Magdalena Larfors

Transcription

1 Uppsala University, Dept. of Theoretical Physics Based on D. Chialva, U. Danielsson, N. Johansson, M.L. and M. Vonk, hep-th/ U. Danielsson, N. Johansson and M.L., hep-th/

2 String theory lives in 10D, we live in 4D. Compactify. Fluxes. Branes. (universe-review.ca) string theory landscape of vacua

3 Natural questions: How many vacua? Distribution? Continuously connected? Barriers? (universe-review.ca) Effects from topography Tunneling, domain walls, inflation, finiteness...

4 1 Flux vacua Ingredients: manifolds, fluxes, branes... enormous landscape! A landscape model Type IIB SUGRA on (conformal) CY 3 fold. 3 fluxes through 3 cycles: fix CS moduli. Generic. Rich structure. Calabi Yau manifolds Complex, Kähler, Ricci flat. h 2,1 Complex structure (CS) moduli. h 1,1 Kähler moduli.

5 1 Flux vacua Ingredients: manifolds, fluxes, branes... enormous landscape! A landscape model Type IIB SUGRA on (conformal) CY 3 fold. 3 fluxes through 3 cycles: fix CS moduli. Generic. Rich structure. Calabi Yau manifolds Complex, Kähler, Ricci flat. h 2,1 Complex structure (CS) moduli. h 1,1 Kähler moduli.

6 1 Flux vacua Ingredients: manifolds, fluxes, branes... enormous landscape! A landscape model Type IIB SUGRA on (conformal) CY 3 fold. 3 fluxes through 3 cycles: fix CS moduli. Generic. Rich structure. Calabi Yau manifolds Complex, Kähler, Ricci flat. h 2,1 Complex structure (CS) moduli. h 1,1 Kähler moduli.

7 Fixing the complex structure Complex structure holomorphic 3-form Ω(z). 3 cycles 3-cycles basis A I, B J. A I α J = B I β J = α I β J = δ IJ Periods Π i (z) = C i Ω(z) z: CS moduli Π(z) = (Π 1 (z), Π 2 (z),...π N (z)) 3 flux IIB: RR F and NS H G = F τh Quantized: C i F F i, C i H H i, F i, H i Z D3 tadpole condition: CY F H = N D3

8 The potential for CS moduli Fluxes wrapping non-trivial cycles potential V. V = e K ( DW 2 3 W 2) Kähler potential e K 1 = Im(ρ) 3 Imτ Π Q Π Superpotential W = G Π(z) CS moduli and τ fixed at of potential. No-scale: Kähler moduli unfixed perturbatively.

9 Paths between flux vacua (hep-th/ ) CS moduli space is complicated: singularities, branch cuts, non trivial loops monodromies of 3-cycles. Idea: Use monodromies to continuously connect vacua.

10 Monodromies Period monodromies Π(z) T Π(z) T M Sp(N, Z) E.g. Mirror Quintic h 2,1 = 1 CS modulus h 1,1 = 101 Kähler moduli Π i Im(z) Re(z) 0 1

11 Recall: V = e K ( DW 2 3 W 2), W = G Π(z) Thus Π(z) T Π(z) V has branch cuts in CS moduli space. Traverse cuts paths between. 150 Π T Π or G G T 100 T M Sp(N, Z) 50 F H unchanged V Starting flux configuration: F0 = (2, 9, 2, 10) H0 = (1, 10, 4, 1) Re(z) Im(z)

12 V V Series of Several continuously connected found: Starting flux configuration: F0 = (1, 9,!7, 1) H0 = (1, 0, 8, 0) Starting flux configuration: F0 = (1, 9,!7, 1) H0 = (1, 0, 8, 0) Im(z)! !1 0 Re(z) No infinite series of found. What about flux not related by monodromies ( islands )? !1! Re(z) 2 2.5!1!0.5 0 Im(z) 0.5 1

13 An extended landscape model Monodromies: important for topography. Larger moduli space more monodromies. : Moduli spaces of different Calabi Yau 3-folds are connected Idea: extend M 101,1 CS moduli space. Connect it to what?

