Mirrored K3 automorphisms and non-geometric compactifications Dan Israe l, LPTHE New Frontiers in String Theory, Kyoto 2018 Based on: D.I. and Vincent Thie ry, arxiv:1310.4116 D.I., arxiv:1503.01552 Chris Hull, D.I. and Alessandra Sarti, arxiv:1710.00853
Introduction Non-geometric superstring compactifications Generic SUSY compactifications non-geometric? Non-geometric compactifications very few massless moduli Only sporadic classes known T-folds,... Three view-points on non-geometry Worldsheet : asymmetric 2d superconformal field theories Non-geometric symmetries (e.g. T-duality) quotient of geometric solutions Four-dimensional low-energy SUGRA gauging by non-geometric fluxes Scope of this presentation Asymmetric K3 T 2 Gepner models Mathematical understanding Mirrored K3 automorphisms Corresponding 4d gauged supergravities Dan Israël Mirrored K3 automorphisms 1 / 20
A simple example T 2 compactification ds 2 = ρ2 τ 2 dx 1 + τ dx 2 2, ρ 1 = B 12 Factorized moduli space: SL(2, R) SL(2, Z) τ U(1) }{{} complex structure τ SL(2, R) SL(2, Z) ρ U(1) }{{} Kähler ρ Exchanged by T-duality along x 1 Order 4 symmetry { x 1 x σ 4 : 2 x 2 x 1 Induced SL(2, Z) τ action on complex structure moduli space: τ 1/τ Fixed point τ = i square torus Generalized Scherk-Schwarz (monodrofolds) σ 4 monodromy twist: (x i, y) (σ 4 x i, y + 2πR) breaks all SUSY (Dabholkar, Hull) Double T-fold (x i l, y) ( x i l, y + 2πR) { τ 1/τ corresponds to, y y + 2πR ρ 1/ρ SUSY from right-movers only (Hellerman, Walcher) Dan Israël Mirrored K3 automorphisms 2 / 20
Asymmetric Landau-Ginzburg/Gepner orbifolds
Gepner models/lg orbifolds for K3 surfaces Landau-Ginzburg models N = (2, 2) QFTs in two dimensions, chiral multiplets Z l L = d 4 θ K(Z l, Z l ) + d 2 θ W (Z l ) + h.c. Quasi-homogeneous polynomial with an isolated critical point: W (λ w l Z l ) = λ d W (Z l ) Flows to a (2, 2) SCFT in the IR Some LG orbifold models for K3 surfaces Quantum non-linear sigma-model for a K3 surface with volume : LG model W = Z p1 1 + + Zp4 4 Quotient by j W : Z l e 2iπ/p l Z l, of order K = lcm (p 1,, p 4 ) ( GSO ) fields in twisted sectors γ = 0,..., K 1 IR fixed point: N = (2, 2) SCFT with c = c = 6 and Q R, Q R Z Gepner model: solvable (2, 2) SCFT Dan Israël Mirrored K3 automorphisms 3 / 20
K3 Gepner/Landau-Ginzurg orbifolds Some symmetries of Gepner models W = Z p1 1 + + Zp4 4 discrete symmetry group ( Z p1 Z p4 ) /JW Z l e 2iπr l p l Z l with l r l p l Z group SL W of SUSY-preserving symmetries Quantum symmetry of LG orbifold σ Q K : φ γ e 2iπγ/K φ γ, γ = 0,..., K 1 Orbifolds of Gepner models Supersymmetric orbifold of a K3 Gepner model by G SL W other point in K3 NLSM moduli space Quotient by σ pl, with σ pl : Z l e 2iπ/p l Z l for given l breaks all space-time SUSY Latter case: space-time SUSY can be partially restored using discrete torsion Dan Israël Mirrored K3 automorphisms 4 / 20
