Tops and G2 Mirror Symmetry♦

Home , G2 manifold

♦ Tops and G2 mirror symmetry♦

based on

• [1602.03521] • [1701.05202] with Michele del Zotto (Stony Brook)

Tops and G2 mirror symmetry ♦ Tops and G2 mirror symmetry♦

based on

• [1602.03521] • [1701.05202] with Michele del Zotto (Stony Brook)

Tops and G2 mirror symmetry ♦ 1) Mirror Symmetry for Calabi-Yau and G2 manifolds ♦

Tops and G2 mirror symmetry • They have to 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 !

... this duality has amazing implications ... [Candelas, de la Ossa, Green, Parkes; .... ]

a few words about Calabi-Yau ...

• Mirror symmetry is a feature of compactiﬁcation of type II strings. • On Calabi-Yau threefolds, the CFTs associated to IIA and IIB become isomorphic after reversing the left-moving U(1) charge for a pair of appropriatey chosen manifolds

X ↔ X∨

which are called a mirror pair.

Tops and G2 mirror symmetry ... this duality has amazing implications ... [Candelas, de la Ossa, Green, Parkes; .... ]

a few words about Calabi-Yau ...

X ↔ X∨

which are called a mirror pair. • They have to 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 !

Tops and G2 mirror symmetry a few words about Calabi-Yau ...

X ↔ X∨

which are called a mirror pair. • They have to 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 !

... this duality has amazing implications ... [Candelas, de la Ossa, Green, Parkes; .... ]

Tops and G2 mirror symmetry • CY n-folds have ﬁbration by special lagrangian T n. Mirror symmetry is T-duality along the n S1s. • For three-folds: BPS states related to D0-brane in type IIA on X have moduli

space = X of dimR = 6. for type IIB on X∨, these are mapped to D3-branes wrapped on ! sLag 3-cycles L, dimR of moduli space is 2b1(L) = 6, corresponding to three deformations and three Wilson lines.

mirror symmetry is T-duality

[Strominger,Yau,Zaslow] more ‘physical’ picture generalzing CFT story from study of BPS D-branes:

Tops and G2 mirror symmetry • For three-folds: BPS states related to D0-brane in type IIA on X have moduli

mirror symmetry is T-duality

[Strominger,Yau,Zaslow] more ‘physical’ picture generalzing CFT story from study of BPS D-branes: • CY n-folds have ﬁbration by special lagrangian T n. Mirror symmetry is T-duality along the n S1s.

Tops and G2 mirror symmetry for type IIB on X∨, these are mapped to D3-branes wrapped on ! sLag 3-cycles L, dimR of moduli space is 2b1(L) = 6, corresponding to three deformations and three Wilson lines.

mirror symmetry is T-duality

space = X of dimR = 6.

Tops and G2 mirror symmetry ! dimR of moduli space is 2b1(L) = 6, corresponding to three deformations and three Wilson lines.

mirror symmetry is T-duality

space = X of dimR = 6. for type IIB on X∨, these are mapped to D3-branes wrapped on sLag 3-cycles L,

Tops and G2 mirror symmetry mirror symmetry is T-duality

Tops and G2 mirror symmetry • We may in particular consider compactifications of type II strings on G2 manifolds to 2 + 1 dimensions. • Manifolds of G2 holonomy are 7-dimensional and have a Ricci-flat metric → interesting for 4D N = 1 compactification of M-Theory. • No simple algebraic construction as for Calabi-Yau; can do orbifolds T 7/Γ [Joyce]!

•∃ closed forms Φ3 and Ψ4 = ∗Φ3 (analogue of Kähler form and Ω3,0); they calibrate 3 and 4-cycles; non-trivial Betti numbers are b2 = b5 and b3 = b4. • In this case, there are similar automorphisms of the (extended) superconformal algebra, and the CFT can only detect b2 + b3. [Shatashvili, Vafa; O’Farrill; Becker, Becker, Morrison, Ooguri, Oz, Yin; Roiban, Romelsberger, Walcher; Gaberdiel, Kaste; Papadopoulos, Townsend] 4 • Arguments similar to SYZ imply coassociative T ﬁbration for G2 manifolds. [Acharya]

Mirror Symmetry: the G2 story

• More generally, we may ask if a given CFT allows several ‘geometric interpretations’.

Tops and G2 mirror symmetry • Manifolds of G2 holonomy are 7-dimensional and have a Ricci-ﬂat metric → interesting for 4D N = 1 compactiﬁcation of M-Theory. • No simple algebraic construction as for Calabi-Yau; can do orbifolds T 7/Γ [Joyce]!

Mirror Symmetry: the G2 story

Tops and G2 mirror symmetry • No simple algebraic construction as for Calabi-Yau; can do orbifolds T 7/Γ [Joyce]!

