Orbifold and Local Heterotic Flux Geometry Li-Sheng Tseng (with - PowerPoint PPT Presentation

Orbifold and Local Heterotic Flux Geometry Li-Sheng Tseng (with S.-T. Yau) Harvard University String Phenomenology 2008 University of Pennsylvania 1

Heterotic Models ** Compactifications: phenomenologically interesting, Natural gauge group and Standard Model fields (Works of Penn Math/Physics Group and many speak- ers in this conference.) ** Add Fluxes and Branes, geometry backreacts and becomes no longer Calabi-Yau. ** Lift scalar moduli [See M. Becker talk] 2

Outline I. Review heterotic SUSY constraint & mathematical motivation Background and References [See M. Becker talk] II. Orbifold solutions Geometric quotient of T 2 bundle over K 3 solution. III. Local non-compact solutions A heterotic model on ALE space: Eguchi-Hanson space T ∗ P 1 . Works with M. Becker and J.X. Fu, to appear. 3

I. Heterotic SUGRA N = 1 SUSY Costraints Physical fields: ( g, H 3 , φ, F 2 ) Geometry: ( X 6 , E ) [ M 3 , 1 × X 6 & gauge bundle] 1. SU (3) structure ( J, Ω) z ¯ z ¯ b dz a ∧ d ¯ b = ig a ¯ b dz a ∧ d ¯ b J = J a ¯ Ω 3 , 0 ← defines an almost complex structure Ω = 4 3 � Ω � 2 J ∧ J ∧ J � Ω � = e − 2 φ i Ω ∧ ¯ 2. Complex d Ω = 0 Hence, class c 1 ( X 6 ) = 0 holomorphically. 4

3. Balanced Metric (conformal) d ( � Ω � ∗ J ) = 0 4. Hermitian Yang-Mills (equivalent to the condition that E is a “stable” bundle, Li-Yau ’86) F 2 , 0 = F 0 , 2 = 0 , F ∧ J ∧ J = 0 ⇔ F mn J mn = 0 5. Anomaly Equation ( H = i (¯ ∂ − ∂ ) J ) ∂J = α ′ 2 i ∂ ¯ 4 (tr R ∧ R − tr F ∧ F ) 5

Balanced Manifold (Michelsohn ’82) K¨ ahler: dJ = 0 d ( ∗ J ) = 1 Balanced: 2 d ( J ∧ J ) = J ∧ dJ = 0 ** weaker, relaxation of K¨ ahler condition ** Preserved under smooth blowing down in 6D (Alessandrini-Bassanelli ’95). ** Preserved under conifold transitions (Fu-Li-Yau ’08). Important for the Reid conjecture connecting CY 3 . Examples : Iwasawa manifolds ( T 2 bundle over T 4 ), twistor spaces, connected sums of ( S 3 × S 3 ). 6

Anomaly Condition C 1 ( X 6 ) = 0 i Ricci-flat: c 1 ( X 6 ) = 2 π tr R a ¯ b = λJ with λ = 0. 1 1 i 4 π 2 ∂ ¯ Anomaly: p 1 ( X 6 ) = 8 π 2 tr R ∧ R = 8 π 2 tr F ∧ F + ∂J ** Cohomlogy class level, [ c 1 ( X 6 )] = 0 and and [ p 1 ( X 6 )] = [ p 1 ( E )]. ** Involves analysis of 4-form characteristic invariants ** Highly non-linear 7

Summary: Heterotic N = 1 Constraints (Strominger system) Let X 6 be a hermitian manifold with a stable gauge bundle E . Topologically, we require 1. C 1 ( X 6 ) = 0 2. P 1 ( X 6 ) = P 1 ( E ) 3. ∃ a positive (2 , 2) form on X 6 Solve, for ( J, Ω) on X 6 and F the curvature on E A. d Ω = 0 B. d ( � Ω � ∗ J ) = 0 C. F 2 , 0 = F 0 , 2 = 0 , F ∧ J ∧ J = 0 ∂J = α ′ D. 2 i ∂ ¯ 4 (tr R ∧ R − tr F ∧ F ) 8

� � � � II. Orbifolds of T 2 bundle over K 3 (FSY) solutions We start with the FSY solution � X 6 Geometry: T 2 E X X 6 � K 3 K 3 E K 3 Gauge bundle: E X stable bundle on X 6 lifted from K 3 J = e u J K 3 + i 2 θ ∧ ¯ ( J, Ω) : θ Ω = Ω K 3 ∧ θ where θ = dz + α = ( dx + α 1 ) + τ ( dy + α 2 ) is defined to be a (1 , 0)-form. � Ω � 2 = e − 2 u = e − 4 φ ω = dθ = ω 1 + τω 2 , 9