Previous works Asymmetric K3 Gepner models Asymmetric models from simple currents (Schellekens & Yankielowicz 90) LG orbifolds (Intriligator & Vafa 90) Asymmetric K3 T 2 models in type II (DI & Thiéry 13) Asym. CY 3 models from simple currents & fractional mirror sym. (DI 15) More general CY 3 models (Blumenhagen, Fuchs & Plauschinn 16) A simple class of asymmetric K3 LG orbifolds Quotient of W = Z p1 1 + + Zp4 4 by σ p1 : Z 1 e 2iπ/p1 Z 1 twisted sectors r = 0,..., p 1 1 Order p subgroup of the quantum sym. group generated by σp Q 1 := (σ Q ) K/p1 charge Q Q p 1 := γ p 1 Modified Z p1 orbifold charge: ˆQp1 = Q p1 + γ } p 1 mod 1 discrete torsion Modified Z k orbifold charge: ˆQK = Q K r p 1 mod 1 Orbifold theory: Q R Z but Q R / Z WS chiral space-time SUSY Dan Israël Mirrored K3 automorphisms 5 / 20
K3 fibrations with non-geometric monodromies Asymmetric K3 T 2 Gepner models in type IIA/IIB (DI, Thiéry) K3 Gepner model W = Z p 1 1 + Zp 2 2 + Zp 3 3 + Zp 4 4 times R 2 (x, y) in type IIA/B Freely-acting Z p1 Z p2 quotient { { Z1 e 2iπ/p1 Z 1 Z2 e 2iπ/p2 Z 2 x x + 2πR 1 y y + 2πR 2 Each quotient modified by the discrete torsion discussed before Dan Israël Mirrored K3 automorphisms 6 / 20
Main features Supersymmetry breaking All space-time supercharges from left-movers non-geometric Breaking N = 4 N = 2 in 4d, gravitini masses: M 2 = ρ2 τ 2 + (ρ1±1)2 ρ 2τ 2 SUSY-breaking can be achieved much below string scale can be analyzed reliably as spontaneous breaking in SUGRA Moduli space ρ and τ moduli of the T 2 and axio-dilaton S always massless For about 50% of the models: all K3 moduli become massive Low-energy 4d theory Axio-dilaton and torus moduli in vector multiplets N = 2 ST U SUGRA Surviving K3 moduli (if any): hypermultiplets N = 2 vacua of a gauged N = 4 supergravity? Dan Israël Mirrored K3 automorphisms 7 / 20
Roadmap N=4 Supergravity With 22 vector multiplets Low-energy limit Type IIA on K3 x T 2 Small-volume limit Worldsheet CFT: (K3 Gepner model x T 2 ) Gauge algebra Monodromy twists? Asymmetric orbifold Gauged N=4 SUGRA with N=2 vacua K3 fibration over T 2 ( "mirrorfold" ) Non-geometric compactification: Asymmetric Gepner model Dan Israël Mirrored K3 automorphisms 8 / 20
Non-linear sigma models on K3 and mirrored automorphisms
K3-surfaces K3 surfaces: elementary facts K3 surface X: Kähler 2-fold with a nowhere vanishing holomorphic 2-form Ω Hodge diamond: h 0,0 h 1,0 h 0,1 h 2,0 h 1,1 h 0,2 = h 2,1 h 1,2 h 2,2 1 0 0 1 20 1 0 0 1 Inner product: (α, β) H 2 (X, Z) H 2 (X, Z) α, β = α β Z H 2 (X, Z) isomorphic to unique even, unimodular lattice of signature (3, 19): Γ 3,19 = E8 E 8 U U U, U = Lattice of total cohomology H (X, Z): Moduli space of Ricci-flat metrics on K3 Γ 4,20 = E8 E 8 U U U U ( 0 1 1 0 Ricci-flat metric on X space-like oriented 3-plane Σ = (Re(Ω), Im(Ω), J) R 3,19 = H 2 (X, R), modulo large diffeomorphisms M ke = O(Γ3,19) \ O(3, 19) / O(3) O(19) R + ) Dan Israël Mirrored K3 automorphisms 9 / 20
String theory compactifications on K3 Non-linear sigma-models on K3 surfaces ( {g i j z φ i z φ j + z φ i φ j) ( z + b i j z φ i z φ j z φ i φ j)} z Σ d2 z φ(σ) b 22 real parameters Moduli space of NLSMs: Choice of metric & B-field choice of space-like oriented 4-plane Π R 4,20 M σ = O(Γ4,20) \ O(4, 20) / O(4) O(20) R+ (Seiberg, Aspinwall-Morrison) K3 surfaces hyper-kähler what does mirror symmetry mean? Dan Israël Mirrored K3 automorphisms 10 / 20
