Heterotic Vector Bundles, Deformations and Geometric Transitions
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1 Heterotic Vector Bundles, Deformations and Geometric Transitions Lara B. Anderson Harvard University Work done in collaboration with: J. Gray, A. Lukas and B. Ovrut arxiv: , , , 1207.???? String Math 2012 Bonn July 18th, 2012 Lara Anderson (Harvard) Heterotic Vector Bundles, Deformations and Geometric TransitionsString Math - July 18th, 12 1 / 14
2 A smooth E 8 E 8 heterotic model: The geometric ingredients include: A Calabi-Yau 3-fold, X A holomorphic vector bundle, V, on X (with structure group G E 8) Compactifying on X leads to N = 1 SUSY in 4D, while V breaks E 8 G H, where H is the Low Energy GUT group G = SU(n), n = 3, 4, 5 leads to H = E 6, SO(10), SU(5) Matter and Moduli H-charged matter, H 1 (X, V ), H 1 (X, V ), H 1 (X, 2 V ),... X h 1,1 (X ) - Kähler moduli and h (X ) - Comple structure moduli V h 1 (X, End 0(V )) Bundle moduli A susy vacuum must satsify the Hermitian Yang-Mills Equations F ab = Fā b δχ = 0 = 0 g ab F ab = 0 Lara Anderson (Harvard) Heterotic Vector Bundles, Deformations and Geometric TransitionsString Math - July 18th, 12 2 / 14
3 Begin with a holomorphic bundle, F ab = Fā b = 0, w.r.t a fied comple structure. What happens as we vary the comple structure? Must a bundle stay holomorphic for any variation δz I v I h (X )? No. Infinitesimally, we must solve: δz I v c I [ā] F (0) c b] + 2D(0) [ā δa b] = 0 For the 4d Theory: Superpotential W = X Ω H where H = db 3α 2 ( ω 3YM ω 3L). ω 3YM = A da 1 3 A A A. In Minkowski vacuum, F-terms: F Ci δ(f Ci ) = ɛā c b ɛ abc Ω (0) abc 2 ωi c tr(t T y ) X = W C i = 3α 2 X Ω ω3ym C i ) ( δz I v c I [ā F (0)y c b] + 2D(0) [ā δaȳ b] Physics Idea Use bundle holomorphy to constrain C.S. Moduli of X. New moduli stabilization tool. Math Question Given a CY X, how to find/engineer bundles that are holomorphic only at special loci (ideally isolated pts) in CS moduli space? Lara Anderson (Harvard) Heterotic Vector Bundles, Deformations and Geometric TransitionsString Math - July 18th, 12 3 / 14
4 There are three objects in Deformation Theory that we need: Def (X ): Deformations of X as a comple manifold. First order defs parameterized by the vector space H 1 (TX ) = H (X ). These are the comple structure deformations of X. Def (V ): The deformation space of V (changes in connection, δa) for fied C.S. moduli. Infinitesimally H 1 (End(V )). The bundle moduli of V. Def (V, X ): Simultaneous holomorphic deformations of V and X. The tangent space is H 1 (X, Q) where 0 End(V ) Q π TX 0 Q is defined by the projectivized total space of the bundle P(V ) X. 0 H 1 (End(V )) H 1 (Q) dπ H 1 (TX ) α H 2 (End(V ))... Must determine: H 1 (X, Q) = H 1 (X, End(V )) Ker(α) where α = [F 1,1 ] H 1 (End(V ) TX ) is the Atiyah Class. These are the moduli of a heterotic theory. Lara Anderson (Harvard) Heterotic Vector Bundles, Deformations and Geometric TransitionsString Math - July 18th, 12 4 / 14
5 A simple eample Consider an SU(2) bundle defined by a etension in Et 1 (L, L): 0 L V L 0 In principle, such a bundle can stabilize arbitrarily many moduli. For eample, consider L = O( 2, 2, 1, 1) on the CY, X = P 1 2 P 1 2 P 1 2 P 1 2 Why this one? Here Et 1 (L, L) = H 1 (X, O( 4, 4, 2, 2)) = 0 generically. Hence cannot define the bundle for general comple structure! Happily, cohomology can jump at higher co-dimensional loci in Comple Structure moduli space. This is a easy eample of structural C.S. dependence in V. 4,68 Lara Anderson (Harvard) Heterotic Vector Bundles, Deformations and Geometric TransitionsString Math - July 18th, 12 5 / 14