14 Mirror symmetry M (86,2) GT M (101,1) M (86,2) M (101,1) shrink 16 3-cycles A i blow up 16 2-cycles a i A 1 A 2 = δd 1... ai = δb A 15 A 16 = δd 15 δd i = 0

15 Complex (101,1) M Kahler M (86,2) Complex Complex

16 We need to: Construct M (86,2) (using toric geometry) Complex (101,1) M Kahler Compute periods of M (86,2) M (86,2) Complex Embed M (101,1) in M (86,2) Compute new monodromies with flux? string theory vacua? Complex

17 Toric geometry Toric geometry Construct CY: zero locus of polynomial equations on toric variety. (C Toric variety: ) n Z G Fans and polytopes toric variety and equations. Batyrev s mirror construction M (2,86) : CI in P 1 P 4 Polytope for M (2,86) : = Mirror construction: Polytope for M (86,2) : = 1 + 2, k, j δ k,j M (86,2) f 1 1 a 1 t 1 a 2 t 3 a 3 t 4 a 4 t 5 a 5 /t 2 t 3 t 4 t 5 f 2 1 a 6 /t 1 a 7 t 2, CS moduli a i : φ 1 = a 1 a 6, φ 2 = a 2 a 3 a 4 a 5 a 7.

18 Toric geometry Toric geometry Construct CY: zero locus of polynomial equations on toric variety. (C Toric variety: ) n Z G Fans and polytopes toric variety and equations. Batyrev s mirror construction M (2,86) : CI in P 1 P 4 Polytope for M (2,86) : = Mirror construction: Polytope for M (86,2) : = 1 + 2, k, j δ k,j M (86,2) f 1 1 a 1 t 1 a 2 t 3 a 3 t 4 a 4 t 5 a 5 /t 2 t 3 t 4 t 5 f 2 1 a 6 /t 1 a 7 t 2, CS moduli a i : φ 1 = a 1 a 6, φ 2 = a 2 a 3 a 4 a 5 a 7.

19 Toric geometry Toric geometry Construct CY: zero locus of polynomial equations on toric variety. (C Toric variety: ) n Z G Fans and polytopes toric variety and equations. Batyrev s mirror construction M (2,86) : CI in P 1 P 4 Polytope for M (2,86) : = Mirror construction: Polytope for M (86,2) : = 1 + 2, k, j δ k,j M (86,2) f 1 1 a 1 t 1 a 2 t 3 a 3 t 4 a 4 t 5 a 5 /t 2 t 3 t 4 t 5 f 2 1 a 6 /t 1 a 7 t 2, CS moduli a i : φ 1 = a 1 a 6, φ 2 = a 2 a 3 a 4 a 5 a 7.

20 M (86,2) periods: 1 The fundamental period ω 0 = 1 1 dt 1 (2πi) γ 5 f 1f 2 t 1... dt5 t 5 Near φ 1 = φ 2 = 0 ω 0 (φ) = n 1,n 2 c(n 1, n 2 )φ n1 1 φn2 2, where c(n 1, n 2 ) = (n1+4n2)!(n1+n2)! (n 1!) 2 (n 2!) 5 Picard Fuchs equations Recall Π i = C i Ω H 3 = H 3,0 H 2,1 H 1,2 H 0,3 is finite L k Ω = dη L k Π i = C i L k Ω = C i dη = 0 We get: 2 DE of degree 2 and 3 6 linearly indep. solutions 6 periods.

21 M (86,2) periods: 1 The fundamental period ω 0 = 1 1 dt 1 (2πi) γ 5 f 1f 2 t 1... dt5 t 5 Near φ 1 = φ 2 = 0 ω 0 (φ) = n 1,n 2 c(n 1, n 2 )φ n1 1 φn2 2, where c(n 1, n 2 ) = (n1+4n2)!(n1+n2)! (n 1!) 2 (n 2!) 5 Picard Fuchs equations Recall Π i = C i Ω H 3 = H 3,0 H 2,1 H 1,2 H 0,3 is finite L k Ω = dη L k Π i = C i L k Ω = C i dη = 0 We get: 2 DE of degree 2 and 3 6 linearly indep. solutions 6 periods.

22 M (86,2) periods: 2 The Frobenius method φ 1 = φ 2 = 0 regular singular point five solutions with logarithmic singularities: Using ω(ρ, φ) = n 1,n 2 c(n 1 + ρ 1, n 2 + ρ 2 )φ n1+ρ1 1 φ n2+ρ2 2 we get all periods as: (hep-th/ ) ω 1 = ρ1 ω ρ=0, ω 2 = ρ2 ω ρ=0, ω 3 = κ 1jk ρj ρk ω ρ=0, ω 4 = κ 2jk ρj ρk ω ρ=0, ω 5 = κ ijk ρi ρj ρk ω ρ=0 κ ijk = J i J j J k classical intersection numbers. Want integral and symplectic monodromies: change basis. (No details here).