Mirror Symmetry: the G2 story

• More generally, we may ask if a given CFT allows several ‘geometric interpretations’. • We may in particular consider compactifications of type II strings on G2 manifolds to 2 + 1 dimensions. • Manifolds of G2 holonomy are 7-dimensional and have a Ricci-flat metric → interesting for 4D N = 1 compactification of M-Theory.

Tops and G2 mirror symmetry •∃ closed forms Φ3 and Ψ4 = ∗Φ3 (analogue of Kähler form and Ω3,0); they calibrate 3 and 4-cycles; non-trivial Betti numbers are b2 = b5 and b3 = b4. • In this case, there are similar automorphisms of the (extended) superconformal algebra, and the CFT can only detect b2 + b3. [Shatashvili, Vafa; O’Farrill; Becker, Becker, Morrison, Ooguri, Oz, Yin; Roiban, Romelsberger, Walcher; Gaberdiel, Kaste; Papadopoulos, Townsend] 4 • Arguments similar to SYZ imply coassociative T ﬁbration for G2 manifolds. [Acharya]

Mirror Symmetry: the G2 story

Tops and G2 mirror symmetry • In this case, there are similar automorphisms of the (extended) superconformal algebra, and the CFT can only detect b2 + b3. [Shatashvili, Vafa; O’Farrill; Becker, Becker, Morrison, Ooguri, Oz, Yin; Roiban, Romelsberger, Walcher; Gaberdiel, Kaste; Papadopoulos, Townsend] 4 • Arguments similar to SYZ imply coassociative T ﬁbration for G2 manifolds. [Acharya]

Mirror Symmetry: the G2 story

•∃ closed forms Φ3 and Ψ4 = ∗Φ3 (analogue of Kähler form and Ω3,0); they calibrate 3 and 4-cycles; non-trivial Betti numbers are b2 = b5 and b3 = b4.

Tops and G2 mirror symmetry 4 • Arguments similar to SYZ imply coassociative T ﬁbration for G2 manifolds. [Acharya]

Mirror Symmetry: the G2 story

Tops and G2 mirror symmetry Mirror Symmetry: the G2 story

Tops and G2 mirror symmetry ± ± • There are four possible choices of T-dualities I3 and I4 corresponding to automorphisms of the extended chiral algebra.

Figure taken from [Gaberdiel, Kaste]

Mirror Symmetry: the G2 story

7 3 This has been studied in detail only for T /(Z2) orbifolds [Joyce] and their smoothings Yl, l = 0..8 [Gaberdiel, Kaste]. Here

b2(Yl) = 8 + l b3(Yl) = 47 − l

Tops and G2 mirror symmetry Mirror Symmetry: the G2 story

7 3 This has been studied in detail only for T /(Z2) orbifolds [Joyce] and their smoothings Yl, l = 0..8 [Gaberdiel, Kaste]. Here

b2(Yl) = 8 + l b3(Yl) = 47 − l

± ± • There are four possible choices of T-dualities I3 and I4 corresponding to automorphisms of the extended chiral algebra.

Figure taken from [Gaberdiel, Kaste]

Tops and G2 mirror symmetry The mirror family is obtained by modding out a discrete (non-freely acting) group action and resolving singularities [Greene, Plesser]. E.g. this pencil of quintics in P4:

5 5 5 5 5 X : x1 + x2 + x3 + x4 + x5 − 5ψx1x2x3x4x5 = 0

3 has a mirror obtained by modding out (and resolving) a (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.

How to construct the mirror: CY manifolds

Classic construction of CY mirror manifolds (besides orbifolds) works for hypersurfaces in weighted projective space [related to superconformal minimal models]:

P P ki (xi) = 0 in Pk1,··· ,kn .

Tops and G2 mirror symmetry E.g. this pencil of quintics in P4:

5 5 5 5 5 X : x1 + x2 + x3 + x4 + x5 − 5ψx1x2x3x4x5 = 0

3 has a mirror obtained by modding out (and resolving) a (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.

How to construct the mirror: CY manifolds

Classic construction of CY mirror manifolds (besides orbifolds) works for hypersurfaces in weighted projective space [related to superconformal minimal models]:

P P ki (xi) = 0 in Pk1,··· ,kn . The mirror family is obtained by modding out a discrete (non-freely acting) group action and resolving singularities [Greene, Plesser].

Tops and G2 mirror symmetry How to construct the mirror: CY manifolds

Classic construction of CY mirror manifolds (besides orbifolds) works for hypersurfaces in weighted projective space [related to superconformal minimal models]:

P P ki (xi) = 0 in Pk1,··· ,kn . The mirror family is obtained by modding out a discrete (non-freely acting) group action and resolving singularities [Greene, Plesser]. E.g. this pencil of quintics in P4:

5 5 5 5 5 X : x1 + x2 + x3 + x4 + x5 − 5ψx1x2x3x4x5 = 0

3 has a mirror obtained by modding out (and resolving) a (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.

