Non-geometric Calabi-Yau backgrounds and heterotic/type II duality - PowerPoint PPT Presentation

Non-geometric Calabi-Yau backgrounds and heterotic/type II duality Dan Isra¨ el, Univ. Sorbonne GGI String Workshop, April 2019 ‹ Non-geometric Calabi-Yau Backgrounds and K3 automorphisms , Chris Hull, D.I., Alessandra Sarti, arXiv:1710.00853, JHEP 1711 (2017) 084 ‹ Heterotic/type II duality and non-geometric compactifications Yoan Gautier, Chris Hull and D.I., to appear

Introduction What are the generic (SUSY) string compactifications? ➥ One may expect that most are not of geometrical nature Non-geometric compactifications have few massless moduli Interesting underlying mathematics Only sporadic classes known ➥ T-folds,... Many view-points on non-geometry Worldsheet : asymmetric 2d CFTs Quotient of geometric solutions with stringy symmetries Generalized geometry 4d supergravity String dualities ... Dan Isra¨ el Non-geometric Calabi-Yaus GGI String Workshop, April 2019 1 / 28

‹ Motivations Genuine non-geometric string backgrounds apart from free-fields ? How to construct mirror-folds ? General N “ 2 vacua in 4d and string dualities Scope of this presentation Supersymmetric vacua from non-geometric Calabi-Yau automorphisms Mathematical framework: Mirrored K3 automorphisms String backgrounds: Asymmetric K 3 ˆ T 2 Gepner models New type of heterotic/type II duality Moduli spaces and quantum corrections Dan Isra¨ el Non-geometric Calabi-Yaus GGI String Workshop, April 2019 2 / 28

Non-geometric Calabi-Yau backgrounds

Generalized Scherk-Schwarz reductions (Dabholkar, Hull ’02) String theory on compact manifolds: moduli space of vacua M “ O p Γ qz G { H O p Γ q Ă G isometry group of a charge lattice Γ O p Γ q contains ”stringy” symmetries as T-dualities Those symmetries can appear in transition functions ➥ T-folds, U-folds,... Fibration over S 1 with (non-geometric) monodromy twist: 2 πR φ p x µ q , M “ e N P O p Γ q Ny φ p x µ , y q “ e M of finite order ➥ critical points with Minkowski vacuum Critical point corresponds to fixed points of M ➥ orbifold CFTs Dan Isra¨ el Non-geometric Calabi-Yaus GGI String Workshop, April 2019 3 / 28

A simple toroidal model T 2 compactification Order 4 automorphism d s 2 “ T 2 " x 1 ÞÑ ´ x 2 U 2 | d x 1 ` U d x 2 | 2 , T 1 “ B 12 σ 4 : x 2 ÞÑ x 1 Moduli space: Induced O p 2 , 2; Z q action: U ÞÑ ´ 1 { U SL p 2 , R q SL p 2 , R q ˆ SL p 2 , Z q ˆ U p 1 q SL p 2 , Z q ˆ U p 1 q Fixed point U “ i Ø square torus loooooooooomoooooooooon loooooooooomoooooooooon complex structure U K¨ ahler T Orbifold by x σ 4 y breaks all susy T-dual ò Supersymmetric T-fold reduction (Hellerman, Walcher ’06) Ñ M 3 Ñ S 1 with O p 2 , 2; Z q monodromy Fibration T 2 ã p x i l , x i r ; y q „ p´ x i l , x i r ; y ` 2 πR q " U ÞÑ ´ 1 { U ➥ Monodromy twist ÞÑ ´ 1 { T T Half- susy vacua with spacetime susy from right-movers Dan Isra¨ el Non-geometric Calabi-Yaus GGI String Workshop, April 2019 4 / 28

N “ 2 vacua from type IIA on K 3 ˆ T 2 (Hull,D.I., Sarti ’17) Ñ M 6 Ñ T 2 fibrations with monodromy twists Type IIA superstrings on K 3 ã Low-energy limit of type IIA on K 3 ˆ T 2 N “ 4 SUGRA in four dimensions Field content: SUGRA multiplet p g µν , ψ i µ , A 1 ,..., 6 , χ i , τ q µ 22 vector multiplets p A a µ , λ a i , M q O p 6 qˆ O p 22 q ˆ SL p 2 q O p 6 , 22 q Scalars M , τ take value in the coset O p 2 q Moduli space of K3 compactifications O p Γ 4 , 20 qz O p 4 , 20 q{ O p 4 q ˆ O p 20 q ➥ Consider monodromies M P O p Γ 4 , 20 q Ă O p 4 , 20 q Goal: N “ 4 Ñ N “ 2 spontaneous SUSY breaking Dan Isra¨ el Non-geometric Calabi-Yaus GGI String Workshop, April 2019 5 / 28