( ω, F, u ) fixed by the following requirements: 1. Complex: ω = dθ ∈ H 2 , 0 ( K 3 , Z ) ⊕ H 1 , 1 ( K 3 , Z ) 2. Conformally balanced: ω ∧ J K 3 = 0 3. F : the hermitian Yang-Mills curvature associated with the stable bundle on E X . 4. Anomaly condition: � K 3 ω ∧ ¯ ω + � K 3 tr F ∧ F = � K 3 tr R ∧ R = 24 Fu-Yau showed that there exists a solution to the non- linear system of differential equations. 10

Constructing new solutions by orbifolding FSY For T 2 bundle over K 3 geometries with a discrete sym- metry, we construct new solutions by quotienting the geometry, X 6 / Γ, where Γ is a finite group action. Require the discrete symmetry to leave invariant the physical fields g mn = J mr J rn H = d c J e − 4 φ = � Ω � 2 This is satisfied as long as J is invariant and Ω → ζ Ω where | ζ | = 1. If ζ � = 1, then the resulting orbifold solution breaks all supersymmetry. 11

Discrete symmetry action can have two components, one acting on the fiber and the other on the base. Let N be the order of the finite group. Separately, we have Fiber T 2 (1) shift ρ : z → z + c Nc = a + bτ ζ N = 1 (2) rotation ρ : z → ζz Base K 3: ρ : Ω 2 , 0 → Ω (1) symplectic ζ N = 1 ρ : Ω 2 , 0 → ζ Ω (2) non-symplectic Must be algebraic K 3 surfaces Classification: Nikulin ( Z 2 ); Artebani & Sarti, Taki ( Z 3 ) 12

Construct Solutions: (1) Start with K 3 surfaces with discrete symmetry ρ (2) The curvature twist ω 1 , ω 2 of T 2 sits in the lattice L of H 2 ( K 3 , Z ) such that (a) choose primitive ω 1 , ω 2 that transforms similarly to the action on the torus action such that (b) ω = ω 1 + τω 2 ∈ H 2 , 0 ( K 3 , Z ) ⊕ H 1 , 1 ( K 3 , Z ) (c) � K 3 ω ∧ ¯ ω = 24. Construct examples below. First consider torus action 1. Shift : z → z + c **no fixed points, always smooth **SUSY N=2,1,0 **Reduce size of the torus fiber along fixed points on the base 13

Example: K 3 as a triple cover of P 1 × P 1 branched over a curve K 3 as the interesection of two hypersurfaces. P 4 : [ z 0 , z 1 , z 2 , z 3 , z 4 ] f 1 = z 0 z 3 − z 1 z 2 (embed P 1 × P 1 in P 3 ) f 2 = g 3 ( z 0 , z 1 , z 2 , z 3 ) + z 3 4 Z 3 action: ρ ( z 0 , z 1 , z 2 , z 3 , z 4 ) = ( z 0 , z 1 , z 2 , z 3 , ζz 4 ) One fixed genus 4 curve at g 3 ( z 0 , z 1 , z 2 , z 3 ) = 0 ρ : Ω 2 , 0 → ζ Ω 2 , 0 with ζ 3 = 1. ω ∼ ω A − ω B invariant 14

� 2. Rotations: z → ζz **Since θ = dz + α is a global 1-form, action must be non-trivial on the base. ** Fixed locus set is non-empty. ***Generically, must resolve singularities (points and curves) to get a smooth manifold. For triple cover K3, can choose ω that transforms non- � X ′ T 2 trivially and obtain a SUSY solution 6 P 1 × P 1 but must resolve the singularities along the branched curve. 15

Example: K 3 surface with Z 3 action with only fixed points As before, intersection of a degree 2 and a degree 3 hypersurface in P 4 : [ z 0 , z 1 , z 2 , z 3 , z 4 ]. f 1 = f 2 ( z 0 , z 1 ) + b 1 z 2 z 3 + b 2 z 2 z 4 f 2 = f 3 ( z 0 , z 1 )+ b 3 z 3 2 + g 3 ( z 3 , z 4 )+ z 2 f 1 ( z 0 , z 1 ) g 1 ( z 3 , z 4 ) f 1 = z 2 0 + z 2 For example, 1 + z 2 ( z 3 + z 4 ) = 0 f 2 = z 3 1 + z 3 2 + z 3 3 − z 3 4 = 0 Z 3 action: ρ ( z 0 , z 1 , z 2 , z 3 , z 4 ) = ( ζ 2 z 0 , ζ 2 z 1 , ζz 2 , z 3 , ζz 4 ) 3 fixed points at ( z 0 , z 1 , z 2 ) = (0 , 0 , 0) and g 3 ( z 3 , z 4 ) = 0 16

