G2 Mirror Symmetry

Home , G2 (mathematics), G2 manifold, K3 surface

♦ 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)\O(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 ∈ Uv : ( ωˆi = ωi − (ωi · B) v Σ4 = ˆ 2 2 B = B + v0 + v(ωi − B )

ωi · ωj = δij

The ωi 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) ( ωˆi = ωi − (ωi · B) v Σ4 = ˆ 2 2 B = B + v0 + v(ωi − 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 {h1,1(X), h2,1(X)}, 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 ◦ m∈∆ νi∈∆

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  −1 0 0 0 1   −1 −1 −1 −1 4   −1 0 0 1 0   −1 −1 −1 4 −1  ◦     ∆ ∼   ∆ ∼    −1 0 1 0 0   −1 −1 4 −1 −1  −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 ◦ m∈♦ νi∈♦ 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 ∈♦ Θ ∈♦ Θ ∈♦ 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 ∈ 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.

20 comments

Strenght of using polytopes is in resolving and analysing situations which degenerate K3 ﬁbres. Can have large

2 2 |K| = |ker(H (Z, Z) → H (S, Z))/[S]|

giving large b2(J) for resulting TCS G2 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,Nordstr¨om,Pacini].

21 Mirror Building Blocks

◦ Inverting the roles of ♦ and ♦ gives us ‘mirror building blocks’ Z and Z∨ with h2,1(Z) = |K(Z∨)|

2 2 where K = ker H (Z, Z) → H (S0, Z) /[S].

Recall that (for orthogonal gluings):

2,1 2,1 b2 + b3 = 23 + 2 h (Z+) + h (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 S0±.

Hence: for a G2 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 K¨ahlerrotation used to glue S0± is found from that for S0±.

If T+ ∩ T− ⊃ U: 2 4 2 ∨ 4 ∨ H (J, Z) ⊕ H (J, Z) = H (J , Z) ⊕ H (J , Z) 3 5 3 ∨ 5 ∨ H (J, Z) ⊕ H (J, Z) = H (J , Z) ⊕ H (J , Z) for any matching. 23 G2 Mirrors

∨ The asymptotic K3 ﬁbres S0± of Z± are mapped to S0±. This exchanges N ↔ T˜ where N = im(ρ) and T = U ⊕ T˜ = N ⊥ ∈ Γ3,19

∨ Depending on the choice of ω and Ω, S0± 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 ⊕ (−E8) h1,1(Z) = 11 h2,1(Z) = 240 |N(Z)| = 10 |K(Z)| = 0 1,1 ∨ 2,1 ∨ ∨ ∨ h (Z ) = 251 h (Z+) = 0 |N(Z )| = 10 |K(Z )| = 240

We can glue such that N+ ∩ N− = 0 = T˜+ ∩ T˜− to ﬁnd smooth mirrors with b2 + b3 = 983 ∨ b2(J) = 0 b2(J ) = 480 ∨ b3(J) = 983 b3(J ) = 503

However, perpendicularly gluing to building blocks with quartic K3 ﬁbre

N± = (4) gives ∨ ∨ ⊕2 N+ ∩ N− ⊂ (−E8)

25 Comparing to an example of Joyce

7 3 Consider again the smoothings Yl of the orbifold T /Z2. They can be realized as a TCS [Nordstr¨om,Kovalev] with building blocks

4 1 ((T /Z2) × S × R+)/Z2

± ± We can now compare our construction to the mirror maps I3 and I4 of [Gaberdiel,Kaste].

+ Our mirror map is I4 : Yl → Yl which is trivial in this case and acts as Z+ ↔ Z−.

As an aside : the discrete torsion in the CFT description precisely 3 8−l corresponds to H (Yl, Z) = Z2 .

26 The Curious Case of three T-dualities

What about other mirror maps in TCS picture ? − 3 I3 : Yl → Y8−l corresponds to performing three T-dualities along T SYZ ﬁbre of Z+ and elliptic ﬁbre of Z− !

S0+ S0− × × 1 1 X+ S S X− × − − − − × − − − − × − − − − × S1 S1 S1 S1 × × I I

∨3 ∨ Hence: for a TCS J built from Z+ and Z−, we construct J from Z+ and Z−. If T+ ∩ N− ⊃ U: • • ∨3 H (J, Z) = H (J , Z)

For orthogonal gluings b2 + b3 trivially invariant as

2,1 2,1 27 b2 + b3 = 23 + 2 h (Z+) + h (Z−) + 2 [|K(Z+)| + |K(Z−)|] . Summary

We have motivated our construction by a ‘physics picture’ of SYZ

ﬁbrations and their generalization to G2 as proposed by [Acharya], exploiting the structure of TCS G2 manifolds in the Kovalev limit.

Our construction stands on its own and gives many pairs of G2 manifolds with the same b2 + b3, as expected for G2 mirrors from a CFT analysis [Shatashvili, Vafa]

Interestingly, mirrors can be singular; TCS G2 manifolds are (real) K3 ﬁbrations over S3 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 Z2 ? • Are all G2 manifolds with the same b2 + b3 (or H (J, Z)) dual as suggested by Shatashvili,Vafa?

28 Finally

→ Want to exploit physics to learn about G2 ←

Thank You !

29 comments

Need better understanding of CFT picture ! B-ﬁeld and geometric singularities ? 1 We know the CFT for some examples of the type CY × S /Z2, compare to TCS examples ?

Categories of D-branes ?

1 1 TCS G2’s vs. CY × S /Z2; Gepner models ? S ﬁbrations and M-Theory - IIA duality ?

Topological G2 strings [de Boer,Naqvi,Shomer]?