6 Eplicitly: Let A = P 1 P 1 P 1 P 1. Given, p H 0 (A, O(2, 2, 2, 2)), the Koszul sequence for X gives us 0 O( 2, 2, 2, 2) L A 2 p LA 2 L X 2 0 H 1 (X, L 2 ) = ker(p), p : H 2 (A, O( 2, 2, 2, 2) L A) p H 2 (A, L 2 ) H 2 (X, L 2 ) = coker(p) Question: How can we vary p = p 0 + δp so that ker(p) 0? In field theory: E 7 Singlets: C + H 1 (L 2 ), C H 1 (L 2 ). ( Jump together) Superpotential: W = λ ia (z) C i +C a λ ia z I Choose Vacuum: < C + > 0 and < C >= 0. Non-trivial F-term: ( In fluctuation δ W C b W C a ) = λ ia (z) < C i + >= 0 = λ ib < C z I + i > δz I vanishes along locus with λ = 0. to locus, δz I gets a mass. All agree with Atiyah Computation. Lara Anderson (Harvard) Heterotic Vector Bundles, Deformations and Geometric TransitionsString Math - July 18th, 12 6 / 14
7 Global questions... Everything we have discussed so far involves fluctuation around a point in C.S. moduli space. Big limitiation: You have to know where to start. Hard to find isolated solutions (wanted for moduli stabilization) this way. New Idea: Represent CS Loci (vacuum solutions to the F-terms) as an algebraic variety. From Koszul Sequence: H 1 (X, L 2 ) = ker(p) forms an ideal in the combined moduli space p(comple structure)s(bundle moduli) = 0 in cohomology Using computational algebraic geometry (i.e. Groebner Basis methods) Form the Ideal Primary Decompose Elimination to remove s Now we can scan over all possible starting points! Lara Anderson (Harvard) Heterotic Vector Bundles, Deformations and Geometric TransitionsString Math - July 18th, 12 7 / 14
8 In order to describe the ideal on one slide, quotient by freely acting Z 2 Z 4 symmetry. Then we have p(c)s(b) = 0 H 2 (A, L 2 ), where p = c 1 1,0 1,1 2,0 3,0 3,1 4,0 4,1 + c 9 ( 1,0 2 ) 3,0 3,1 4,0 4,1 2 2, ,1 3,0 3,1 4,0 4,1 2 2, ,0 2 3,0 3,1 4,0 4, ,1 2 3,0 3,1 4,0 4,1 + c 3 ( 1,1 2 ) 2,0 4,0 4,1 2 3,0 + 1,0 1,1 2 3,1 2 4,0 3,0 + 1,0 1,1 2 2,0 3,1 2 4,1 3, ,0 2,0 2 3,1 4,0 4,1 + c 4 ( 1,0 1,1 2,0 2 ) 4,0 4,1 2 3, ,1 2,0 3,1 2 4,0 3, ,0 2,0 3,1 2 4,1 3,0 + 1,0 1, ,1 4,0 4,1 + c 5 ( 1,0 1,1 2 ) 4,0 4,1 2 3, ,0 2,0 3,1 2 4,0 3, ,1 2,0 3,1 2 4,1 3,0 + 1,0 1,1 2 2,0 2 3,1 4,0 4,1 + c 6 ( 1,0 2 ) 2,0 4,0 4,1 2 3,0 + 1,0 1,1 2 2,0 3,1 2 4,0 3,0 + 1,0 1,1 2 3,1 2 4,1 3, ,1 2,0 2 3,1 4,0 4,1 + c 7 ( 1,1 2 ) 2 2 3,0 2 4, , ,1 2 4, ,1 2 2,0 2 3,0 2 4, ,0 2 2,0 2 3,1 2 4,1 + ( c 8 1,0 2 ) 2 2 3,0 2 4, ,0 2 2,0 2 3,1 2 4, , ,0 2 4, ,1 2 2,0 2 3,1 2 4,1 + c 2 ( 1,0 1,1 2,0 3,0 2 ) 2 4,0 + 1,0 1,1 2,0 2 3,1 2 4,0 + 1,0 1,1 2,0 2 3,0 2 4,1 + 1,0 1,1 2,0 2 3,1 2 4,1 + ( c 10 1,1 2 ) 2 2,0 2 3,0 2 4, , ,1 2 4, ,0 2 2,0 2 3,0 2 4, , ,1 2 4,1 + c 11 ( 1,0 2 ) 2 2,0 2 3,0 2 4, ,1 2 2,0 2 3,1 2 4, , ,0 2 4, , ,1 2 4,1 s = b 1 1, ,1 2 2,0 2 + b 2 1,0 1,1 3 2,0 3 + b 2 1,0 3 1,1 3 2,0 + b 3 1, , ,0 b 2 1,0 1, ,0 + b 4 1, ,0 b 4 1, ,0 b 3 1, ,0 2 + b 3 1, ,1 4 + a,i -homogeneous coordinates on P 1 P 1 P 1 P 1 b 3 1, ,0 2 + b 4 1, b 2 1,0 3 1,1 2,0 3 + b 4 4 1,1 4 Lara Anderson (Harvard) Heterotic Vector Bundles, Deformations and Geometric TransitionsString Math - July 18th, 12 8 / 14
9 Disconnected Loci Let s perform the analysis for the eample... L = O( 4, 4, 2, 2) on X 4,68 /Z 2 Z 4. Primary Decomposing and Eliminating gives 27 distinct loci in Comple Structure Moduli Space Branches to the solution space range in dimension from 7 to zero (i.e. from 3 to all of the comple structure moduli fied by V ). Having found isolated point-like solutions, we might think we can declare victory... But for the given values of C.S., we still have to check transversality of the CY: p = 0 = dp By Bertini s Theorem, a generic CICY is smooth, but once we are fied to very special points in CS, singularities are a real concern... Lara Anderson (Harvard) Heterotic Vector Bundles, Deformations and Geometric TransitionsString Math - July 18th, 12 9 / 14