ρ (Ω 2 , 0 ) = ζ 2 Ω hence, ρ ( θ ) = ζθ if we want to preserve SUSY Ω = Ω 2 , 0 ∧ θ . Take τ = e 2 πi/ 3 . ω 1 , ω 2 are in the N ⊥ ρ = U (1) ⊕ U (3) ⊕ A 5 2 ⊂ L, chosen such that ω = ω 1 + τω 2 ∈ H 1 , 1 ( K 3 , Z ) i.e. orthognal to Ω 2 , 0 and ¯ Ω 0 , 2 . Resolution: Blow up fixed points with boundary C 3 / Z 3 . 17

III. Local Model with Eguchi-Hanson base Metric ansatz: J = e u J CY 2 + i 2 θ ∧ ¯ θ Take the base CY 2 to be an ALE space. Simplest is the Eguchi-Hanson space: blow up of C 2 / Z 2 at the origin of the Z 2 action σ ( z 1 , z 2 ) = ( − z 1 , − z 2 ). Alternatively, B = O 1 P ( − 2) = T ∗ P 1 . There is a Ricci-flat metric ∂r 2 + k ′ ( r 2 ) ∂r 2 ∧ ¯ J EH = i 2 ( k ( r 2 ) ∂ ¯ ∂r 2 ) � 1 + a 4 r 4 and r 2 = | z 1 | 2 + | z 2 | 2 radius on C 2 . k = a is the size of the blow-up P 1 . 18

On EH, there is a single anti-self dual (1,1)-form. We can use this to twist the torus and as U (1) gauge fields. ∂r 2 + h ′ ( r 2 ) ∂r 2 ∧ ¯ ω ∼ i ( h ( r 2 ) ∂ ¯ ∂r 2 ) 1 where h ( r 2 ) = . � 1+ r 4 a 2 r 2 a 4 We need to satisfy the anomaly equation. Much simpli- fication due to dependence only on the radial coordinate for all quantities on C 2 / Z 2 . The differential equation can be written as 0 = dH − α ′ 4 [tr R ∧ R − tr F ∧ F ] A ( r 2 ) r 4 � ′ � = dz 1 ∧ d ¯ z 1 ∧ dz 2 ∧ d ¯ z 2 r 2 19

where n 2 � a 4 + α ′ ( | n | 2 + i 2 ) A ( r 2 ) = − u ′ e u a 2 1 + r 4 r 2 r 4 (1+ r 4 a 4 )     u ′ a 4 ) 2 + ( u ′ ) 2 + α ′ | n | 2 e − u 3 4 − α ′ a 4 ) 3 / 2 +  r 4 (1+ r 4  a 2 r 2 (1+ r 4 a 6 (1+ r 4   a 4 ) 5 / 2 = 0 where | n | 2 = n 2 1 + n 2 2 , n 1 , n 2 , and n ′ i are the first Chern number of the torus bundle and U (1) gauge bundle. We find a smooth solution for | n | 2 + n 2 i = 3, which 2 corresponds to matching characteristic classes on the EH base. 20

Convergent solution for α ′ /a 2 sufficiently small. e u = � ∞ a k k =0 k (1+ r 4 a 4 ) 2 � α ′ � α ′ � 2 � 3 ( | n | 2 +9 / 7) | n | 2 = 1 − α ′ 1 + a 4 ) 2 + + . . . a 2 3 a 2 (1+ r 4 a 2 7 (1+ r 4 (1+ r 4 a 4 ) 2 a 4 ) 2 Physical Implications **Solution has non-zero fractional H 3 charge, sourced by the twist of the T 2 and gauge fields. **Five-brane charge is generated when wrapped on twisted T 2 bundle. **Expect higher order in α ′ corrections of the differential equation and solution. 21

In Summary ** The study of heterotic torsional solutions are phe- nomenologically important and provides a good frame- work for investigating new mathematics. ** The space and structure of solutions is currently not well-understood, except for specific cases. ** Expect new exciting results in the future. 22