Mirror Symmetry for G2 Manifolds: Twisted Connected Sums and Dual Tops

Michele Del Zotto Simons Center for Geometry and Physics New York State University at Stony Brook

Presented at TAMU Texas, February 6th, 2017 arXiv:1701.05202

Joint work with Andreas Braun (Mathematical Institute, University of Oxford, UK) arXiv:1701.05202

Joint work with Andreas Braun (Mathematical Institute, University of Oxford, UK) Joint work with Andreas Braun (Mathematical Institute, University of Oxford, UK)

arXiv:1701.05202

Among the known dualities, the mirror symmetry for compactiﬁcations of Type II superstrings on Calabi-Yau (CY) manifolds is one of the most powerful.

The full quantum duality is expected to give rise to an isomorphism for the whole quantum physics of mirror compactiﬁcations with a wide array of applications.

Dualities along the landscape of superstring compactiﬁcations are one of the most important features of string theory. The full quantum duality is expected to give rise to an isomorphism for the whole quantum physics of mirror compactiﬁcations with a wide array of applications.

Dualities along the landscape of superstring compactiﬁcations are one of the most important features of string theory.

Among the known dualities, the mirror symmetry for compactiﬁcations of Type II superstrings on Calabi-Yau (CY) manifolds is one of the most powerful. Dualities along the landscape of superstring compactiﬁcations are one of the most important features of string theory.

Among the known dualities, the mirror symmetry for compactiﬁcations of Type II superstrings on Calabi-Yau (CY) manifolds is one of the most powerful.

The full quantum duality is expected to give rise to an isomorphism for the whole quantum physics of mirror compactiﬁcations with a wide array of applications. In this talk I am going to survey a recent result about mirror symmetry for G2 manifolds.

Similar dualities have been conjectured for manifolds with special holonomy G2 and Spin(7).

Shatashvili-Vafa 94 Similar dualities have been conjectured for manifolds with special holonomy G2 and Spin(7).

Shatashvili-Vafa 94

In this talk I am going to survey a recent result about mirror symmetry for G2 manifolds. Consider compactiﬁcations of Type II superstrings on J. Network:

h IIAJ o / IIAJ ∨ O O

v v

IIB o / IIB ∨ J vhv J

Papadopoulos-Townsend 95, Acharya 96,97

In this talk, h.

Let J denote one such manifold. Network:

h IIAJ o / IIAJ ∨ O O

v v

IIB o / IIB ∨ J vhv J

Papadopoulos-Townsend 95, Acharya 96,97

In this talk, h.

Let J denote one such manifold. Consider compactiﬁcations of Type II superstrings on J. In this talk, h.

Let J denote one such manifold. Consider compactiﬁcations of Type II superstrings on J. Network:

h IIAJ o / IIAJ ∨ O O

v v

IIB o / IIB ∨ J vhv J

Papadopoulos-Townsend 95, Acharya 96,97 Let J denote one such manifold. Consider compactiﬁcations of Type II superstrings on J. Network:

h IIAJ o / IIAJ ∨ O O

v v

IIB o / IIB ∨ J vhv J

Papadopoulos-Townsend 95, Acharya 96,97

In this talk, h. As in the CY case, the origin of mirror symmetry from the CFT perspective is the presence of a non-trivial order two mirror automorphism of the = 1 algebra. N Odake 88, Roiban-Romelsberger-Walcher 02, Gaberdiel-Kaste 04

7 3 In particular, the case of the Joyce T /(Z2) orbifolds has been analysed in detail.

Original formulation in the context of appropriate 2d = 1 SCFTs describing strings N propagating on G2-holonomy manifolds.

Shatashvili-Vafa 94, Figueroa O’Farrill 96 7 3 In particular, the case of the Joyce T /(Z2) orbifolds has been analysed in detail.

Original formulation in the context of appropriate 2d = 1 SCFTs describing strings N propagating on G2-holonomy manifolds.

Shatashvili-Vafa 94, Figueroa O’Farrill 96

As in the CY case, the origin of mirror symmetry from the CFT perspective is the presence of a non-trivial order two mirror automorphism of the = 1 algebra. N Odake 88, Roiban-Romelsberger-Walcher 02, Gaberdiel-Kaste 04 Original formulation in the context of appropriate 2d = 1 SCFTs describing strings N propagating on G2-holonomy manifolds.