Tops and G2 mirror symmetry 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∈∆

How to construct the mirror: CY manifolds

This has a beautiful generalization to toric hypersurfaces [Batyrev].

Tops and G2 mirror symmetry 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∈∆

How to construct the mirror: CY manifolds

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

Tops and G2 mirror symmetry • 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∈∆

How to construct the mirror: CY manifolds

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:

Tops and G2 mirror symmetry ◦ • 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∈∆

How to construct the mirror: CY manifolds

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.

Tops and G2 mirror symmetry • Each lattice point m on ∆ gives a Monomial and the hypersurface equation is

X Y hm,νii+1 X(∆,∆◦) : cm xi = 0 ◦ m∈∆ νi∈∆

How to construct the mirror: CY manifolds

This has a beautiful generalization to toric hypersurfaces [Batyrev]. A pair of lattice polytopes (in lattices M and N) satisfying

h∆, ∆◦i ≥ −1

Tops and G2 mirror symmetry How to construct the mirror: CY manifolds

This has a beautiful generalization to toric hypersurfaces [Batyrev]. A pair of lattice polytopes (in lattices M and N) satisfying

h∆, ∆◦i ≥ −1

X Y hm,νii+1 X(∆,∆◦) : cm xi = 0 ◦ m∈∆ νi∈∆

Tops and G2 mirror symmetry Combinatorial formulas for Hodge numbers

More abstract point of view: ∆ deﬁnes a toric variety PΣn(∆) via its ◦ normal fan Σn(∆)=Σ f (∆ ) as well as a divisor (our CY !) on it.

h∆, ∆◦i ≥ −1

Tops and G2 mirror symmetry More abstract point of view: ∆ deﬁnes a toric variety PΣn(∆) via its ◦ normal fan Σn(∆)=Σ f (∆ ) as well as a divisor (our CY !) on it.

h∆, ∆◦i ≥ −1 Combinatorial formulas for Hodge numbers

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]

Tops and G2 mirror symmetry More abstract point of view: ∆ deﬁnes a toric variety PΣn(∆) via its ◦ normal fan Σn(∆)=Σ f (∆ ) as well as a divisor (our CY !) on it.

h∆, ∆◦i ≥ −1 Combinatorial formulas for Hodge numbers satisfy: 1,1 2,1 h (X(∆,∆◦)) = h (X(∆◦,∆)) 2,1 1,1 h (X(∆,∆◦)) = h (X(∆◦,∆)) ∨ !!!! → X(∆,∆◦) = X(∆◦,∆) ←!!!!

Tops and G2 mirror symmetry e.g. two-dimensional faces of ∆◦ looks like this:

Extra points ∼ reﬁnement Σ → Σf ∼ resolution of orbifold singularities

example: 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

3 3 ◦ For the mirror, patches of PΣn(∆ ) = PΣf (∆) are C /(Z5) : like [Greene, Plesser]!

Tops and G2 mirror symmetry example: the quintic

3 3 ◦ For the mirror, patches of PΣn(∆ ) = PΣf (∆) are C /(Z5) : like [Greene, Plesser]! e.g. two-dimensional faces of ∆◦ looks like this:

Extra points ∼ reﬁnement Σ → Σf ∼ resolution of orbifold singularities

Tops and G2 mirror symmetry • Glue a G2 manifold from algebraic ‘building blocks’ •O (100s of millions) of examples available

♦

2) G2 manifolds as twisted connected Sums ♦

Tops and G2 mirror symmetry •O (100s of millions) of examples available

♦

2) G2 manifolds as twisted connected Sums ♦

• Glue a G2 manifold from algebraic ‘building blocks’

Tops and G2 mirror symmetry ♦

2) G2 manifolds as twisted connected Sums ♦

• Glue a G2 manifold from algebraic ‘building blocks’ •O (100s of millions) of examples available

Tops and G2 mirror symmetry this is called a ‘building block’ *terms and conditions apply easiest example: hypersurface of degree (4, 1) in P3 × P1

z0P4(x) + z1Q4(x) = 0

Twisted Connected Sum G2 manifolds (TCS)

[Kovalev; Corti, Haskins, Nordström, Pacini] Take a K3-ﬁbred Kähler threefold∗ Z (with base P1) such that

c1(Z) = [S]

Tops and G2 mirror symmetry easiest example: hypersurface of degree (4, 1) in P3 × P1

z0P4(x) + z1Q4(x) = 0

Twisted Connected Sum G2 manifolds (TCS)