Picard and transcendental lattices Lattice-polarized K3 surfaces Picard lattice S(X) = H 2 (X, Z) H 1,1 (X) rank ρ(x) 1 for an algebraic surface, signature (1, ρ 1) Transcendental lattice T (X) = H 2 (X, Z) S(X), signature (2, 20 ρ) Moduli spaces of polarized K3 surfaces Lattice M of signature (1, r 1) with primitive embedding in S(X) M-polarized surface (X, M) Complex structure moduli space: M M = O(M ) \ O(2, 20 r) / O(2) O(20 r) Aspinwall-Morrison description of mirror symmetry Consider an algebraic K3 surface polarized by its Picard lattice S(X) Splitting of the NSLM moduli space (complex/kähler) ( ) ( O(T (X)) \ O(2, 20 ρ) / O(2) O(ρ) O(S(X) U) \ O(2, ρ) / ) O(2) O(20 ρ) R+ A-M mirror symmetry: exchange of the two factors Dan Israël Mirrored K3 automorphisms 11 / 20
Lattice-polarized vs. Berglund-Hübsch mirror symmetry Lattice-polarized mirror symmetry M-polarized surface (X, M) and M-polarized surface ( X, M) LP-mirror if Γ 3,19 M = U M (Dolgachev) Berglund-Hübsch mirror symmetry (physicists Greene-Plesser mirror) Hypersurface X W in weighted projective space P [wl ] : W = 4 4 a x i j j = 0 i=1 j=1 Group of SUSY symmetries SL W : {g l, W (e 2iπg l x l ) = W (x l ), g l Z} J W G SL W K3 surface (X W, G) (resolution of X W /(G/J W ) ) Berglund-Hübsch mirror surface: (X W T, G T ) with W T = 4 4 (a x T ) i j j and G T = {g G W T, g(a i j)h T Z, h G} i=1 j=1 Dan Israël Mirrored K3 automorphisms 12 / 20
Example Self-mirror K3 surface Fermat hypersurface w 2 + x 3 + y 7 + z 42 = 0 in P [21,14,6,1] Picard lattice: S(X) = E 8 U Transcendental lattice T (X) = E 8 U U S(X)-polarized surface is LP self-mirror! Dual group for a Fermat surface: G T = SL W In this particular example, SL W /J W = 1 This K3 surface is also its own Berglund-Hübsch mirror In this example, LP mirror of the S(X)-polarized surface BH mirror Is is true in general? no! Dan Israël Mirrored K3 automorphisms 13 / 20
Automorphisms of K3 surfaces Symplectic automorphisms of K3 surfaces A K3 automorphism σ is symplectic if it preserves the (2, 0) form: σ (Ω) = Ω Central role in the Mathieu moonshine Non-symplectic automorphisms Non-symplectic order p automorphism σ p : σp (Ω) = e 2iπ p Ω Invariant sublattice of Γ 3,19 : S(σ p ) S(X) Orthogonal complement T (σ p ) = S(σ p ) Γ 3,19 Ex: self-mirror K3 surface w 2 + x 3 + y 7 + z 42 = 0 σ 2 : w e iπ w σ 3 : x e 2iπ/3 x σ 7 : y e 2iπ/7 y All cases: S(σ p ) = S(X) T (σ p ) = T (X) Dan Israël Mirrored K3 automorphisms 14 / 20
Non-symplectic automorphisms and mirror symmetry Non-symplectic automorphisms of prime order p-cyclic algebraic K3 surface X W,G: W = w p + f(x, y, z) Berglund-Hübsch mirror X W T,G T : W T = w p + f( x, ỹ, z) σ p : w e 2iπ p w σ p : w e 2iπ p Theorem (Artebani et al., Comparin et al.): For prime p {2, 3, 5, 7, 13}, the S(σ p )-polarized surface X W,G and the S( σ p )-polarized surface X W T,G T are lattice-polarized mirrors. w Corollary: lattice decompositions { T (σp) := S(σ p) = U S( σ p) (Hull, DI, Sarti) T ( σ p) := S( σ p) = U S(σ p) T ( σ p ) is the orthogonal complement of T (σ p ) in Γ 4,20 : T ( σ p ) = T (σ p ) Γ 4,20. We obtain then the orthogonal decomposition over R (and Q): Γ 4,20 R ( ) = T ( σ p ) T (σ p ) R. Dan Israël Mirrored K3 automorphisms 15 / 20