Gauged supergravity analysis ( Ried-Edwards, Spanjaard ’08, Horst, Louis, Smyth ’12) K 3 ˆ T 2 with monodromy twists M i “ e N i P O p Γ 4 , 20 q along T 2 J J ➥ structure constants t “ N of N “ 4 gauged supergravity iI iI Potential and SUSY breaking mass terms computed from t MNP Vacua with spontaneous SUSY breaking N “ 4 Ñ N “ 2 Gravitini transform in p 2 , 1 , 1 q ‘ p 1 , 2 , 1 q of t SU p 2 q ˆ SU p 2 q – SO p 4 qu ˆ SO p 20 q Ă O p 4 , 20 q Minkowski vacua from elliptic monodromies in t SO p 4 q ˆ SO p 20 qu X O p Γ 4 , 20 q Ă O p 4 , 20 q Half-SUSY vacua from monodromies in t SU p 2 q ˆ SO p 20 qu X O p Γ 4 , 20 q Ă O p Γ 20 q Ă O p 4 , 20 q Such solutions, if any, are necessarily non-geometric (as K 3 diffeos in O p 3 , 19 q Ă O p 4 , 20 q ) ➥ mirror-folds? Their construction relies on recent works on mirror symmetry of K 3 surfaces Dan Isra¨ el Non-geometric Calabi-Yaus GGI String Workshop, April 2019 6 / 28

Non-linear sigma models on K3 and mirrored automorphisms

K3 surfaces: elementary facts K3-surfaces K3 surface X : K¨ ahler 2-fold with a nowhere vanishing holomorphic 2-form Ω h 0 , 0 1 h 1 , 0 h 0 , 1 0 0 Hodge diamond: “ h 2 , 0 h 1 , 1 h 0 , 2 1 20 1 h 2 , 1 h 1 , 2 0 0 h 2 , 2 1 Inner product: p α, β q P H 2 p X, Z q ˆ H 2 p X, Z q ÞÑ x α, β y “ ş α ^ β P Z H 2 p X, Z q isomorphic to unique even, unimodular lattice of signature p 3 , 19 q : ˆ ˙ 0 1 Γ 3 , 19 – E 8 ‘ E 8 ‘ U ‘ U ‘ U , U “ 1 0 Lattice of total cohomology H ‹ p X, Z q : Γ 4 , 20 – E 8 ‘ E 8 ‘ U ‘ U ‘ U ‘ U Moduli space of Ricci-flat metrics on K3 Ricci-flat metric on X Ø space-like oriented 3-plane Σ “ p Ω , J q Ă R 3 , 19 – H 2 p X, R q , modulo large diffeos M ke – O p Γ 3 , 19 q z O p 3 , 19 q { O p 3 q ˆ O p 19 q ˆ R ` Dan Isra¨ el Non-geometric Calabi-Yaus GGI String Workshop, April 2019 7 / 28

String theory compactifications on K3 Non-linear sigma-models on K3 surfaces !  ˘)  ` B ¯  ´ B ¯ ş Σ d 2 z ` B z φ i B ¯ z φ ¯ z φ i B z φ ¯  ˘ ` B z φ i B ¯ z φ ¯ z φ i B z φ ¯ ` b i ¯ g i ¯   g Ricci-flat and d b “ 0 ➥ CFT ş φ p Σ q b ➥ 22 real parameters Moduli space of NLSMs Choice of metric & B-field Ø choice of space-like oriented 4-plane Π Ă R 4 , 20 M σ – O p Γ 4 , 20 qz O p 4 , 20 q { O p 4 q ˆ O p 20 q (Seiberg, Aspinwall-Morrison) O p Γ 4 , 20 q contains non-geometric symmetries as mirror symmetry K3 surfaces hyper-K¨ ahler ➥ what does mirror symmetry mean? ➥ how to define mirror-folds? Dan Isra¨ el Non-geometric Calabi-Yaus GGI String Workshop, April 2019 8 / 28