[Kovalev; Corti, Haskins, Nordström, Pacini] Take a K3-ﬁbred Kähler threefold∗ Z (with base P1) such that

c1(Z) = [S]

this is called a ‘building block’ *terms and conditions apply

Tops and G2 mirror symmetry Twisted Connected Sum G2 manifolds (TCS)

[Kovalev; Corti, Haskins, Nordström, Pacini] Take a K3-ﬁbred Kähler threefold∗ Z (with base P1) such that

c1(Z) = [S]

this is called a ‘building block’ *terms and conditions apply easiest example: hypersurface of degree (4, 1) in P3 × P1

z0P4(x) + z1Q4(x) = 0

Tops and G2 mirror symmetry these are open asymptotically cylindrical Calabi-Yau threefolds !

Twisted Connected Sum G2 manifolds (TCS)

• cut out a generic ﬁbre X = Z/S0

Tops and G2 mirror symmetry Twisted Connected Sum G2 manifolds (TCS)

• cut out a generic ﬁbre X = Z/S0 these are open asymptotically cylindrical Calabi-Yau threefolds !

Tops and G2 mirror symmetry Twisted Connected Sum G2 manifolds (TCS)

• cut out a generic ﬁbre X = Z/S0 these are open asymptotically cylindrical Calabi-Yau threefolds ! 1 • take two building blocks Z+ and Z− and glue X+ × S and 1 X− × S like this:

congratulations: here is your shiny new G2 manifold J !

Tops and G2 mirror symmetry the cohomology of the resuting G2 manifold J is

1 H (J, Z) = 0 2 H (J, Z) = N+ ∩ N− ⊕ K+ ⊕ K− 3 3,19 H (J, Z) = Z[S] ⊕ Γ /(N+ + N−) ⊕ (N− ∩ T+) ⊕ (N+ ∩ T−) 3 3 ⊕ H (Z+) ⊕ H (Z−) ⊕ K+ ⊕ K− 4 4 3,19 3,19 H (J, Z) = H (S) ⊕ (T+ ∩ T−) ⊕ Γ /(N− + T+) ⊕ Γ /(N+ + T−) 3 3 ∗ ∗ ⊕ H (Z+) ⊕ H (Z−) ⊕ K+ ⊕ K−

For a speciﬁc class of gluings (hyper Kähler rotations) φ : S+ 7→ S−:

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

TCS:Cohomology

Letting ρ = H2(Z, Z) → H2(S, Z) and K = ker(ρ)/[S] N = im(ρ)

Tops and G2 mirror symmetry For a speciﬁc class of gluings (hyper Kähler rotations) φ : S+ 7→ S−:

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

TCS:Cohomology

Letting ρ = H2(Z, Z) → H2(S, Z) and K = ker(ρ)/[S] N = im(ρ)

the cohomology of the resuting G2 manifold J is

Tops and G2 mirror symmetry TCS:Cohomology

Letting ρ = H2(Z, Z) → H2(S, Z) and K = ker(ρ)/[S] N = im(ρ)

the cohomology of the resuting G2 manifold J is

For a speciﬁc class of gluings (hyper Kähler rotations) φ : S+ 7→ S−:

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

Tops and G2 mirror symmetry four T-dualities correspond to

∨ ∨ ∨ X+ → X+ X− → X− S → S together with T-dualities along the S1 factors.

Given J as a TCS, can we construct♦ a mirror in this sense and check that b2 + b3 stays invariant ?

Mirror Symmetry for TCS G2 manifolds: heuristics

S × 1 X+ S X− × − − − − × − − − − × S1 S1 S1 × I We can exploit the various SYZ ﬁbrations to ﬁnd a coassociative T 4;

Tops and G2 mirror symmetry ∨ ∨ ∨ X+ → X+ X− → X− S → S together with T-dualities along the S1 factors.

Given J as a TCS, can we construct♦ a mirror in this sense and check that b2 + b3 stays invariant ?

Mirror Symmetry for TCS G2 manifolds: heuristics

S × 1 X+ S X− × − − − − × − − − − × S1 S1 S1 × I We can exploit the various SYZ ﬁbrations to ﬁnd a coassociative T 4; four T-dualities correspond to

Tops and G2 mirror symmetry Given J as a TCS, can we construct♦ a mirror in this sense and check that b2 + b3 stays invariant ?

Mirror Symmetry for TCS G2 manifolds: heuristics

S × 1 X+ S X− × − − − − × − − − − × S1 S1 S1 × I We can exploit the various SYZ ﬁbrations to ﬁnd a coassociative T 4; four T-dualities correspond to

∨ ∨ ∨ X+ → X+ X− → X− S → S together with T-dualities along the S1 factors.

