Stable Bundles for Heterotic String Models Bj¨ orn Andreas Department of Mathematics, University of Salamanca Philadelphia, 2008 B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 1 / 20

A standard model compactification of the E 8 × E 8 heterotic string requires: B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 2 / 20

A standard model compactification of the E 8 × E 8 heterotic string requires: a smooth Calabi-Yau threefold X which admits a free involution 1 τ X : X → X , B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 2 / 20

A standard model compactification of the E 8 × E 8 heterotic string requires: a smooth Calabi-Yau threefold X which admits a free involution 1 τ X : X → X , a stable τ X invariant rank 5 vector bundle V on X , 2 B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 2 / 20

A standard model compactification of the E 8 × E 8 heterotic string requires: a smooth Calabi-Yau threefold X which admits a free involution 1 τ X : X → X , a stable τ X invariant rank 5 vector bundle V on X , 2 c 1 ( V ) = 0, 3 B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 2 / 20

A standard model compactification of the E 8 × E 8 heterotic string requires: a smooth Calabi-Yau threefold X which admits a free involution 1 τ X : X → X , a stable τ X invariant rank 5 vector bundle V on X , 2 c 1 ( V ) = 0, 3 c 2 ( TX ) − c 2 ( V ) = [ W ] an effective curve class, 4 B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 2 / 20

A standard model compactification of the E 8 × E 8 heterotic string requires: a smooth Calabi-Yau threefold X which admits a free involution 1 τ X : X → X , a stable τ X invariant rank 5 vector bundle V on X , 2 c 1 ( V ) = 0, 3 c 2 ( TX ) − c 2 ( V ) = [ W ] an effective curve class, 4 c 3 ( V ) / 2 = 6. 5 B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 2 / 20

A standard model compactification of the E 8 × E 8 heterotic string requires: a smooth Calabi-Yau threefold X which admits a free involution 1 τ X : X → X , a stable τ X invariant rank 5 vector bundle V on X , 2 c 1 ( V ) = 0, 3 c 2 ( TX ) − c 2 ( V ) = [ W ] an effective curve class, 4 c 3 ( V ) / 2 = 6. 5 B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 2 / 20

A standard model compactification of the E 8 × E 8 heterotic string requires: a smooth Calabi-Yau threefold X which admits a free involution 1 τ X : X → X , a stable τ X invariant rank 5 vector bundle V on X , 2 c 1 ( V ) = 0, 3 c 2 ( TX ) − c 2 ( V ) = [ W ] an effective curve class, 4 c 3 ( V ) / 2 = 6. 5 → SU (5) Z 2 V E 8 → SU (3) C × SU (2) L × U (1) Y B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 2 / 20

A standard model compactification of the E 8 × E 8 heterotic string requires: a smooth Calabi-Yau threefold X which admits a free involution 1 τ X : X → X , a stable τ X invariant rank 5 vector bundle V on X , 2 c 1 ( V ) = 0, 3 c 2 ( TX ) − c 2 ( V ) = [ W ] an effective curve class, 4 c 3 ( V ) / 2 = 6. 5 → SU (5) Z 2 V E 8 → SU (3) C × SU (2) L × U (1) Y The search for pairs ( X , V ) which satisfy these conditions has inspired various constructions (cf. talks of V. Bouchard and M. H. Saito). Example X is given by Schoen’s Calabi-Yau threefold dP 9 × P 1 dP 9 and V is given by an extension of Z 2 -invariant spectral cover bundles (Donagi-Ovrut-Pantev-Waldram, 2000; Bouchard-Donagi, 2005). B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 2 / 20

Motivation Explore the ”mathematical landscape” of vector bundle constructions on CY threefolds. Search for perturbative heterotic models (GUT and SM), that is, look for solutions without five-branes. Search for CY threefolds which admit free acting groups. B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 3 / 20

I will report on joint work with G. Curio: math.AG/0611762, hep-th/0611762, 0611309, 0703210, 0706.1158 motivated by the search for solutions to the above conditions; and some recent work with D. Hern´ andez Ruip´ erez, D. S´ anchez G´ omez: math.AG/08022903 which studies the construction of stable vector bundles on K 3 fibered CY threefolds. Elliptic Fibrations with Two Sections 1 Spectral Cover Bundles 2 Pullback Bundles 3 Stable Bundle Extensions 4 Bundles on K 3 Fibrations 5 B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 4 / 20

Elliptic Fibrations with Two Sections Restricting to elliptic CY threefolds, we search for a free involution τ X which preserves the fibration structure and holomorphic threeform of X . If there is some involution τ X preserving the fibration structure (i.e., it sends fibers to fibers), then this must project to some (not necessarily free acting) involution in the base τ B : B → B . On a smooth elliptic curve there exists a translation symmetry acting free suggesting to choose an elliptically fibered CY threefold with a fiber translation symmetry. One way to do this is to require that X admits two sections. B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 5 / 20