Lattice-polarized mirror symmetry Picard lattice S p X q “ H 2 p X, Z q X H 1 , 1 p X q Ă Γ 3 , 19 ➥ rank ρ p X q ě 1 for an algebraic surface, signature p 1 , ρ ´ 1 q Polarized K3 surfaces Lattice M of signature p 1 , r ´ 1 q with primitive embedding in S p X q ➥ M -polarized surface p X, M q Moduli space of complex structures compatible with polarization: M M – O p M K qz O p 2 , 20 ´ r q { O p 2 q ˆ O p 20 ´ r q Lattice-polarized mirror symmetry (Dolgachev, Nikulin) M -polarized surface p X, M q and ˜ M -polarized surface p ˜ X, ˜ M q LP-mirror if Γ 3 , 19 X M K “ U ‘ ˜ M Dan Isra¨ el Non-geometric Calabi-Yaus GGI String Workshop, April 2019 9 / 28

Greene-Plesser mirror symmetry Is lattice-polarized mirror symmetry related to ”physicist’s” mirror symmetry? Example of Greene-Plesser construction (Greene, Plesser ’90) Hypersurface w 2 ` x 3 ` y 8 ` z 24 “ 0 Ă P r 12 , 8 , 3 , 1 s Greene-Plesser mirror surface: quotient of the same hypersurface by the group G of supersymmetry-preserving automorphisms " w ÞÑ ´ w Here G » Z 2 generated by g : y ÞÑ ´ y More general case (non-Fermat): Berglund-H¨ ubsch (Berglund-H¨ ubsch ’91) The key point, to compare both notions, is the choice of lattice polarization Dan Isra¨ el Non-geometric Calabi-Yaus GGI String Workshop, April 2019 10 / 28

Automorphisms of K3 surfaces (Nikulin) 2 iπ Non-symplectic order p automorphism σ p : σ ‹ p Ω p p Ω q “ e Invariant sublattice of Γ 3 , 19 : S p σ p q Ď S p X q Orthogonal complement T p σ p q “ S p σ p q K X Γ 3 , 19 Previous example Hypersurface w 2 ` x 3 ` y 8 ` z 24 “ 0 Ă P r 12 , 8 , 3 , 1 s Order 3 automorphism σ 3 : x ÞÑ e 2 iπ { 3 x Sub-lattices S p σ 3 q – E 6 ‘ U and T p σ 3 q – E 8 ‘ A 2 ‘ U ‘ U Greene-Plesser mirror surface w 2 ` ˜ x 3 ` ˜ y 8 ` ˜ z 24 “ 0 Ă P r 12 , 8 , 3 , 1 s { Z 2 Orbifold ˜ x ÞÑ e 2 iπ { 3 ˜ Order 3 automorphism ˜ σ 3 : ˜ x Sub-lattices S p ˜ σ 3 q – E 8 ‘ A 2 ‘ U and T p σ 3 q – E 6 ‘ U ‘ U ➥ Lattice-polarized mirror symmetry relates the first surface polarized by S p σ 3 q to the second surface polarized by S p ˜ σ 3 q Dan Isra¨ el Non-geometric Calabi-Yaus GGI String Workshop, April 2019 11 / 28

The general story Non-symplectic automorphisms and mirror symmetry W “ w p ` f p x, y, z q 2 iπ p w p-cyclic K3 surface X : σ p : w ÞÑ e ö w p ` ˜ 2 iπ ubsch mirror ˜ X : ˜ p ˜ Berglund-H¨ W “ ˜ f p ˜ x, ˜ y, ˜ z q{ G σ p : ˜ ˜ w ÞÑ e w ö Theorem (Artebani et al., Comparin et al., Bott et al.): σ p q -polarized surface ˜ The S p σ p q -polarized surface X and the S p ˜ X are lattice-polarized mirrors. Corollary: lattice decomposition (Hull, DI, Sarti) T p ˜ σ p q is the orthogonal complement of T p σ p q in Γ 4 , 20 : σ p q – T p σ p q K X Γ 4 , 20 . T p ˜ Orthogonal decomposition over R (and over Q ): ´ ¯ Γ 4 , 20 b R – T p σ p q ‘ T p ˜ σ p q b R Dan Isra¨ el Non-geometric Calabi-Yaus GGI String Workshop, April 2019 12 / 28