Tops and G2 mirror symmetry Mirror Symmetry for TCS G2 manifolds: heuristics

S × 1 X+ S X− × − − − − × − − − − × S1 S1 S1 × I We can exploit the various SYZ ﬁbrations to ﬁnd a coassociative T 4; four T-dualities correspond to

∨ ∨ ∨ X+ → X+ X− → X− S → S together with T-dualities along the S1 factors.

Given J as a TCS, can we construct♦ a mirror in this sense and check that b2 + b3 stays invariant ?

Tops and G2 mirror symmetry ♦ 3) Building Blocks from Tops and G2 mirrors ♦

Tops and G2 mirror symmetry ◦ ◦ If ∆ ∩ F is again a reﬂexive polytope ∆F for a 3d plane F , → X ◦ is ﬁbred by K3 surfaces X ◦ . (∆,∆ ) (∆F ,∆F ) [Candelas, Font; Klemm, Lerche, Mayr; Hosono, Lian, Yau; Avram,Kreuzer,Mandelberg,Skarke] ◦ ◦ ◦ • F cuts ∆ into two halves called ‘tops’, ♦a, ♦b ◦ • If the projection of a top to F sits inside ∆F , a top is called ‘projecting’.

• In this case, ∆ is cut by F into ♦a and ♦b and X(∆◦,∆) is ﬁbred by X ◦ . (∆F ,∆F ) • Furthermore, X(∆,∆◦) enjoys a limit in which it splits into two algebraic threefolds Z ◦ and Z ◦ with c1(Z) = [S]. (♦a,♦a) (♦b,♦b ) ◦ ◦ We may think of the tops ♦a, ♦b as each capturing ‘half’ of the singular ﬁbres of X(∆,∆◦).

Tops

Consider a Calabi-Yau threefold X(∆,∆◦).

Tops and G2 mirror symmetry → X ◦ is ﬁbred by K3 surfaces X ◦ . (∆,∆ ) (∆F ,∆F ) [Candelas, Font; Klemm, Lerche, Mayr; Hosono, Lian, Yau; Avram,Kreuzer,Mandelberg,Skarke] ◦ ◦ ◦ • F cuts ∆ into two halves called ‘tops’, ♦a, ♦b ◦ • If the projection of a top to F sits inside ∆F , a top is called ‘projecting’.

Tops

Consider a Calabi-Yau threefold X(∆,∆◦). ◦ ◦ If ∆ ∩ F is again a reﬂexive polytope ∆F for a 3d plane F ,

Tops and G2 mirror symmetry ◦ ◦ ◦ • F cuts ∆ into two halves called ‘tops’, ♦a, ♦b ◦ • If the projection of a top to F sits inside ∆F , a top is called ‘projecting’.

Tops

Tops and G2 mirror symmetry ◦ • If the projection of a top to F sits inside ∆F , a top is called ‘projecting’.

Tops

Consider a Calabi-Yau threefold X(∆,∆◦). ◦ ◦ If ∆ ∩ F is again a reﬂexive polytope ∆F for a 3d plane F , → X ◦ is ﬁbred by K3 surfaces X ◦ . (∆,∆ ) (∆F ,∆F ) [Candelas, Font; Klemm, Lerche, Mayr; Hosono, Lian, Yau; Avram,Kreuzer,Mandelberg,Skarke] ◦ ◦ ◦ • F cuts ∆ into two halves called ‘tops’, ♦a, ♦b

Tops and G2 mirror symmetry • In this case, ∆ is cut by F into ♦a and ♦b and X(∆◦,∆) is ﬁbred by X ◦ . (∆F ,∆F ) • Furthermore, X(∆,∆◦) enjoys a limit in which it splits into two algebraic threefolds Z ◦ and Z ◦ with c1(Z) = [S]. (♦a,♦a) (♦b,♦b ) ◦ ◦ We may think of the tops ♦a, ♦b as each capturing ‘half’ of the singular ﬁbres of X(∆,∆◦).

Tops

Tops and G2 mirror symmetry • Furthermore, X(∆,∆◦) enjoys a limit in which it splits into two algebraic threefolds Z ◦ and Z ◦ with c1(Z) = [S]. (♦a,♦a) (♦b,♦b ) ◦ ◦ We may think of the tops ♦a, ♦b as each capturing ‘half’ of the singular ﬁbres of X(∆,∆◦).

Tops

• In this case, ∆ is cut by F into ♦a and ♦b and X(∆◦,∆) is ﬁbred by X ◦ . (∆F ,∆F )

Tops and G2 mirror symmetry ◦ ◦ We may think of the tops ♦a, ♦b as each capturing ‘half’ of the singular ﬁbres of X(∆,∆◦).

