Toric Resolution of Heterotic Orbifolds Stefan Groot Nibbelink - PowerPoint PPT Presentation

Toric Resolution of Heterotic Orbifolds Stefan Groot Nibbelink (Bielefeld / Heidelberg University) based on collaborations with Michele Trapletti (Orsay / Ecole Polytechnique, Paris) Tae-Won Ha, Denis Klevers, Hans-Peter Nilles, Felix Pl¨ oger, Patrick Vaudrevange, Martin Walter (Bonn University) JHEP 0703 (2007) 035 [hep-th/0701227] Phys.Lett.B652 (2007) 124 [hep-th/0703211] Phys.Rev.D77 (2008) 026002 [arXiv:0707.1597] JHEP 0804 (2008) 060 [arXiv:0802.2809] 1

Contents Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Contents Orbifold Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . 4 Calabi–Yau Phenomenology . . . . . . . . . . . . . . . . . . . . . . 7 Blowup of orbifold singularities . . . . . . . . . . . . . . . . . . . . . . . . 10 Toric resolution of C 3 / Z 3 . . . . . . . . . . . . . . . . . . . . . . . . 12 Multiple anomalous U(1)s? . . . . . . . . . . . . . . . . . . . . . . . 16 Blowup of MSSM Z 3 model . . . . . . . . . . . . . . . . . . . . . . . 17 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2

One of the aims of String Phenomenology is to find the Standard Model of Particle Physics from String constructions. We focus on heterotic String compactifications that could lead to the Supersymmetric Motivation Standard Model (i.e. the MSSM). Two approaches are most often considered to achieve this goal: • orbifold constructions; • smooth Calabi–Yau compactifications with gauge bundles. Both approaches have their advantages and disavantages. 3

Orbifolds can be chosen such that we get N = 1 supersymmetry in 4D:  Z i ∼ Z i + R i , Translations : Orbifold Phenomenology  T 6 / Z 3 : ( Z 1 , Z 2 , Z 3 ) ∼ e 2 πi/ 3 ( Z 1 , Z 2 , Z 3 ) . Rotations :  Orbifolds look like pillows: T 2 / Z 3 : T 2 / Z 2 : Orbifolds are flat spaces except for the orbifold fixed points, where there are curvature singularities. It is precisely these singularities that lead to the breaking of supersymme- try and allow for the possibility of obtaining a chiral spectrum. 4

A extensive classification of heterotic E 8 × E 8 models with gauge shift v only (i.e. without Wilson lines) can be found in Katsuki,Kawamura,Kobayashi,Ohtsubo’89 . Classifications of heterotic SO(32) models without Wilson lines was considered much later Giedt’03 , Orbifold Phenomenology Choi,SGN,Trapletti’04 , Nilles,Ramos-Sanchez,Vaudrevange,Wingerter’06 One can generate a “landscape” of models and search for MSSM–like models. Promis- ing candidates have been found in this way: • A few Z 3 orbifold models with two Wilson lines; Ibanez,Mas,Nilles,Quevedo’88 , Casas,Mondragon,Munoz’89 • A few hundred Z 6 -II models with two Wilson lines. Kobayashi,Raby,Zhang’04 , Buchmuller,Hamaguchi,Lebedev,Ratz’04 Lebedev,Nilles,Raby,Ratz,Ramos-Sanchez,Vaudrevange,Wingerter’06 These models look like orbifold GUTs from a 6D perspective. Moreover, all exotic states that are charged under the Standard Model group can be decoupled by Higgs mechanisms. • Also Z 12 -I models have been constructed. Kim 2 ,Kyae’07 5

Heterotic strings on orbifolds have as advantages: • they described by free CFTs, Orbifold Phenomenology • allow for a systematic classification, • full spectrum (not only massless states) computable, • even interactions can (in principle) be calculated completely, • give rise to a large pool of possible MSSMs. Their disadvantages are: • singular spaces with curvature singularities, • field theory on them ill–defined, • perturbative so only ’small’ VEVs allowed, • when a anomalous U(1) is present, some VEVs need to be large, • they define a special corner in full moduli space. 6

Calabi–Yau manifolds can be constructed in the following ways: Calabi–Yau Phenomenology • as the bundle M → B with an elliptically fibered torus over the basis B , • as Complete Intersections CY, i.e. hypersurfaces in projective spaces. Physically acceptable gauge backgrounds, i.e. stable bundles, can then be obtained • as spectral covers over elliptically fibered CYs, Friedman,Morgan,Witten’97,Donagi’97 Donagi,Lukas,Ovrut,Waldram’99 . • as monad constructions, Blumenhagen,Schimmrigk,Wisskirchen’96,Anderson,He,Lukas’07 • using the method of extensions. Donagi,Ovrut,Pantev,Waldram’00,Andreas,Curio’06 Using such constructions MSSM–like models have been constructed. Braun,He,Ovrut,Pantev’05,Bouchard,Donagi’07 7

