New Toric Swiss Cheese Solutions

Transcription

1 String Phenomenology 2017 (Based on arxiv: [hep-th]) Northeastern University 1/31

2 Outline Motivation 1 Motivation 2 CY Geometry Large Volume Scenario 3 Explicit Toric 4 2/31

3 Abundance of Moduli Type IIB: Calabi-Yau 3-fold moduli Non-physical, free scalar fields, which must be stabilized Axion-dilaton: S Kähler moduli: T 1,..., T h 1,1 Complex structure moduli: U 1,..., U h 2,1 We focus on the Large Volume Scenario (LVS) mechanism Kreuzer-Skarke 4D reflexive polyhedra Largest class of CY 3-fold vacua (more than half a billion!) [Kreuzer, Skarke ] 3/31

4 Abundance of Moduli 22 ( h 1,1 + h 2,1) 502 [Kreuzer, Skarke ] 4/31

5 Outline Motivation CY Geometry Large Volume Scenario 1 Motivation 2 CY Geometry Large Volume Scenario 3 Explicit Toric 4 5/31

6 Calibrated Geometry CY Geometry Large Volume Scenario Moduli are geometric properties of the CY vacuum. No explicit metric known for CY manifolds. Volume of 2n-cycles fixed by Kähler form: Curves (2-cycles): vol(c) := 1 1! J Divisors (4-cycles): vol(d) := 1 2! J J D Compact CY (6-cycle): vol(x) := 1 3! J J J C X 6/31

7 Kähler Moduli Motivation CY Geometry Large Volume Scenario Choose a Z-basis {J i } H 1,1 (X) and a dual basis of curves {C i } such that C i J j = δ i j. The 2n-cycle volume moduli are: t i := vol(c i ) = 1 1! t j J j = t i C i τ i := vol(j i ) = 1 2! J i t j J j t k J k = 1 2 tj t k κ ijk X V := vol(x) = 1 3! t i J i t j J j t k J k = 1 6 ti t j t k κ ijk X where κ ijk = J i J j J k is the triple intersection tensor. X 7/31

8 Outline Motivation CY Geometry Large Volume Scenario 1 Motivation 2 CY Geometry Large Volume Scenario 3 Explicit Toric 4 8/31

9 Tree Level Motivation CY Geometry Large Volume Scenario Effective F-term scalar potential V(S, T i, U i ) = e K [ (K 1 ) a b Da WD b W 3 W 2 ] with tree level Kähler potential K = ln ( S + S ) 2 ln (V) ln ( ) i Ω Ω X By turning on gauge fluxes, we generate a superpotential W GVW = Ω G 3 In the presence of a warped conifold, fixes S, U 1,..., U h 2,1. X [Gukov, Vafa, Witten ], [Giddings, Kachru, Polchinski ] 9/31

10 α Correction Motivation CY Geometry Large Volume Scenario We break the no-scale structure w.r.t. T 1,..., T h 1,1 and stabilize the volume modulus by adding the leading order α correction to the Kähler potential ( ) 3/2 S + S, ξ = χ(x)ζ(3) V V + ξ V α (T i ) = 3e K W GVW 2 ξ (ξ2 +7ξV+V 2 ) (V ξ)(2v+ξ) 2 [Bobkov ] 10/31

11 Non-perturbative Corrections CY Geometry Large Volume Scenario We need non-perturbative corrections to W to stabilize the remaining Kähler moduli. Euclidean D3 instantons or low energy dynamics on D7 branes. [Witten ], [Gorlich, Kachru, Tripathy,Trivedi ] Must be localized at small volume (τ s := vol(j s ) V), rigid (χ h (J s ) = 1) blowup cycles. For one blowup cycle: [Balasubramanian, Berglund, Conlon, and Quevedo ] W W + A s e asts V np (T s ) = 2ca2 s A s 2 e 2asτs τ s V 2as AsW GVW e asτs τ s V W GVW 2 ξ 8V 3 11/31