Tops

Tops and G2 mirror symmetry Tops

Tops and G2 mirror symmetry Z ◦ (♦,♦ ) has a direct construction as a hypersurface in PΣ, where Σ is a reﬁnement of Σn(♦), given by

X hν0,mi Y hνi,mi+1 Z ◦ : c x x = 0 (♦,♦ ) m e i ◦ m∈♦ νi∈♦ Starting from ♦, this is the same as Batyrev’s construction and results in a building block for G2 manifolds. There is combinatorial computation of Hodge numbers (and lattices N and K) using toric stratiﬁcation [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 <♦

Building Blocks From Projecting Tops

A pair of dual 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.

Tops and G2 mirror symmetry Starting from ♦, this is the same as Batyrev’s construction and results in a building block for G2 manifolds. There is combinatorial computation of Hodge numbers (and lattices N and K) using toric stratiﬁcation [AB]:

Building Blocks From Projecting Tops

A pair of dual tops is a pair of lattice polytopes which satisfy ◦ h♦, ♦ i ≥ −1 ◦ h♦, ν0i ≥ 0 hm0, ♦ i ≥ 0

ν m ⊥ F hm , ν i = −1 Z ◦ with 0 and 0 , 0 0 . (♦,♦ ) has a direct construction as a hypersurface in PΣ, where Σ is a reﬁnement of Σn(♦), given by

X hν0,mi Y hνi,mi+1 Z ◦ : c x x = 0 (♦,♦ ) m e i ◦ m∈♦ νi∈♦

Tops and G2 mirror symmetry Building Blocks From Projecting Tops

A pair of dual tops is a pair of lattice polytopes which satisfy ◦ h♦, ♦ i ≥ −1 ◦ h♦, ν0i ≥ 0 hm0, ♦ i ≥ 0

ν m ⊥ F hm , ν i = −1 Z ◦ with 0 and 0 , 0 0 . (♦,♦ ) has a direct construction as a hypersurface in PΣ, where Σ is a reﬁnement of Σn(♦), given by

Tops and G2 mirror symmetry • For a speciﬁc class of gluings φ:

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

We are in business !

◦ ♦ ↔ ♦

More exciting: ◦ • swapping the roles of ♦ ↔ ♦ gives the SYZ mirror of the open, X = Z ◦ /S asymptotically cylindrical Calabi-Yau (♦,♦ ) . • in particular: the mirror map exchanges h2,1 with |ker(H2(Z, Z) → H2(S, Z))/[S]|:

2,1 h (Z ◦ ) = |K(Z ◦ )| (♦,♦ ) (♦ ,♦)

Tops and G2 mirror symmetry We are in business !

◦ ♦ ↔ ♦

2,1 h (Z ◦ ) = |K(Z ◦ )| (♦,♦ ) (♦ ,♦)

• For a speciﬁc class of gluings φ:

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

Tops and G2 mirror symmetry ◦ ♦ ↔ ♦

2,1 h (Z ◦ ) = |K(Z ◦ )| (♦,♦ ) (♦ ,♦)

• For a speciﬁc class of gluings φ:

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

We are in business !

Tops and G2 mirror symmetry 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 K3 ﬁbres of Z ◦ and its mirror Z ◦ are from mirror (♦±,♦±) (♦±,♦±) families and there is a ‘mirror gluing’ φ∨ = µ ◦ φ ◦ µ−1. • It is straightforward to show that ∨ ∨ b2(J) + b3(J) = b2(J ) + b3(J ) .

• In fact 2 4 H (J, Z) ⊕ H (J, Z) is invariant under this mirror map !

Mirror G2 manifolds

From our discussion of SYZ, we should trade both building blocks for their duals !

Tops and G2 mirror symmetry • The K3 ﬁbres of Z ◦ and its mirror Z ◦ are from mirror (♦±,♦±) (♦±,♦±) families and there is a ‘mirror gluing’ φ∨ = µ ◦ φ ◦ µ−1. • It is straightforward to show that ∨ ∨ b2(J) + b3(J) = b2(J ) + b3(J ) .

• In fact 2 4 H (J, Z) ⊕ H (J, Z) is invariant under this mirror map !

Mirror G2 manifolds

From our discussion of SYZ, we should trade both building blocks for their duals ! 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 + (♦+,♦+) + − (♦−,♦− −

Tops and G2 mirror symmetry • It is straightforward to show that ∨ ∨ b2(J) + b3(J) = b2(J ) + b3(J ) .

• In fact 2 4 H (J, Z) ⊕ H (J, Z) is invariant under this mirror map !

Mirror G2 manifolds

From our discussion of SYZ, we should trade both building blocks for their duals ! 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 K3 ﬁbres of Z ◦ and its mirror Z ◦ are from mirror (♦±,♦±) (♦±,♦±) families and there is a ‘mirror gluing’ φ∨ = µ ◦ φ ◦ µ−1.

