1 and η2 can be packed into two pure spinors on T T Z e3aφ η1 η2t IIA T eφ η1 η2 e.g. for Z = Z0 Z2 Z4 Z6 T = T1 T3 T5 = CY3 : Z = eij, T = eφ Ω3,0

20 Bosonic fields and spinors flux Supersymmetry generators space-filling Full information on NS-NS sector, η1 and η2 can be packed into two pure spinors on T T Z e3aφ η1 η2t IIB Z = Z1 Z3 Z5 T = T0 T2 T4 T6 T eφ η1 η2 e.g. for = CY3 : Z = Ω3,0, T = eφ eij

21 Potential and pure spinors BPS form of the potential V = e 4A{ e 2φ R 1 2 H2 4(dφ) A 20(dA) 2] 1 } 2 F el 2 ( τ i e 4Aφ ) det(g Σi F i ) C e F i Σ i Σ i i

22 Potential and pure spinors BPS form of the potential V = V V BPS dh (e 4A Re T ) e 4A F RR ] 2 dh (e 2A Im T )] 2 dh Z 2 T,dH Z 2 (... ) 2 d H := d H

23 Potential and pure spinors BPS form of the potential V = V V BPS dh (e 4A Re T ) e 4A F RR ] 2 dh (e 2A Im T )] 2 dh Z 2 T,dH Z 2 (... ) d H := d H

24 Potential and pure spinors BPS form of the potential V = V V BPS dh (e 4A Re T ) e 4A F RR ] 2 dh (e 2A Im T )] 2 dh Z 2 T,dH Z 2 (... ) 2 d H := d H D INTERPRETATION (brane BPS-bound) F Z 2 D 2 F T 2 F T 2 D F T 2 Koerber & L.M. `07

25 Potential and pure spinors Simplest solution of EoM s V = V V BPS dh (e 4A Re T ) e 4A F RR ] 2 dh (e 2A Im T )] 2 dh Z 2 T,dH Z 2 (... ) 2

26 Potential and pure spinors Simplest solution of EoM s V = V V BPS dh (e 4A Re T ) e 4A F RR ] 2 dh (e 2A Im T )] 2 dh Z 2 T,dH Z 2 (... ) 2 d H (e 4A Re T )=e 4A F RR d H (e 2A Im T ) d H Z

27 Potential and pure spinors Simplest solution of EoM s V = V V BPS dh (e 4A Re T ) e 4A F RR ] 2 dh (e 2A Im T )] 2 dh Z 2 T,dH Z 2 (... ) 2 calibration for space-filling d H (e 4A Re T )=e 4A F RR d H (e 2A Im T ) d H Z

28 Potential and pure spinors Simplest solution of EoM s V = V V BPS dh (e 4A Re T ) e 4A F RR ] 2 dh (e 2A Im T )] 2 dh Z 2 T,dH Z 2 (... ) 2 calibration for space-filling d H (e 4A Re T )=e 4A F RR d H (e 2A Im T ) d H Z plus calibrated spacefilling

29 Potential and pure spinors Simplest solution of EoM s V = V V BPS dh (e 4A Re T ) e 4A F RR ] 2 dh (e 2A Im T )] 2 dh Z 2 T,dH Z 2 (... ) 2 calibration for space-filling d H (e 4A Re T )=e 4A F RR d H (e 2A Im T ) d H Z plus calibrated spacefilling

30 Potential and pure spinors Simplest solution of EoM s V = V V BPS dh (e 4A Re T ) e 4A F RR ] 2 dh (e 2A Im T )] 2 dh Z 2 T,dH Z 2 (... ) 2 calibration for space-filling d H (e 4A Re T )=e 4A F RR d H (e 2A Im T ) d H Z plus calibrated spacefilling SUSY conditions! Graña, Minasian, Petrini & Tomasiello `05 L.M. & Smyth `05

31 Potential and pure spinors Simplest solution of EoM s V = V V BPS dh (e 4A Re T ) e 4A F RR ] 2 dh (e 2A Im T )] 2 dh Z 2 T,dH Z 2 (... ) 2 calibration for space-filling d H (e 4A Re T )=e 4A F RR d H (e 2A Im T ) d H Z plus calibrated spacefilling SUSY conditions! Graña, Minasian, Petrini & Tomasiello `05 L.M. & Smyth `05 All EoM s are manifestly satisfied!