10 Singularities in the CY Dimension in CS Dimension of Singularities in X 0 0 smooth For this eample, of the 27 branches to the solution space, all but one force the CY to be singular. The 4-dimensional locus given by c 10 c 11 = 0, c 8 c 11 = 0, c 7 c 11 = 0, c 4 + c 5 = 0, c 3 + c 6 = 0, c 2c 9 c 1c 11 = 0 leads to a smooth CY. Can we do anything with the singular solutions? Locally, for dim(sing) 1 we can imagine resolving (i.e. blowing-up) the singularities. Unfortunately, for the case at hand, all isolated point-like solutions are too badly singular. However, we can consider one of the larger loci... Lara Anderson (Harvard) Heterotic Vector Bundles, Deformations and Geometric TransitionsString Math - July 18th, / 14
11 Blowing up and Splitting CICYs Consider the 5-dimensional locus with pt-like singularities in CY Locally, we can resolve these singular pts. But have to worry about global issues: CY condition? What happens to the bundle? Symmetries? Happily, there are some resolutions of singular CYs that we have good control over: Conifold Transitions. CY defining poly takes the form p = f 1 f 3 f 2 f 4 = 0. Topologically a cone over S 3 S 2. Can be resolved by introducing new P 1 direction f1 f2 0 = 0 f 3 f 4 1 This resolved manifold can be described as a new CICY, called a split of the initial singular one. Can eplicitly track divisors, etension bundles, and symmetries through this geometric transition. Lara Anderson (Harvard) Heterotic Vector Bundles, Deformations and Geometric TransitionsString Math - July 18th, / 14
12 Consider the following split of the Tetraquadric: P 1 2 P 1 2 X = P 1 2 P 1 2 [p (2,2,2,2) f 1 (2,0,2,0) f 3 (0,2,0,2) f 2 (2,0,2,0) f 4 ] (0,2,0,2) = 0 X split = P P P P P Quotienting both sides by Z 2 Z 4, this split locus intersects the dim 5 CS locus above, leads to 8 singular pts on the CY. Resolving such pts, X X split. To check, can repeat the analysis independently with L = O(0, 2, 2, 1, 1) on X split. This time, 14 branches ranging from dimension 3 to 0. The resolution X gives a 2 dimensional locus in the CS space of a smooth X split. All CICYs connected by such transitions. Reid s Fantasy? Not yet dynamical transitions, but provides interesting web of stabilizing bundles on CYs... Lara Anderson (Harvard) Heterotic Vector Bundles, Deformations and Geometric TransitionsString Math - July 18th, / 14
13 Conclusions The presence of a holomorphic vector bundle constrains C.S. moduli can be used as a hidden sector mechanism for moduli stabilization The C.S. can be stabilized at the perturbative level without moving away from a CY manifold Avoids problems of naive KKLT scenarios in heterotic Allows us to keep geometric model-building toolkit Stabilized values fully determined for use in physical couplings The full moduli-dependent vacuum space can be determined using computational algebraic geometry Some singular solutions can be resolved Gives insight into connections between possible base manifold CYs. Lara Anderson (Harvard) Heterotic Vector Bundles, Deformations and Geometric TransitionsString Math - July 18th, / 14
14 The End Lara Anderson (Harvard) Heterotic Vector Bundles, Deformations and Geometric TransitionsString Math - July 18th, / 14
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