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G2 Mirror Symmetry ♦ Mirror symmetry for G2 manifolds ♦ based on • [1602.03521] • [1701.05202]+[1706.xxxxx] with Michele del Zotto (Stony Brook) 1 ♦ Strings, T-duality & Mirror Symmetry ♦ 2 Type II String Theories and T-duality Superstring theories on different backgrounds can give rise to equivalent physics: `string dualities' `T-duality': 1;8 1 1;8 1 IIA on R × S =∼ IIB on R × S −1 1 where rIIA = rIIB. It is crucial that strings can wind around S ! For type II strings on T 2: T-duality along one S1 swaps volume with complex structure. This can be discussed at various levels: • effective field theory • worldsheet CFT • full string theory ⊃ CFT • topological String Theory 3 String Theory on K3 Surfaces CFT's have an (intrinsically defined) `moduli space' = moduli space of background (metric plus B-field) on which string propagates The CFT of type IIA string theory on a K3 surface S has a moduli space 4;20 which is a Grassmanian of four-planes Σ4 ! Γ ⊗ R O(Γ4;20)nO(4; 20)=O(4) × O(20) `Geometric Interpretation': 4;20 3;19 2 0 4 Γ = Γ ⊕ Uv = H (S; Z) ⊕ H (S; Z) ⊕ H (S; Z) pick v0 2 Uv : ( !^i = !i − (!i · B) v Σ4 = ^ 2 2 B = B + v0 + v(!i − B ) !i · !j = δij The !i give the hyper K¨ahlerstructure of S and B is the two-form B-field [Aspinwall, Morrison] 4 Mirror Symmetries 4;20 3;19 2 0 4 Γ = Γ ⊕ Uv = H (S; Z) ⊕ H (S; Z) ⊕ H (S; Z) ( !^i = !i − (!i · B) v Σ4 = ^ 2 2 B = B + v0 + v(!i − B ) Isometries of Γ4;20 correspond to identical physics; this involves • Diffeomorphisms of S • Mirror Maps: Uv $ Uw Mirror maps can associate smooth with singular geometries ! Physics stays smooth: strings wrapped on vanishing P1s correspond to massive states (with mass ∼ B), just as for finite volume ! Mirror maps arise from two T-dualities along a sLag fibration [Strominger, Yau, Zaslow; Gross]! This is stronger than the equivalence at the level of the CFT and includes states originating from wrapped D-branes; note: we map IIA ! IIA here 5 Calabi-Yau threefolds On a suitably chosen pair of mirror Calabi-Yau threefolds X and X_, the worldsheet CFTs associated to IIA and IIB are isomorphic. The Hodge numbers must satisfy h1;1(X) = h2;1(X_) h2;1(X) = h1;1(X_) The CFT just sees the unordered set fh1;1(X); h2;1(X)g, but can't decide which one is which ! The exchange h1;1 $ h2;1is realized via an automorphism of the symmetry group of the CFT. This duality has amazing implications [Candelas, de la Ossa, Green, Parkes; ... ] Analyzing states from wrapped branes led to the conjecture of a sLag T 3 fibration for Calabi-Yau threefolds, mirror symmetry in the full string theory ∼ three T-dualities aling this T 3 fibre [Strominger, Yau, Zaslow]. 6 how to find X_ For some Calabi-Yau threefolds, the exact CFT is known at special point in moduli space, the `Gepner point', allowing to construct the mirror geometry [Greene, Plesser] Example: the quintic: 5 5 5 5 5 4 X : x1 + x2 + x3 + x4 + x5 = 0 in P _ 3 The mirror X is found as a (resolution) of a quotient of X by Z5 acting with weights (1; 0; 0; 0; 4) (0; 1; 0; 0; 4) (0; 0; 1; 0; 4) Indeed h1;1(X) = h2;1(X_) = 1 and h2;1(X) = h1;1(X_) = 101. 