Tops and G2 mirror symmetry • In fact 2 4 H (J, Z) ⊕ H (J, Z) is invariant under this mirror map !

Mirror G2 manifolds

From our discussion of SYZ, we should trade both building blocks for their duals ! 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 + (♦+,♦+) + − (♦−,♦− −

Tops and G2 mirror symmetry Mirror G2 manifolds

From our discussion of SYZ, we should trade both building blocks for their duals ! 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 + (♦+,♦+) + − (♦−,♦− −

• In fact 2 4 H (J, Z) ⊕ H (J, Z) is invariant under this mirror map !

Tops and G2 mirror symmetry The corresponding pair of dual tops is  −1 0 0 0 1   −1 −1 3 3 −1 −1 −1 −1   −1 0 0 1 0   −1 −1 −1 −1 3 3 −1 −1    ,    −1 0 1 0 0   −1 −1 −1 −1 −1 −1 3 3  0 1 0 0 0 −1 0 0 −1 0 −1 0 −1

h1,1(Z) = 2 h1,1(Z∨) = 53 h1,1(Z) = 33 h2,1(Z∨) = 0 |K(Z)| = 0 |K(Z∨)| = 33 note here |K| = h1,1(Z) − |P ic(S)| − 1 For a G2 manifold J glued from two copies of Z we have

b2(J) = 0 , b3(J) = 155 .

The associated mirror G2 manifold has ∨ ∨ b2(J ) = 84 , b3(J ) = 71 .

example: the simplest TCS G2 manifold and its mirror

Consider again a building block Z described by a hypersurface of degree (4, 1) in P3 × P1.

Tops and G2 mirror symmetry  −1 0 0 0 1   −1 −1 3 3 −1 −1 −1 −1   −1 0 0 1 0   −1 −1 −1 −1 3 3 −1 −1    ,    −1 0 1 0 0   −1 −1 −1 −1 −1 −1 3 3  0 1 0 0 0 −1 0 0 −1 0 −1 0 −1

h1,1(Z) = 2 h1,1(Z∨) = 53 h1,1(Z) = 33 h2,1(Z∨) = 0 |K(Z)| = 0 |K(Z∨)| = 33 note here |K| = h1,1(Z) − |P ic(S)| − 1 For a G2 manifold J glued from two copies of Z we have

b2(J) = 0 , b3(J) = 155 .

The associated mirror G2 manifold has ∨ ∨ b2(J ) = 84 , b3(J ) = 71 .

example: the simplest TCS G2 manifold and its mirror

Consider again a building block Z described by a hypersurface of degree (4, 1) in P3 × P1. The corresponding pair of dual tops is

Tops and G2 mirror symmetry h1,1(Z) = 2 h1,1(Z∨) = 53 h1,1(Z) = 33 h2,1(Z∨) = 0 |K(Z)| = 0 |K(Z∨)| = 33 note here |K| = h1,1(Z) − |P ic(S)| − 1 For a G2 manifold J glued from two copies of Z we have

b2(J) = 0 , b3(J) = 155 .

The associated mirror G2 manifold has ∨ ∨ b2(J ) = 84 , b3(J ) = 71 .

example: the simplest TCS G2 manifold and its mirror

Consider again a building block Z described by a hypersurface of degree (4, 1) in P3 × P1. The corresponding pair of dual tops is  −1 0 0 0 1   −1 −1 3 3 −1 −1 −1 −1   −1 0 0 1 0   −1 −1 −1 −1 3 3 −1 −1    ,    −1 0 1 0 0   −1 −1 −1 −1 −1 −1 3 3  0 1 0 0 0 −1 0 0 −1 0 −1 0 −1

Tops and G2 mirror symmetry For a G2 manifold J glued from two copies of Z we have

b2(J) = 0 , b3(J) = 155 .

The associated mirror G2 manifold has ∨ ∨ b2(J ) = 84 , b3(J ) = 71 .

example: the simplest TCS G2 manifold and its mirror

h1,1(Z) = 2 h1,1(Z∨) = 53 h1,1(Z) = 33 h2,1(Z∨) = 0 |K(Z)| = 0 |K(Z∨)| = 33 note here |K| = h1,1(Z) − |P ic(S)| − 1

Tops and G2 mirror symmetry The associated mirror G2 manifold has ∨ ∨ b2(J ) = 84 , b3(J ) = 71 .

example: the simplest TCS G2 manifold and its mirror

h1,1(Z) = 2 h1,1(Z∨) = 53 h1,1(Z) = 33 h2,1(Z∨) = 0 |K(Z)| = 0 |K(Z∨)| = 33 note here |K| = h1,1(Z) − |P ic(S)| − 1 For a G2 manifold J glued from two copies of Z we have

b2(J) = 0 , b3(J) = 155 .

