Mirror symmetry for G 2 manifolds

Mirror symmetry for G 2 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 : IIA on R 1,8 S 1 = IIB on R 1,8 S 1 where r IIA = r 1 IIB. It is crucial that strings can wind around S1! For type II strings on T 2 : T-duality along one S 1 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 which is a Grassmanian of four-planes Σ 4 Γ 4,20 R Geometric Interpretation : O(Γ 4,20 )\O(4, 20)/O(4) O(20) Γ 4,20 = Γ 3,19 U v = H 2 (S, Z) H 0 (S, Z) H 4 (S, Z) pick v 0 U v : {ˆωi = ω i (ω i B) v Σ 4 = ˆB = B + v 0 + v(ωi 2 B2 ) ω i ω j = δ ij The ω i give the hyper Kähler structure of S and B is the two-form B-field [Aspinwall, Morrison] 4

Mirror Symmetries Γ 4,20 = Γ 3,19 U v = H 2 (S, Z) H 0 (S, Z) H 4 (S, Z) {ˆωi = ω i (ω i B) v Σ 4 = ˆB = B + v 0 + v(ωi 2 B2 ) Isometries of Γ 4,20 correspond to identical physics; this involves Diffeomorphisms of S Mirror Maps: U v U w Mirror maps can associate smooth with singular geometries! Physics stays smooth: strings wrapped on vanishing P 1 s 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 h 1,1 (X) = h 2,1 (X ) h 2,1 (X) = h 1,1 (X ) The CFT just sees the unordered set {h 1,1 (X), h 2,1 (X)}, but can t decide which one is which! The exchange h 1,1 h 2,1 is 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: X : x 5 1 + x 5 2 + x 5 3 + x 5 4 + x 5 5 = 0 in P 4 The mirror X is found as a (resolution) of a quotient of X by Z 3 5 acting with weights (1, 0, 0, 0, 4) (0, 1, 0, 0, 4) (0, 0, 1, 0, 4) Indeed h 1,1 (X) = h 2,1 (X ) = 1 and h 2,1 (X) = h 1,1 (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, 1 are called reflexive and determine a CY hypersurface as follows: Via an appropriate triangulation, defines a fan Σ and a toric variety P Σ. Each lattice point ν i on except the origin gives rise to a homogeneous coordinate x i and a divisor D i. Each lattice point m on gives a Monomial and the hypersurface equation is m,ν X (, ) : i +1 = 0 c m x i m ν i 8

Batyrev Mirrors More abstract point of view: a polytope defines a toric variety P Σn( ) via its normal fan Σ n ( ) = Σ f ( ) a line bundle O( ); is the Newton polytope of a generic section, 1 Combinatorial formulas for Hodge numbers [Danilov,Khovanskii; Batyrev]: h 1,1 (X (, )) = l( ) 5 Θ [3] h 2,1 (X (, )) = l( ) 5 Θ [3] l (Θ [3] ) + Θ [2] l (Θ [3] ) + Θ [2] l (Θ [1] )l (Θ [2] ) l (Θ [2] )l (Θ [1] ) h 1,1 (X (, )) = h 2,1 (X (, )) h 2,1 (X (, )) = h 1,1 (X (, )) X (, ) = X (, ) 9

Examples The Quintic 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 1 1 4 1 1 1 4 1 1 1 4 1 1 1 4 1 1 1 For the mirror, P Σn( ) = P Σf ( ) is P 4 /(Z 5 ) 3 as in [Greene, Plesser]! e.g. two-dimensional faces of look like this: 01 00 11 00 11 01 0 00 11 00 00 11 00 11 Extra points refinement Σ Σ f 1 11 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 G 2 Story We can put IIA or IIB string theory on a manifold of G 2 holonomy to compactify to 10 7 = 3 dimensions. The CFT can only detect b 2 + b 3 but cannot discriminate [Shatashvili,Vafa]. Arguments similar to SYZ imply coassociative T 4 fibration for G 2 manifolds. [Acharya] Discussed in detail for (few) examples of Joyce [Shatashvili,Vafa; Acharya; Gaberdiel,Kaste] And G 2 manifolds of the type [ CY S 1] /Z 2 [Eguchi,Sugawara; Roiban, Romelsberger, Walcher; Pioline, Blumenhagen, V.Braun] 11