7 Batyrev Mirrors This has a beautiful generalization to toric hypersurfaces [Batyrev].A pair of lattice polytopes (in lattices M and N) satisfying h∆; ∆◦i ≥ −1 are called reflexive and determine a CY hypersurface as follows: • Via an appropriate triangulation, ∆◦ defines a faN Σ and a toric variety PΣ. ◦ • Each lattice point νi on ∆ except the origin gives rise to a homogeneous coordinate xi and a divisor Di. • Each lattice point m on ∆ gives a Monomial and the hypersurface equation is X Y hm;νii+1 X(∆;∆◦) : cm xi = 0 ◦ m2∆ νi2∆ 8 Batyrev Mirrors More abstract point of view: a polytope ∆ defines ◦ • a toric variety PΣn(∆) via its normal fan Σn(∆)=Σ f (∆ ) • a line bundle O(∆); ∆ is the Newton polytope of a generic section h∆; ∆◦i ≥ −1 Combinatorial formulas for Hodge numbers [Danilov,Khovanskii; Batyrev]: 1;1 ◦ X ∗ ◦[3] X ∗ [1] ∗ ◦[2] h (X(∆;∆◦)) = `(∆ ) − 5 − ` (Θ ) + ` (Θ )` (Θ ) Θ◦[3] Θ◦[2] 2;1 X ∗ [3] X ∗ [2] ∗ ◦[1] h (X(∆;∆◦)) = `(∆) − 5 − ` (Θ ) + ` (Θ )` (Θ ) Θ[3] Θ[2] 1;1 2;1 h (X(∆;∆◦)) = h (X(∆◦;∆)) 2;1 1;1 h (X(∆;∆◦)) = h (X(∆◦;∆)) _ X(∆;∆◦) = X(∆◦;∆) 9 Examples The Quintic 0 −1 0 0 0 1 1 0 −1 −1 −1 −1 4 1 B −1 0 0 1 0 C B −1 −1 −1 4 −1 C ◦ B C B C ∆ ∼ B C ∆ ∼ B C @ −1 0 1 0 0 A @ −1 −1 4 −1 −1 A −1 1 0 0 0 −1 4 −1 −1 −1 4 3 ◦ For the mirror, PΣn(∆ ) = PΣf (∆) is P =(Z5) as in [Greene, Plesser]! e.g. two-dimensional faces of ∆◦ look like this: Extra points ∼ refinement Σ ! Σf ∼ resolution of orbifold singularities Algebraic K3 Surfaces: T (S) = U ⊕ T~(S); mirror symmetry swaps N $ T~. This is realized by Batyrev's construction using 3D polytopes 10 Mirror Symmetry: the G2 Story We can put IIA or IIB string theory on a manifold of G2 holonomy to compactify to 10 − 7 = 3 dimensions. • The CFT can only detect b2 + b3 but cannot discriminate [Shatashvili,Vafa]. 4 • Arguments similar to SYZ imply coassociative T fibration for G2 manifolds. [Acharya] • Discussed in detail for (few) examples of Joyce [Shatashvili,Vafa; Acharya; Gaberdiel,Kaste] 1 • And G2 manifolds of the type CY × S =Z2 [Eguchi,Sugawara; Roiban, Romelsberger, Walcher; Pioline, Blumenhagen, V.Braun] 11 7 3 T =Z2 7 3 Consider T =Z2 with action [Joyce] α :(x1; x2; x3; x4; x5; x6; x7) 7! (−x1; −x2; −x3; −x4; x5; x6; x7) 1 β :(x1; x2; x3; x4; x5; x6; x7) 7! (−x1; 2 − x2; x3; x4; −x5; −x6; x7) 1 γ :(x1; x2; x3; x4; x5; x6; x7) 7! ( 2 − x1; x2; −x3; x4; −x5; x6; −x7) Different smoothings ∼ `discrete torsion' in the orbifold CFT [Joyce; Acharya; Gaberdiel,Kaste] give b2(Yl) = 8 + l b3(Yl) = 47 − l 12 7 3 T =Z2 Different smoothings ∼ `discrete torsion' in the orbifold CFT give b2(Yl) = 8 + l b3(Yl) = 47 − l Action of various 'mirror maps' ∼ T-dualities: Figure taken from [Gaberdiel, Kaste] 13 ♦ Twisted Connected Sums, Tops & Mirror Symmetry ♦ 14 Twisted Connected Sum (TCS) G2 Manifolds [Kovalev; Corti, Haskins, Nordstr¨om, Pacini] Can we find `mirror geometries' for a given TCS G2 manifold ? Is there an SYZ picture ? 