Shatashvili-Vafa 94, Figueroa O’Farrill 96

7 3 In particular, the case of the Joyce T /(Z2) orbifolds has been analysed in detail. The 2d theories have a conformal manifold of dimension

b2(J) + b3(J) = b2(J) + b4(J),

where bn denotes the n-th Betti number of the manifold, and the equality follows from the fact that G2 holonomy entails dim(J) = 7.

As for the K¨ahler moduli spaces of CYs, the moduli spaces of these 2d = 1 SCFTs are generically larger than theN geometric moduli spaces. where bn denotes the n-th Betti number of the manifold, and the equality follows from the fact that G2 holonomy entails dim(J) = 7.

As for the K¨ahler moduli spaces of CYs, the moduli spaces of these 2d = 1 SCFTs are generically larger than theN geometric moduli spaces.

The 2d theories have a conformal manifold of dimension

b2(J) + b3(J) = b2(J) + b4(J), and the equality follows from the fact that G2 holonomy entails dim(J) = 7.

As for the K¨ahler moduli spaces of CYs, the moduli spaces of these 2d = 1 SCFTs are generically larger than theN geometric moduli spaces.

The 2d theories have a conformal manifold of dimension

b2(J) + b3(J) = b2(J) + b4(J), where bn denotes the n-th Betti number of the manifold, As for the K¨ahler moduli spaces of CYs, the moduli spaces of these 2d = 1 SCFTs are generically larger than theN geometric moduli spaces.

The 2d theories have a conformal manifold of dimension

b2(J) + b3(J) = b2(J) + b4(J), where bn denotes the n-th Betti number of the manifold, and the equality follows from the fact that G2 holonomy entails dim(J) = 7. Notice that this agrees with what is expected from the reduction of the 9+1 dimensional Type II supergravities to 2+1 dimensions on a G2 holonomy manifold preserving 4 supercharges.

Therefore, G2-mirror pairs must satisfy the Shatashvili-Vafa relation

b2(J) + b4(J) = b2(J ∨) + b4(J ∨). Therefore, G2-mirror pairs must satisfy the Shatashvili-Vafa relation

b2(J) + b4(J) = b2(J ∨) + b4(J ∨).

Notice that this agrees with what is expected from the reduction of the 9+1 dimensional Type II supergravities to 2+1 dimensions on a G2 holonomy manifold preserving 4 supercharges. At least 50 million can be easily generated by means of twisted connected sums (TCS) of asymptotically cylindrical CY three-folds.

Kovalev 03, Corti-Pacini-Haskins-Nordstr¨om12

The physical implications of this fact are stunning. In particular, it is very natural to ask about the G2-mirror map in this context.

Recently, lots of progress has been made in producing examples of compact G2 holonomy manifolds. The physical implications of this fact are stunning. In particular, it is very natural to ask about the G2-mirror map in this context.

Recently, lots of progress has been made in producing examples of compact G2 holonomy manifolds.

At least 50 million can be easily generated by means of twisted connected sums (TCS) of asymptotically cylindrical CY three-folds.

Kovalev 03, Corti-Pacini-Haskins-Nordstr¨om12 In particular, it is very natural to ask about the G2-mirror map in this context.

Recently, lots of progress has been made in producing examples of compact G2 holonomy manifolds.

At least 50 million can be easily generated by means of twisted connected sums (TCS) of asymptotically cylindrical CY three-folds.

Kovalev 03, Corti-Pacini-Haskins-Nordstr¨om12

The physical implications of this fact are stunning. Recently, lots of progress has been made in producing examples of compact G2 holonomy manifolds.

At least 50 million can be easily generated by means of twisted connected sums (TCS) of asymptotically cylindrical CY three-folds.

Kovalev 03, Corti-Pacini-Haskins-Nordstr¨om12

The physical implications of this fact are stunning. In particular, it is very natural to ask about the G2-mirror map in this context. Consider a pair X+ and X of CY threefolds which are asymptotically cylindrical,− meaning that they have one “corner” which asymptotically has the form + 1 R S S , × × ± where S are smooth K3 surfaces. ±

Let us begin with a quick informal review of the construction of TCS G2-holonomy manifolds. meaning that they have one “corner” which asymptotically has the form + 1 R S S , × × ± where S are smooth K3 surfaces. ±

Let us begin with a quick informal review of the construction of TCS G2-holonomy manifolds.