32 Potential and pure spinors Natura SUSY-breaking ansatz V = V V BPS dh (e 4A Re T ) e 4A F RR ] 2 dh (e 2A Im T )] 2 dh Z 2 T,dH Z 2 (... ) 2

33 Potential and pure spinors Natura SUSY-breaking ansatz V = V V BPS dh (e 4A Re T ) e 4A F RR ] 2 dh (e 2A Im T )] 2 dh Z 2 T,dH Z 2 (... ) 2 d H (e 4A Re T )=e 4A F RR d H (e 2A Im T )

34 Potential and pure spinors Natura SUSY-breaking ansatz V = V V BPS dh (e 4A Re T ) e 4A F RR ] 2 dh (e 2A Im T )] 2 dh Z 2 T,dH Z 2 (... ) 2 calibration for space-filling d H (e 4A Re T )=e 4A F RR d H (e 2A Im T )

35 Potential and pure spinors Natura SUSY-breaking ansatz V = V V BPS dh (e 4A Re T ) e 4A F RR ] 2 dh (e 2A Im T )] 2 dh Z 2 T,dH Z 2 (... ) 2 calibration for space-filling d H (e 4A Re T )=e 4A F RR d H (e 2A Im T ) plus calibrated

36 Potential and pure spinors Natura SUSY-breaking ansatz V = V V BPS dh (e 4A Re T ) e 4A F RR ] 2 dh (e 2A Im T )] 2 dh Z 2 T,dH Z 2 (... ) 2 calibration for space-filling d H (e 4A Re T )=e 4A F RR d H (e 2A Im T ) plus calibrated

37 Potential and pure spinors Natura SUSY-breaking ansatz V = V V BPS dh (e 4A Re T ) e 4A F RR ] 2 dh (e 2A Im T )] 2 dh Z 2 T,dH Z 2 (... ) 2 > 0 < 0 calibration for space-filling d H (e 4A Re T )=e 4A F RR d H (e 2A Im T ) d H Z 0 plus calibrated choose d H Z 0 such that they cancel each other SUSY-breaking compactification to Mink 4

38 A concrete recipe Take fibration with calibrated fibers Π B

39 A concrete recipe Take fibration with calibrated fibers Π B Choose SUSY-breaking of the form d H Z = re R vol B SUSY-breaking parameter twisting two-form dr = H Π

40 A concrete recipe Take fibration with calibrated fibers Π B Choose SUSY-breaking of the form d H Z = re R vol B SUSY-breaking parameter twisting two-form dr = H Π In the wcy case: {fibers Π} = {points in } B = Z = Ω CY d H Z =d H Ω CY = r vol,

41 A concrete recipe Take fibration with calibrated fibers Π B Choose SUSY-breaking of the form d H Z = re R vol B SUSY-breaking parameter twisting two-form dr = H Π Explicit examples on twisted tori dω 3,0 r vol 2,2 T 4 see also Camara & Graña `08 Blaback, Danielsson, Junghans, Van Riet, Wrase & Zagermann `10 E.g.

42 Open problems Existence theorems? as in SUSY-case Higher order corrections? Combination with quantum effects? no-scale, KKLT-like scenarios Extension of the strategy to de Sitter? Andriot, Goi, Minasian & Petrini `08...

43 Potential and pure spinors CALIBRATIONS for D- space-filling strings d H (e 4A Re T )=e 4A F RR d H (e 2A Im T ) d H Z domain-walls plus calibrated spacefilling SUSY conditions! Graña, Minasian, Petrini & Tomasiello `05 L.M. & Smyth `05

44 Potential and pure spinors CALIBRATIONS for D- space-filling d H (e 4A Re T )=e 4A F RR strings d H (e 2A Im T ) d H Z 0 (DWSB) domain-walls plus calibrated SUSY-breaking compactification to Mink 4

45 The problem Find non-supersymmetric flux compactifications Break supersymmetry in a controlled way In particular, keep back-reaction of localized sourced (D and orientifold) under control