Two possibilities to realize an elliptically fibered CY threefold with two sections have been investigated: use a different elliptic curve than the usually taken P 2 , 3 , 1 (6) (A.-Curio-Klemm, hep-th/9903052) ( x , y , z ) ∈ Elliptic Curve y 2 + x 3 + z 6 + λ xz 4 = 0 P 2 , 3 , 1 (6) y 2 + x 4 + z 4 + λ x 2 z 2 = 0 P 1 , 2 , 1 (4) x 3 + y 3 + z 3 + λ xyz = 0 P 1 , 1 , 1 (3) x 2 + y 2 + λ zw = 0 , z 2 + w 2 + λ xy = 0 P 3 (2 , 2) B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 6 / 20

Two possibilities to realize an elliptically fibered CY threefold with two sections have been investigated: use a different elliptic curve than the usually taken P 2 , 3 , 1 (6) (A.-Curio-Klemm, hep-th/9903052) ( x , y , z ) ∈ Elliptic Curve y 2 + x 3 + z 6 + λ xz 4 = 0 P 2 , 3 , 1 (6) y 2 + x 4 + z 4 + λ x 2 z 2 = 0 P 1 , 2 , 1 (4) x 3 + y 3 + z 3 + λ xyz = 0 P 1 , 1 , 1 (3) x 2 + y 2 + λ zw = 0 , z 2 + w 2 + λ xy = 0 P 3 (2 , 2) specialize the Weierstrass model to force a second section and resolve a curve of A 1 singularities that occur in this process. (Donagi et.al. hep-th/9912208) B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 6 / 20

Two possibilities to realize an elliptically fibered CY threefold with two sections have been investigated: use a different elliptic curve than the usually taken P 2 , 3 , 1 (6) (A.-Curio-Klemm, hep-th/9903052) ( x , y , z ) ∈ Elliptic Curve y 2 + x 3 + z 6 + λ xz 4 = 0 P 2 , 3 , 1 (6) y 2 + x 4 + z 4 + λ x 2 z 2 = 0 P 1 , 2 , 1 (4) x 3 + y 3 + z 3 + λ xyz = 0 P 1 , 1 , 1 (3) x 2 + y 2 + λ zw = 0 , z 2 + w 2 + λ xy = 0 P 3 (2 , 2) specialize the Weierstrass model to force a second section and resolve a curve of A 1 singularities that occur in this process. (Donagi et.al. hep-th/9912208) The possibility to consider elliptic fibrations with different elliptic curve representations has been also investigated in F-theory compactifications (c.f. Klemm-Mayr-Vafa, 1996). B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 6 / 20

Model (A.-Curio-Klemm, 1999) Let π : X → B be an elliptically fibered CY threefold with generic fiber described by P 1 , 2 , 1 (4) . X can be described by a generalized Weierstrass equation y 2 + x 4 + ax 2 z 2 + bxz 3 + cz 4 = 0 where x , y , z and a , b , c are sections of K − i with i resp. given by 1 , 2 , 0 B and 2 , 3 , 4 . ֒ → X admits two sections σ 1 , σ 2 . B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 7 / 20

Model (A.-Curio-Klemm, 1999) Let π : X → B be an elliptically fibered CY threefold with generic fiber described by P 1 , 2 , 1 (4) . X can be described by a generalized Weierstrass equation y 2 + x 4 + ax 2 z 2 + bxz 3 + cz 4 = 0 where x , y , z and a , b , c are sections of K − i with i resp. given by 1 , 2 , 0 B and 2 , 3 , 4 . ֒ → X admits two sections σ 1 , σ 2 . Assume B = F 0 and fix τ B ! To assure that τ B can be lifted to an involution τ X acting freely on X , we impose B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 7 / 20

Model (A.-Curio-Klemm, 1999) Let π : X → B be an elliptically fibered CY threefold with generic fiber described by P 1 , 2 , 1 (4) . X can be described by a generalized Weierstrass equation y 2 + x 4 + ax 2 z 2 + bxz 3 + cz 4 = 0 where x , y , z and a , b , c are sections of K − i with i resp. given by 1 , 2 , 0 B and 2 , 3 , 4 . ֒ → X admits two sections σ 1 , σ 2 . Assume B = F 0 and fix τ B ! To assure that τ B can be lifted to an involution τ X acting freely on X , we impose Conditions choose a , b , c invariant under τ B 1 { Fix ( τ B ) } ∩ { ∆ = 0 } = ∅ 2 B. Andreas (Salamanca) Stable Bundles for Heterotic String Models Philadelphia, 2008 7 / 20

Recommend

More recommend