12 Non-perturbative Corrections II CY Geometry Large Volume Scenario With AdS minimum: τ s 1 ( 3cχ(X)ζ(3) ) 2/3, with χ(x) < 0 (i.e. h 2,1 > h 1,1 ) 4 4 V W GVW τs e asτs 2ca s A s at exponentially large volume. [Balasubramanian, Berglund, Conlon, and Quevedo ] b s = Im(T s ) axion stabilization is more difficult for more than one blowup cycle. [Cicoli, Conlon, Quevedo ] 12/31

13 Outline Motivation Explicit Toric 1 Motivation 2 CY Geometry Large Volume Scenario 3 Explicit Toric 4 13/31

14 Manifolds Explicit Toric The dynamical quantities τ i and V are related implicitly through t i : τ i = 1 2! κ ijkt j t k and V = 1 3! κ ijkt i t j t k Simplest LVS: Swiss cheese with V τ 3/2 L. V = λτ 3/2 L τ si small γ i τ 3/2 s i 14/31

15 (Strong Form) Explicit Toric Strong Form V = τ Li large λ i τ 3/2 L i τ sj small γ j τ 3/2 s j Implies bases J = t Ai Ji A = t Bi Ji B and κ AAA ijk = X JA i Jj A Jk A, κ BAA ijk = X JB i Jj A Jk A where V = 1 2! κaaa ijk t Ai t Aj t Ak = τ B i ( t Ai ) 3 h 1,1 i=1 = vol(j B i ) = 1 2! κbaa ijk t Aj t Ak = ( t Ai) 2 Signs and coefficients fixed by Kähler cone and V > 0. 15/31

16 (Explicit Form) Explicit Toric Explicit Form V = L 3/2 (τ 1,..., τ h 1,1) S 3/2 (τ 1,..., τ h 1,1) Implies bases J = t Ai J A i = t Bi J B i and κ BAA ijk = X JB i J A j J A k where τ B i = vol(j B i ) = 1 2! κbaa ijk t Aj t Ak invertible System of non-linear equations, typically difficult to solve. Explicitly solvable when τ B i = ( t Ai) 2 16/31

17 Explicit Toric Statistics: (Explicit Form) h 1,1 (X) # of Favorable Geometries % of Favorable Geometries Scanned for Explicitness Explicit and Strong Geometries # of Explicit Geometries # of Strong Geometries Table 1: Statistics for explicit- and strong-form scan over favorable Calabi-Yau threefold geometries. (Note: the scan for explicitness has not completed at this time.) 17/31

18 Outline Motivation Explicit Toric 1 Motivation 2 CY Geometry Large Volume Scenario 3 Explicit Toric 4 18/31

19 (Implicit Form) Explicit Toric Implicit Form V = 1 3! κ ijkt i t j t k Implies bases J = t Ai J A i = t Bi J B i where t Ai = { Large, Small, i IL A i IS A and τ B i = { Large, Small, i IL B i IS B 1. (Volume) (i, j, k) I A L I A I A : κ AAA ijk 0 2. (Large Cycle) i I B L, (j, k) I A L I A : κ BAA ijk 0 3. (Small Cycle) (i, j, k) I B S IA L I A : κ BAA ijk = 0 19/31

20 Toric Motivation Explicit Toric Scan would require two arbitrary transformations T A, T B GL(h 1,1, Q). Special (Toric) Case α : I A I Toric and β : I B I Toric J A i D α(i) J B i D β(i) With known toric divisor intersection tensor d ijk = X D i D j D k, we have κ AAA ijk = d α(i)α(j)α(k) and κ BAA ijk = d β(i)α(j)α(k) 20/31

21 Toric II Explicit Toric The conditions become purely combinatoric 1. (Volume) (i, j, k) α(il A ) α(i A ) α(i A ) : d ijk 0 2. (Large Cycle) i β(il B ), (j, k) α(il A ) α(i A ) : d ijk 0 3. (Small Cycle) (i, j, k) β(is B) α(ia L ) α(i A ) : d ijk = 0 21/31