Mirrored K3 automorphisms Abstract definition Let X W,G be a p-cyclic S(σ p )-polarized K3 surface, and X W T,G T mirror, regarded as an S( σ p )-polarized K3 surface. (Hull, DI, Sarti) its LP/BH By properties of K3 surfaces we can extend the diagonal action of (σ p, σ p ) from T (σ p ) T ( σ p ) to the whole lattice Γ 4,20. This defines an O(Γ 4,20 ) element associated with the action of a NLSM automorphism ˆσ p, that we name mirrored automorphism. Properties ˆσ p T (σp) R = σp, ˆσ p T ( σp) = ( σ R p ) Denoting by µ the BH/LP mirror involution, ˆσ p := µ σ p µ σ p Dan Israël Mirrored K3 automorphisms 16 / 20
Mirrored K3 automorphisms vs. asymmetric LG models BH mirror symmetry and quantum symmetry of LG models In the Gepner model construction we have used: 1 order p symmetry group of the superpotential Z e 2iπ/p Z 2 order p subgroup of the quantum sym. group generated by σ Q p := (σ Q ) K/p These symmetries are exchanged by BH mirror symmetry ( Q R Q R ) Non-geometric orbifolds from mirrored automorphisms K3 orbifold with discrete torsion projection Q p + Q Q p Z Corresponds to the diagonal action of (σ p, σ p ) Therefore, a K3 bundle over T 2 with mirrored K3 automorphisms twists give at the fixed points an asymmetric K3 T 2 Gepner model Dan Israël Mirrored K3 automorphisms 17 / 20
Non-geometric monodromies and gauged supergravity
Gaugings from twisted compactifications N = 4 SUGRA from K3 T 2 compactifications Type IIA/B on K3 T 2 N = 4 SUGRA with 22 vector multiplets Field content: SUGRA multiplet (g µν, ψ i µ, A 1,...,6 µ, χ i, τ) 22 vector multiplets (A a µ, λ a i, M) Scalars M, τ take value in the coset Rigid G = O(6, 22) SL(2) symmetry O(6,22) O(6) O(22) SL(2) O(2) Gaugings from Scherk-Schwarz reductions ( Dabholkar, Hull 02; Ried-Edwards, Spanjaard 08) Promote subgroup K G to gauge symmetry structure constants t MNP K3 T 2 with monodromy twists e Ni O(Γ 4,20 ) O(6, 22) structure constants tii J = NiI J Potential and SUSY breaking mass terms from t MNP Dan Israël Mirrored K3 automorphisms 18 / 20
Gauged SUGRA from mirrored K3 automorphisms Gauged SUGRA from mirrored automorphisms Pair of mirrored K3 automorphisms ˆσ pi twisted K3 T 2 compactification From the action of ˆσ pi on H (X, Z) matrices ˆM pi O(Γ 4,20 ) Provide structure constants of the corresponding N = 4 gauged SUGRA Vacua with spontaneous SUSY breaking N = 4 N = 2 Gravitini in (2, 1, 1) (1, 2, 1) of {SU(2) SU(2) = SO(4)} SO(20) Half-SUSY vacua from monodromies M pi {SU(2) SO(20)} O(Γ 4,20 ) Ordinary non-symplectic K3 automorphism: space-like rotation SO(2) SO(4) (as T (σ p ) signature (2, )) no SUSY Mirrored automorphism: SO(2) SO(2) SO(4) with same angle half-susy Dan Israël Mirrored K3 automorphisms 19 / 20
Conclusions
Non-geometric compactifications of superstring theory all likely the most generic ones yet poorly understood Large class of non-geometric compactifications based on Calabi-Yau rather than toroidal geometries first construction of mirrorfolds 1 Worldsheet CFT Analysis from 3 viewpoints: 2 Algebraic geometry 3 Gauged SUGRA Involve new classes of symmetries of CY sigma-models: mirrored CY automorphisms Some open questions (work in progress): 1 Fully explicit gauged SUGRA analysis 2 How do they fit into N = 2 heterotic/type II dualities in 4d 3 Relation with the Mathieu moonshine 4 Extension to CY 3-based constructions Dan Israël Mirrored K3 automorphisms 20 / 20