Tops and G2 mirror symmetry example: the simplest TCS G2 manifold and its mirror

h1,1(Z) = 2 h1,1(Z∨) = 53 h1,1(Z) = 33 h2,1(Z∨) = 0 |K(Z)| = 0 |K(Z∨)| = 33 note here |K| = h1,1(Z) − |P ic(S)| − 1 For a G2 manifold J glued from two copies of Z we have

b2(J) = 0 , b3(J) = 155 .

The associated mirror G2 manifold has ∨ ∨ b2(J ) = 84 , b3(J ) = 71 .

Tops and G2 mirror symmetry • In this limit the non-gravitational d.o.f. in the eﬀective ﬁeld theory has several sectors with enhanced SUSY [Guio, Jockers, Klemm, Yeh]

TCS and EFT

The TCS construction is good in the limit in which the cylindrical region in the middle:

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

becomes very long.

Tops and G2 mirror symmetry TCS and EFT

The TCS construction is good in the limit in which the cylindrical region in the middle:

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

becomes very long. • In this limit the non-gravitational d.o.f. in the eﬀective ﬁeld theory has several sectors with enhanced SUSY [Guio, Jockers, Klemm, Yeh]

Tops and G2 mirror symmetry TCS and EFT

The TCS construction is good in the limit in which the cylindrical region in the middle:

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

becomes very long. • In this limit the non-gravitational d.o.f. in the eﬀective ﬁeld theory has several sectors with enhanced SUSY [Guio, Jockers, Klemm, Yeh] 1 • For M-Theory on CY3 × S get N = 2 in 4D • For M-Theory on K3 × S1 × S1 get N = 4 in 4D

Tops and G2 mirror symmetry TCS and EFT

The TCS construction is good in the limit in which the cylindrical region in the middle:

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

Tops and G2 mirror symmetry TCS and EFT

The TCS construction is good in the limit in which the cylindrical region in the middle:

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

becomes very long. • In this limit the non-gravitational d.o.f. in the eﬀective ﬁeld theory has several sectors with enhanced SUSY [Guio, Jockers, Klemm, Yeh] 1 • For type IIA on CY3 × S get N = 4 in 3D • For type IIA on K3 × S1 × S1 get N = 8 in 3D

Tops and G2 mirror symmetry TCS and EFT

The TCS construction is good in the limit in which the cylindrical region in the middle: S × 1 X+ S X− × − − − − × − − − − × S1 S1 S1 × I becomes very long. • In this limit the non-gravitational d.o.f. in the eﬀective ﬁeld theory has several sectors with enhanced SUSY [Guio, Jockers, Klemm, Yeh] 1 • For type IIA on CY3 × S get N = 4 in 3D • For type IIA on K3 × S1 × S1 get N = 8 in 3D These are coupled together to form an N = 2 theory ... our mirror map acts on all of these ... relation to 3D mirror symmetry ?

Tops and G2 mirror symmetry This orbifold and its smoothings can also be decomposed as twisted connected sums [AB, Michele del Zotto: to appear]... + • Our map is I4 ... although this acts trivially on the Yl, it has a non-trivial action in general ! ∨ − • Swapping only one Z+ ↔ Z+ (but not Z−) realizes I3 .

Sneak Preview: comparing to Orbifolds

7 3 For Joyce orbifold Yl = T /Z2: b2 = 8 + l , b3 = 47 − l , (l = 0..8)

Tops and G2 mirror symmetry + • Our map is I4 ... although this acts trivially on the Yl, it has a non-trivial action in general ! ∨ − • Swapping only one Z+ ↔ Z+ (but not Z−) realizes I3 .

Sneak Preview: comparing to Orbifolds

7 3 For Joyce orbifold Yl = T /Z2: b2 = 8 + l , b3 = 47 − l , (l = 0..8)

This orbifold and its smoothings can also be decomposed as twisted connected sums [AB, Michele del Zotto: to appear]...

Tops and G2 mirror symmetry ∨ − • Swapping only one Z+ ↔ Z+ (but not Z−) realizes I3 .

Sneak Preview: comparing to Orbifolds

7 3 For Joyce orbifold Yl = T /Z2: b2 = 8 + l , b3 = 47 − l , (l = 0..8)

This orbifold and its smoothings can also be decomposed as twisted connected sums [AB, Michele del Zotto: to appear]... + • Our map is I4 ... although this acts trivially on the Yl, it has a non-trivial action in general !

Tops and G2 mirror symmetry Sneak Preview: comparing to Orbifolds

7 3 For Joyce orbifold Yl = T /Z2: b2 = 8 + l , b3 = 47 − l , (l = 0..8)

Tops and G2 mirror symmetry – Thank you –

Tops and G2 mirror symmetry