T 7 /Z 3 2 Consider T 7 /Z 3 2 with action [Joyce] α : (x 1, x 2, x 3, x 4, x 5, x 6, x 7 ) ( x 1, x 2, x 3, x 4, x 5, x 6, x 7 ) β : (x 1, x 2, x 3, x 4, x 5, x 6, x 7 ) ( x 1, 1 2 x 2, x 3, x 4, x 5, x 6, x 7 ) γ : (x 1, x 2, x 3, x 4, x 5, x 6, x 7 ) ( 1 2 x 1, x 2, x 3, x 4, x 5, x 6, x 7 ) Different smoothings discrete torsion in the orbifold CFT [Joyce; Acharya; Gaberdiel,Kaste] give b 2 (Y l ) = 8 + l b 3 (Y l ) = 47 l 12

T 7 /Z 3 2 Different smoothings discrete torsion in the orbifold CFT give b 2 (Y l ) = 8 + l b 3 (Y l ) = 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) G 2 Manifolds [Kovalev; Corti, Haskins, Nordström, Pacini] Can we find mirror geometries for a given TCS G 2 manifold? Is there an SYZ picture? 15

TCS & SYZ X + S 1 S 0+ S 1 S 1 I S 0 S 1 S 1 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 S 0± S 0± X S 1 together with T-dualities along the various S 1 factors. Can we give a construction and check b 2 + b 3 is invariant? 16

Tops and Building Blocks The acyl Calabi-Yau manifolds are X ± = Z ± /S 0±. Z ± are called building blocks [Corti, Haskins, Nordström,Pacini]. In particular, they are K3 fibred and satisfy c 1 (Z) = [S 0 ]. 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 π F ( ) F we call projecting. This implies: X (, ) is fibred by X ( F, F ) and it mirror X (, ) is fibred by algebraic mirror family X ( F, F ) of K3 surfaces. 17

Tops and Building Blocks There are stable degeneration limits into K3 fibred threefolds X (, ) Z ( a, a ) Z ( b, b ) X (, ) Z ( a, a) Z ( b, b ). Z ( a, a ) and Z ( b, b ) each capture half of the twisting in the K3 fibration; Singular fibres of X (over pts p i ) are distributed into two halfs such that Z ( a, a ) µ i = Z ( b, b ) µ i = 1. X = Z (, )/S 0 and X = Z (, )/S0 least in the SYZ sense); are an open mirror pair (at 18

Tops and Building Blocks This motivates: a pair of dual projecting tops is a pair of lattice polytopes which satisfy, 1, ν 0 0 m 0, 0 with ν 0 and m 0 F, m 0, ν 0 = 1 and π F ( ) F. In fact, starting from, Z (, ) is constructed as a hypersurface O( ) in P Σ, Σ Σ n ( ) as in Batyrev s construction: Z (, ) : m c m x ν0,m e x νi,m +1 i = 0 with [x e ] [S 0 ] ν i This allows a combinatorial computation of Hodge numbers [AB]: h 1,1 = 4 + 1 + l (σ n (Θ [2] )) + (l (Θ [1] ) + 1)(l (σ n (Θ [1] ))) Θ [3] h 2,1 = l( ) l( F ) + Θ [2] Θ [2] < Θ [1] l (Θ [2] ) l (σ n (Θ [2] )) Θ [3] < l (Θ [3] ) 19

comments Blowing up the intersection of two anticanonical divisors P = 0 and P = 0 in a semi-fano toric threefold P F with rays F gives a threefold equation z 1 P = z 2 P P 1 P F, 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). includes the ray (0, 0, 0, 1) giving P 1 P F as the ambient space. 20

comments Strenght of using polytopes is in resolving and analysing situations which degenerate K3 fibres. Can have large K = ker(h 2 (Z, Z) H 2 (S, Z))/[S] giving large b 2 (J) for resulting TCS G 2 manifolds. Reducible K3 fibres can easily be found from [Davis et al; AB,Watari]; similar to theory by [Kulikov], but can have multiplicities > 1 for fibre components. Caveat: for arbitrary F, need to make sure moduli space of algebraic K3 surfaces is large enough to tune in order to find gluings. Guaranteed at least for 1009 semi-fano out of 4319 weak Fano options [Corti,Haskins,Nordström,Pacini]. 21

Mirror Building Blocks Inverting the roles of and gives us mirror building blocks Z and Z with h 2,1 (Z) = K(Z ) where K = ker [ H 2 (Z, Z) H 2 (S 0, Z) ] /[S]. Recall that (for orthogonal gluings): b 2 + b 3 = 23 + 2 [ h 2,1 (Z + ) + h 2,1 (Z ) ] + 2 [ K(Z + ) + K(Z ) ]. We are in business! 22

