Calabi-Yau and Non-Calabi- Yau Backgrounds for Heterotic Phenomenology

Calabi-Yau and Non-Calabi- Yau Backgrounds for Heterotic Phenomenology James Gray, LMU, Munich with M. Larfors And D. Lüst: arxiv:1205.6208 with L. Anderson, A. Lukas and E. Palti: arxiv:1202.1757 arxiv:1106.4804

Heterotic Backgrounds: Consider compactification on a six manifold admiting an SU(3) structure. Torsion classes: dj = 3 2 Im(W 1Ω)+W 4 J + W 3 dω = W 1 J J + W 2 J + W 5 Ω SU(3) Holonomy: Calabi-Yau W i =0 i SU(3) Structure N =1vacuum: Strominger System W 1 = W 2 =0 W 4 = 1 2 W 5 = d ˆφ Lopes et al: hep-th/0211118 SU(3) Structure N =1/2 vacuum: Generalized half-flat W 1 = W 2 =0 W 4 = 1 2 W 5 = d ˆφ Lukas et al: hep-th/1005.5302

We will add extra fluxes to the analysis, and provide solutions for the supergravity fields. The setup: Fibration with manifold of SU(3) structure x u {x m } = {x u,y} Domain wall direction y 3D maximally symmetric space x α

Metric and associated field ansatzes ds 2 10 = e2a(xm ) ( ds 2 3 + e2 (xu) dydy + g uv (x m )dx u dx v) H αβγ = f αβγ H αmn = H αβn =0 α ˆφ =0 Three dimensional space is maximally symmetric. New fluxes: f and H yuv Gravitino variation in x α directions = A(x m ) = constant Define Θ = d The Killing spinor equations and Bianchi Identities become...

Consistency at fixed y J dj = J J d ˆφ, dω =2d ˆφ Ω e H y 1 2 fj J, 0= 1 2 f Ω + H 1 2 e H y J J, e d ˆφ = 1 2 H y Ω 1 2 e H J, Flow eqns dh =0, d( e 2 ˆφ H y )=0, df =0 J J = e dω + 1 2 e (H Ω )J J 2e d ˆφ Ω + e Ω + Θ Ω = e dj e (H Ω )Ω 2e d ˆφ J + e J Θ He fe Ω + ˆφ = 1 2 e (H Ω ) H = dh y, ( e 2 ˆφ H y ) = d (e 2 ˆφ+ H), f =0 reduces correctly to previous cases.

Rewrite fluxes and y derivatives Helps with solving equations in a construction independent manner H = A 1+ Ω + + A 1 Ω + A 2+ J + A 3+ H y = B 1 J + B 2 + B 3+. A 3+ Ω ± = 0 and write: such that A 3+ J = 0 B 2 J J = 0. J = γ 1 J + γ 2+ + γ 3 0 = γ 2+ J J = γ 3 J J. Ω = α 1+ Ω + + α 1 Ω + α 2+ J + α 3, Ω + = β 1+ Ω + + β 1 Ω + β 2+ J + β 3, 0 = Ω ± α 3 = J α 3, 0 = Ω ± β 3 = J β 3. The quantities α, β and γ can easily be found in any given example (see paper for many worked cases).

Solving consistency conditions: d ˆφ = W 4 H y = e ( f 2W 1 )J e W 2 + 1 2 e ((2W 4 W 5 )Ω +c.c) Also specifies some of the components of H Setting new fluxes to zero we recover the generalized half-flat conditions W 1 = W 2 =0 W 4 = 1 2 W 5 = d ˆφ In general all but one of these conditions is relaxed. Solving flow equations: H = 1 2 e ˆφ Ω + +( 7 8 + 3 2 W 1 )Ω + ((3W 4 2W 5+ ) J W 3 + e α 3 )

We also get equations for the flow itself. For example: γ 3 = e W 2+ and α 1+ = 3e W 1 15 8 e f The explicit expressions for H allow us to check the Bianchi Identities and form field equations of motion trivially in any case. The equations for the flow yield the y dependence of the parameters in the SU(3) structure when used with any explicit construction. Please see paper for egs: - CY with flux - Cosets - Toric varieties (SCTV s)

Split heterotic standard models Traditional heterotic model building. - Irreducible bundle: V Structure group: SU(n) E 8 SU(5) SU(5) SU(5) Split bundle model building. - Reducible bundle: Structure group: V = i V i S(U(n 1 )... U(n f )) E 8 SU(5) S(U(3) U(2)) SU(5) U(1) Extreme Case: - Sum of line bundles: V = i L i Structure group: S(U(1) n ) E 8 SU(5) S(U(1) 5 ) SU(5) U(1) 4

Conditions for solution: Holomorphic Anomaly cancelation: ch 2 (TX) ch 2 (V ) ch 2 (Ṽ )=[C] Some kind of stability criterion - Irreducible case: V stable and slope zero. - Reducible case: i V i poly-stable and slope zero. - Line bundle case: i L i all pieces slope zero. For Wilson line breaking to standard model we also need a freely acting symmetry which bundle respects. Equivariance

We have taken simplest case of line bundle sums and have scanned for standard models (no exotics) 2122 heterotic standard models Relationship to other backgrounds: Varying moduli and recombining bundles. with L. Anderson and B. Ovrut: arxiv:1012.3179 Reducible upstairs vs reducible downstairs. - Each line bundle individually equivariant: upstairs i L i ˆLi i downstairs - Only sum of line bundles equivariant: upstairs i L i ˆV downstairs Power of line bundles without it being abelian...

easy to see in some examples: Consider U = O(1) 5 on the quintic. Quotient the quintic by a freely acting symmetry (order 25). Z 5 Z 5 Ind(O(1)) = 5 : can t possibly be equivariant Ind(O(1) 5 ) = 25 : potential, and indeed is, equivariant (index divides) U descends to something irreducible on the quotient

Can be important for phenomenology: Consider a sum of line bundles: U = i L i Yukawa coupling can be thought of as: Fixes: 2 U = i<j L i L j H 1 (X, U) H 1 (X, U) H 1 (X, 2 U ) C 10 10 5 - This becomes: H 1 (X, L 1 ) H 1 (X, L 1 ) H 1 (X, 2 U ) C - Rank one Yukawas not possible at Abelian Locus Recombine line bundles (Froggatt-Nielsen) Work with a bundle which is not a sum on quotient

In the latter case: - Apparently no problem when working on quotient. - When working from an upstairs point of view (to use the computational power of having a split bundle): inv(h 1 (X, L 1 ) H 1 (X, L 2 )) inv(h 1 (X, L 1 ) H 1 (X, L 2 )) H 1 (X, L 1 L 2) H 3 (X, O) = C no problem obtaining rank one Yukawas

Summary SU(3) structure backgrounds: - Showed how to generalise the torsion classes giving rise to a good heterotic background. - Gave explicit solutions for supergravity fields: especially important for solving Bianchi Identities. SU(3) holonomy backgrounds: - Discussed some of the simplifications which can be achieved by using split bundles. - Discussed the relationship of these backgrounds to solutions in more general regimes of moduli space. - Gave an example where these relationships can be phenomenologically important.