23 Embed M (101,1) in M (86,2) M (86,2) Recall: f 1 1 a 1 t 1 a 2 t 3 a 3 t 4 a 4 t 5 a 5 /t 2 t 3 t 4 t 5 = 0 f 2 1 a 6 /t 1 a 7 t 2 = 0. M (101,1) Substitute f 2 into f 1 b 0 + b 1 u 1 + b 2 u 2 + b 3 u 3 + b 4 u 4 + b5 u 1u 2u 3u 4 + b6 u 1u 2u 3 = 0 Redefined CS moduli z 1 = b1b2b3b6 b0 4 = φ2 (1 φ 1) 4 and z 2 = b1b2b3b4b5 b 5 0 Take z 1 0: Mirror quintic equation! = φ1φ2 (1 φ 1) 5

24 Embed M (101,1) in M (86,2) M (86,2) Recall: f 1 1 a 1 t 1 a 2 t 3 a 3 t 4 a 4 t 5 a 5 /t 2 t 3 t 4 t 5 = 0 f 2 1 a 6 /t 1 a 7 t 2 = 0. M (101,1) Substitute f 2 into f 1 b 0 + b 1 u 1 + b 2 u 2 + b 3 u 3 + b 4 u 4 + b5 u 1u 2u 3u 4 + b6 u 1u 2u 3 = 0 Redefined CS moduli z 1 = b1b2b3b6 b0 4 = φ2 (1 φ 1) 4 and z 2 = b1b2b3b4b5 b 5 0 Take z 1 0: Mirror quintic equation! = φ1φ2 (1 φ 1) 5

25 The mirror quintic locus The MQ locus z 1 0: Which period vanish? Monodromy around the locus? Analytically continue ω 0 ω 0 = (4m 1+5m 2)! m 1,m 2=0 ((m 1+m 2)!) 3 m 1!(m 2!) z m1 2 1 zm2 2. z 1 0: MQ fundamental period. Other periods: Analytically continue ω i : focus on derivatives.

26 The mirror quintic locus The MQ locus z 1 0: Which period vanish? Monodromy around the locus? Analytically continue ω 0 ω 0 = (4m 1+5m 2)! m 1,m 2=0 ((m 1+m 2)!) 3 m 1!(m 2!) z m1 2 1 zm2 2. z 1 0: MQ fundamental period. Other periods: Analytically continue ω i : focus on derivatives.

27 Embedded periods and monodromies Periods Integral and symplectic basis: Π (86,2) = Π 1 Π 2 Π 3 Π 4 Π 5 Π 6 z 1 0 Π MQ 1 Π MQ 2 Π MQ 3 Π MQ 4 0 ci Π MQ i New paths between vacua 4 new monodromies.. New series of MQ vacua Complex Complex (101,1) M Kahler M (86,2) Complex

28 A: shrinking 3 cycle, B: torn 3 cycle. Need to be careful: M (86,2) monodromy might yield flux through A or B! Flux through A hep-th/ , RR/NS flux through shrinking 3 cycle A: D5/NS5 branes on new 2 cycles. Positions of 5 branes new open string moduli. New period: Π B (t, z) = B Ω V MQ(z) Ṽ MQ (t, z).

29 Flux through B , hep-th/ RR/NS Flux through torn 3-cycle B D1/F1-instantons? No new terms in the MQ potential.

30 Flux through both 3 cycles A and B New open string moduli Tadpole condition: F H might change D3 branes. Examples N RR-flux through A, M NS-flux through B: either N D5 branes and M F1 instantons or compact Klebanov Strassler: N D5 branes, MN D3 branes. N RR-flux through A, M RR-flux through B: N D5 branes, maybe M D1 instantons, no D3 branes.

31 Flux potential at transition Near transition point: V (86,2) (z 1, z 2 ) = V 1 (z 1, z 2 ) + V 2 (z 2 ); V 2 (z 2 ) z1 0 V MQ (z) With flux through A: V 1 (z 1, z 2 ) z1 0 Without flux through A: V 1 (z 1, z 2 ) z1 0 0 No flux through shrinking cycle geometric transition controlled. Look for connected without such flux.

32 Requirements Apply monodromy n times: F 0 F 0 T n If T = 1 + Θ, Θ 2 = 0 F 0 T n = F 0 + nf 0 T Start flux F 0, H 0 F 0 T = F L V F = (1, 1, 0,!2) H = (5, 7, 2,!8),! " V=0 Limit flux F L, H L has minimum F L H L = 0 F L T = F L Im(z)!0.5!1 0 1 Re(z) 2 3 F 0 = F 0 + nf L, H 0 = H 0 + nh L infinite number of. N.B. Kähler moduli not fixed.

33 Semi-discrete landscape. Topography dynamics. Monodromies connect vacua. New, continuous paths. The new paths allow us to connect more vacua continuously. find infinite series of. describe domain walls. use connected moduli spaces. Kähler moduli dynamics. Back reaction. Transition. Tunnelling between. Inflation.