G2 Mirrors In fact, we can do a lot better: From our discussion of SYZ, we should trade both building blocks for their mirrors and use the mirrors of S 0±. Hence: for a G 2 manifold J constructed from X + = Z ( +, + ) /S + and X = Z (, ) /S the mirror J is found using X+ = Z ( +, +)/S+ and X = Z (, /S The hyper Kähler rotation used to glue S0± is found from that for S 0±. If T + T U: for any matching. H 2 (J, Z) H 4 (J, Z) = H 2 (J, Z) H 4 (J, Z) H 3 (J, Z) H 5 (J, Z) = H 3 (J, Z) H 5 (J, Z) 23

G2 Mirrors The asymptotic K3 fibres S 0± of Z ± are mapped to S 0±. This exchanges N T where N = im(ρ) and T = U T = N Γ 3,19 Depending on the choice of ω and Ω, S 0± can have ADE singularities (so TCS construction does not apply... yet? T 7 /Γ gives singular examples of this type!) In physics, we should include a B-field B ± in N + N ; matching condition becomes ω + = Re(Ω ) Im(Ω + ) = Im(Ω ) ω = Re(Ω + ) B + = B preserved by mirror map on K3 surfaces with SYZ fibre calibrated by Im(Ω + ) and Im(Ω ). Singularities come from 2 curves in N + N which will receive a stringy volume from B. 24

example Consider two identical K3 fibred building blocks Z ± = Z with N + = N = U ( E 8 ) h 1,1 (Z) = 11 h 2,1 (Z) = 240 N(Z) = 10 K(Z) = 0 h 1,1 (Z ) = 251 h 2,1 (Z +) = 0 N(Z ) = 10 K(Z ) = 240 We can glue such that N + N = 0 = T + T to find smooth mirrors with b 2 + b 3 = 983 b 2 (J) = 0 b 2 (J ) = 480 b 3 (J) = 983 b 3 (J ) = 503 However, perpendicularly gluing to building blocks with quartic K3 fibre N ± = (4) gives N + N ( E 8 ) 2 25

Comparing to an example of Joyce Consider again the smoothings Y l of the orbifold T 7 /Z 3 2. They can be realized as a TCS [Nordström, Kovalev] with building blocks ((T 4 /Z 2 ) S 1 R + )/Z 2 We can now compare our construction to the mirror maps I ± 3 and I± 4 of [Gaberdiel,Kaste]. Our mirror map is I 4 + Z + Z. : Y l Y l which is trivial in this case and acts as As an aside : the discrete torsion in the CFT description precisely corresponds to H 3 (Y l, Z) = Z 8 l 2. 26

The Curious Case of three T-dualities What about other mirror maps in TCS picture? I 3 : Y l Y 8 l corresponds to performing three T-dualities along T 3 SYZ fibre of Z + and elliptic fibre of Z! X + S 1 S 0+ S 1 S 1 I S 0 S 1 S 1 I X S 1 Hence: for a TCS J built from Z + and Z, we construct J 3 from Z + and Z. If T + N U: H (J, Z) = H (J 3, Z) For orthogonal gluings b 2 + b 3 trivially invariant as b 2 + b 3 = 23 + 2 [ h 2,1 (Z + ) + h 2,1 (Z ) ] + 2 [ K(Z + ) + K(Z ) ]. 27

Summary We have motivated our construction by a physics picture of SYZ fibrations and their generalization to G 2 as proposed by [Acharya], exploiting the structure of TCS G 2 manifolds in the Kovalev limit. Our construction stands on its own and gives many pairs of G 2 manifolds with the same b 2 + b 3, as expected for G 2 mirrors from a CFT analysis [Shatashvili, Vafa] Interestingly, mirrors can be singular; TCS G 2 manifolds are (real) K3 fibrations over S 3 and every K3 fibre has an ADE singularity. Mathematically rigorous treatment of such solutions? Is mirror symmetry the wrong name because we have more than a Z 2? Are all G 2 manifolds with the same b 2 + b 3 (or H (J, Z)) dual as suggested by Shatashvili,Vafa? 28

Finally Want to exploit physics to learn about G 2 Thank You! 29

comments Need better understanding of CFT picture! B-field and geometric singularities? We know the CFT for some examples of the type [ CY S 1] /Z 2, compare to TCS examples? Categories of D-branes? TCS G 2 s vs. CY S 1 /Z 2 ; Gepner models? S 1 fibrations and M-Theory - IIA duality? Topological G 2 strings [de Boer,Naqvi,Shomer]? 30