22 Toric III Explicit Toric There is good reason to believe that some small toric divisors can be isolated: h 1,1 (X) # of Favorable Geometries # of Geometries with Toric Blowup Divisors del Pezzo surface dp n, n Non-Shrinkable Rigid surface dp n, n > Table 2: Number of Favorable Geometries in the database, along with the number that support various kinds of toric blowup divisors. 22/31

23 Explicit Toric Homogeneity of the Effective Potential In the case of one small cycle τ s, there is an additional constraint on the inverse Kähler metric. [Cicoli, Conlon, Quevedo ] The effective potential can only have a stable minimum when its terms are homogeneous in V 1. This implies Homogeneity Condition (N S = 1) ( K 1 ) ss 2Vκssi t i V τ s This is similar to the condition for explicitness of the small cycle only τ B i = ( t Ai) 2 23/31

24 Explicit Toric Homogeneity of the Effective Potential II For N S > 1, the condition is not strictly required, but it simplifies calculations Homogeneity Condition (N S > 1) ( K 1 ) ii 2Vκiij t j V h 1/2 ({τ k } k IS ), i I S In terms of toric intersection numbers, this implies 4. (Homogeneity) i β(i B S ), j α(ia S ) : d iij 0 Finally, we check that V exists within the Kähler cone. 24/31

25 Statistics: Toric Explicit Toric h 1,1 (X) # of Favorable Geometries Toric Geometries N L = 1-32 (22) 86 (84) 173 (171) 603 (577) 1304 (1137) # of Toric Swiss Cheese Geometries (w/ Homogeneity Condition) N L = (23) 17 (17) 12 (10) 17 (13) N L = (1) 0 (0) 0 (0) N L = (0) 0 (0) N L = (0) Table 3: Statistics for toric Swiss cheese scan over favorable Calabi-Yau threefold geometries. 2, 268 (2, 055) new solutions! 70 (64) with N L 2 and 159 (147) with N S 2. 25/31

26 Explicit Toric Example: Explicit, Toric (h 1,1 = 4, N S = 2) h 1,1 (X) = 4, h 2,1 (X) = 94, χ(x) = 180 Small dp 0 blowup cycles J B 3 and JB 4. K3 fiber J B 2. Polytope ID Geometry ID V = 1 [ 9τ B ( 1 τ 2 B τ 2 B τ B 3 + τ B ) ( 4 τ B ) ( 3/2 ( τ B ) 3/2 ( 3 + τ B ) )] 3/2 4 ( V V, τ B 3, τ B ) 6 2a 2 3 A 3 2 τ 3 Be 2a 3 τb 3 4 = V 6 2a 2 4 A 4 2 τ 4 Be 2a 4 τb 4 + V 2a 4 A 4 W GVW τ4 B e a 4 τb 4 V 2 + 3ξ W 2 GVW 8V 3 2a 3 A 3 W GVW τ B 3 e a 3 τb 3 V 2 26/31

27 Explicit Toric Example: Explicit, Toric (h 1,1 = 4, N S = 2) Figure 1: With V Figure 2: With τ B s := τ B 3 = τ B 4 a 3 = a 4 = 2π and A 1 = A 2 = W GVW = 1 27/31

28 Explicit Toric Example: Explicit, Toric (h 1,1 = 4, N S = 2) V (V, τ s) = 12 2a 2 s A s 2 τs B e 2asτ B s V 4as AsWGVW τ s B e asτ B s + 3ξ WGVW 2 V 2 8V 3 There is a minimum at τ B s 12.3 and V And a flat direction in the large cycles ( ) τ τ1 B B 3/2 = 2 6τ B s τ B ( ) τs B 3/2 + 18V 9 τ2 B 28/31

29 Database ( 29/31

30 Database ( 30/31

31 Database ( Thank you! 31/31