Heterotic supergravities on CYs with bundles have as advantages: Calabi–Yau Phenomenology • generic point in moduli space, • chiral spectrum determined by topological data, • fixing of some (K¨ ahler) moduli, • some MSSMs have been constructed in this way. Their disadvantages are: • construction CYs is difficult, • classification of their gauge bundles is complicated, • SUGRA approximation only; not full string theory 8

Connecting orbifolds with smooth CYs The main motivation of our work is to connect both approaches, i.e. taking advantages of their positive points, and bypassing their problems. To execute this program we proceed as follows: • we start from an orbifold point in the string moduli space, i.e. first focus on a single fixed point, • describe the blowup using toric geometry or an explicit construction, • classify the possible (Abelian) gauge backgrounds, • investigate the possible VEVs of twisted states are F– and D–flat. 9

For an introduction to toric geometry see the textbooks: Fulton , Oda , Hori et al.: Mirror symmetry . Discussion of orbifold resolutions using toric geometry can be found in e.g. Erler,Klemm’92 , Lust,Reffert,Scheidegger,Stieberger’06 . Blowup of orbifold singularities The basic idea of using toric resolutions of an orbifold singularity is to replace the Z n orbifold action Z 3 ) → ( e 2 πiφ 1 ˜ Z 1 , e 2 πiφ 2 ˜ Z 2 , e 2 πiφ 3 ˜ θ : ( ˜ Z 1 , ˜ Z 2 , ˜ Z 3 ) by one or more C ∗ = C − 0 complex scaling(s) ( z 1 , z 2 , z 3 ; x 1 , . . . ) ∼ ( λ p 1 z 1 , λ p 2 z 2 , λ p 3 z 3 ; λ q 1 x 1 , . . . ) with λ ∈ C ∗ and p 1 , p 2 , p 3 ; q 1 , . . . some integer “charges” defining the scaling. The additional homogeneous coordinate(s) x 1 , . . . is introduced to keep the dimensionality the same as that of the orbifold. The orbifold action is recovered from the C ∗ scaling(s) by assuming that all additional coordinates are non–zero and are scaled to unity. The required scaling(s) not uniquely defined, and gives precisely ambiguities of Z n phases back. 10

To be able to do heterotic model building on such a resolution we need to investigate the Hermitean Yang–Mills equations Blowup of orbifold singularities � J F V = 0 , where J is a K¨ ahler class and F V an Abelian gauge background. (Loop corrections can be taken into account. Blummenhagen,Honnecker,Weigand’05 ) If the Bianchi identities are satisfied, the spectra can be computed using index theorems: Witten’84 � � � 1 V − 1 � ( tr R 2 − tr F 2 6 F 3 48 tr R 2 F V V ) = 0 , N V = C We need a toric representation of • the gauge background F V → exceptional divisors, • the curvature R → splitting principle, • and their integrals → intersection numbers. 11

The heterotic string on the orbifold C 3 / Z 3 has only one twisted sector, hence we only need one extra coordinate x defining the single exception divisor E . The local coordi- nates Lust,Reffert,Scheidegger,Stieberger’06,SGN,Ha,Trapletti’07 1 1 1 3 , 3 , 3 , Z 1 = z 1 x Z 2 = z 2 x Z 3 = z 3 x Toric resolution of C 3 / Z 3 define the C ∗ scaling C ∗ : ( z 1 , . . . , z 3 , x ) ∼ ( λ − 1 z 1 , . . . , λ − 1 z 3 , λ 3 x ) , and the linear equivalence relations of the divisors: � � D i = z i = 0 , D i ∼ D j , � � 3 D i + E ∼ 0 , E θ = x θ = 0 , From the toric diagram we infer the basic integrals and intersections: � D 1 D 2 E = D 1 · D 2 · E = 1 D 3 � D 2 D 3 E = D 2 · D 3 · E = 1 � E D 3 D 1 E = D 3 · D 1 · E = 1 D 1 D 2 12

The gauge background F is expanded as SGN,Ha,Trapletti’07 F V = − 1 3 E V I H I , and the splitting principle gives Toric resolution of C 3 / Z 3 − 1 2 tr R 2 = c 2 ( R ) = D 1 D 2 + ( D 1 + D 2 ) E. The Bianchi identity on compact cycle E gives the results: � � V 2 = tr ( i F V ) 2 = tr R 2 = 12 , E E and the spectrum is determined by the multiplicity operator SGN,Trapletti,Walter’07 N V = 1 − 1 � � H V = V I H I , 3 H 2 V + 1 H V , 6 which can take the values: N V = 1 9 , 1 , 26 9 = 3 − 1 9 . The multiplicity factors 1 3 9 = 27 refer to untwisted (delocalized) states, while integral multiplicity factors correspond to states localized at the orbifold fixed point. Gmeiner,SGN,Nilles,Olechowski,Walter’03 13