15 TCS & SYZ S0+ S0− × × 1 1 X+ S S X− × − − − − × − − − − × − − − − × S1 S1 S1 S1 × × I I We can exploit the various SYZ fibrations to find a (coassociative) T 4 (at least in the Kovalev limit). Four T-dualities correspond to _ _ _ X+ ! X+ X− ! X− S0± ! S0± together with T-dualities along the various S1 factors. Can we give a construction and check b2 + b3 is invariant ? 16 Tops and Building Blocks The acyl Calabi-Yau manifolds are X± = Z±=S0±. Z± are called 'building blocks' [Corti, Haskins, Nordstr¨om,Pacini]. In particular, they are K3 fibred and satisfy c1(Z) = [S0] : Can think of X as 'half' a compact K3 fibred Calabi-Yau threefold. Such Calabi-Yau threefolds can be constructed from 4D reflexive ◦ ◦ ◦ polytopes ∆ with a 3D subpolytope ∆F = ∆ \ F cutting it into a pair ◦ ◦ of 'tops' ♦a; ♦a [Candelas, Font; Klemm, Lerche, Mayr; Hosono, Lian, Yau; Avram,Kreuzer,Mandelberg,Skarke] ◦ If πF (♦) ⊇ ∆F we call ♦ 'projecting'. This implies: X ◦ is fibred by X ◦ and it mirror X ◦ is fibred by (∆;∆ ) (∆F ;∆F ) (∆ ;∆) algebraic mirror family X ◦ of K3 surfaces. (∆F ;∆F ) 17 Tops and Building Blocks There are stable degeneration limits into K3 fibred threefolds X ◦ ! Z ◦ [ Z ◦ (∆;∆ ) (♦a;♦a) (♦b;♦b ) X ◦ ! Z ◦ [ Z ◦ : (∆ ;∆) (♦a;♦a) (♦b ;♦b) • Z ◦ and Z ◦ each capture half of the `twisting' in the K3 (♦a;♦a) (♦b;♦b ) fibration; Singular fibres of X (over pts pi) are distributed into two Q Q halfs such that µi = µi = 1. Z( ; ◦ ) Z( ; ◦) ♦a ♦a ♦b ♦b _ _ • X = Z ◦ =S and X = Z ◦ =S are an open mirror pair (at (♦;♦ ) 0 (♦ ;♦) 0 least in the SYZ sense); 18 Tops and Building Blocks This motivates: a pair of dual projecting tops is a pair of lattice polytopes which satisfy ◦ h♦; ♦ i ≥ −1 ◦ h♦; ν0i ≥ 0 hm0; ♦ i ≥ 0 ◦ with ν0 and m0 ? F , hm0; ν0i = −1 and πF (♦) ⊇ ∆ \ F . In fact, starting from , Z ◦ is constructed as a hypersurface ∼ O( ) in , ♦ (♦;♦ ) ♦ PΣ Σ ! Σn(♦) as in Batyrev's construction: X hν0;mi Y hνi;mi+1 Z ◦ : c x x = 0 with [x ] ∼ [S ] (♦;♦ ) m e i e 0 ◦ m2♦ νi2♦ This allows a combinatorial computation of Hodge numbers [AB]: 1;1 X X ∗ [2] X ∗ [1] ∗ [1] h = −4 + 1 + ` (σn(Θ )) + (` (Θ ) + 1)(` (σn(Θ ))) [3] [2] [1] Θ 2♦ Θ 2♦ Θ 2♦ 2;1 X ∗ [2] ∗ [2] X ∗ [3] h = `(♦) − `(∆F ) + ` (Θ ) · ` (σn(Θ )) − ` (Θ ) [2] [3] Θ <♦ Θ <♦ 19 comments Blowing up the intersection of two anticanonical divisors P = 0 and 0 ◦ P = 0 in a semi-Fano toric threefold PF with rays ∼ ∆F gives a threefold equation 0 1 z1P = z2P 2 P × PF ; ◦ which precisely corresponds to a `trivial' top, i.e. ♦ is the convex hull of ◦ (∆F ; 0) (0; 0; 0; 1) : Note: the normal fan of ♦, which is the convex hull of (∆F ; 0) (∆F ; −1) : 1 includes the ray (0; 0; 0; −1) giving P × PF as the ambient space.
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