Consider a pair X+ and X of CY threefolds which are asymptotically cylindrical,− Let us begin with a quick informal review of the construction of TCS G2-holonomy manifolds.

Consider a pair X+ and X of CY threefolds which are asymptotically cylindrical,− meaning that they have one “corner” which asymptotically has the form + 1 R S S , × × ± where S are smooth K3 surfaces. ± In particular, the metric, K¨ahlerform, and holomorphic top form converge to

2 2 2 2 ds dt + dθ + dsS± , ± ∼

ω dt dθ + ωS± , ± ∼ ∧ and 3,0 2,0 Ω (dθ idt) ΩS . ± ∼ − ∧ ± on the asymptotic CY cylinders. Each side can be equipped with a G2-structure

ϕ dξ ω + Re(Ω3,0) ± ≡ ∧ ± ± where we have denoted by ξ the coordinate of the extra S1.

Now consider the products

1 1 S X+ and S X × × − Now consider the products

1 1 S X+ and S X × × −

Each side can be equipped with a G2-structure

ϕ dξ ω + Re(Ω3,0) ± ≡ ∧ ± ± where we have denoted by ξ the coordinate of the extra S1. Fix ` > 0 and consider the interval

` (`, ` + 1) R+. I ≡ ⊂ We want a diﬀeomorphism

1 1 1 1 Ξ` : S ` S S+ S ` S S , × I × × → × I × × − X+ X−

such that | {z } | {z } Ξ`∗ϕ ϕ+. − ≡

Consider the asymptotically cylindrical regions of X . ± We want a diﬀeomorphism

1 1 1 1 Ξ` : S ` S S+ S ` S S , × I × × → × I × × − X+ X−

such that | {z } | {z } Ξ`∗ϕ ϕ+. − ≡

Consider the asymptotically cylindrical regions of X . Fix ` > 0 and consider the interval ±

` (`, ` + 1) R+. I ≡ ⊂ such that Ξ`∗ϕ ϕ+. − ≡

Consider the asymptotically cylindrical regions of X . Fix ` > 0 and consider the interval ±

` (`, ` + 1) R+. I ≡ ⊂ We want a diﬀeomorphism

1 1 1 1 Ξ` : S ` S S+ S ` S S , × I × × → × I × × − X+ X− | {z } | {z } Consider the asymptotically cylindrical regions of X . Fix ` > 0 and consider the interval ±

` (`, ` + 1) R+. I ≡ ⊂ We want a diﬀeomorphism

1 1 1 1 Ξ` : S ` S S+ S ` S S , × I × × → × I × × − X+ X− such that | {z } | {z } Ξ`∗ϕ ϕ+. − ≡ i.e. a diﬀeomorphism of K3 surfaces which induces

2 2 2,0 2,0 g∗ds = ds , g∗Im(Ω ) = Im(Ω ), S− S+ S− − S+ 2,0 2,0 g∗Re(ΩS− ) = ωS+ , g∗ωS− = Re(ΩS+ ).

In local coordinates Ξ` is given by

Ξ :(ξ, t, θ, Z) (θ, 2` + 1 t, ξ, g(Z)), ` 7→ − where g : S+ S is a hyperk¨ahlerrotation, → − In local coordinates Ξ` is given by

Ξ :(ξ, t, θ, Z) (θ, 2` + 1 t, ξ, g(Z)), ` 7→ − where g : S+ S is a hyperk¨ahlerrotation, i.e. a diﬀeomorphism→ − of K3 surfaces which induces

2 2 2,0 2,0 g∗ds = ds , g∗Im(Ω ) = Im(Ω ), S− S+ S− − S+ 2,0 2,0 g∗Re(ΩS− ) = ωS+ , g∗ωS− = Re(ΩS+ ). For suﬃciently large `, the manifold J so obtained is a G2-holonomy manifold, the twisted connected sum of X . ± Corti-Pacini-Haskins-Nordstr¨om12

Truncating both manifolds S1 X at t = ` + 1 one obtains compact manifolds× with± boundaries S1 S1 S which can be glued via the × × ± diffeomorphism Ξ`. Truncating both manifolds S1 X at t = ` + 1 one obtains compact manifolds× with± boundaries S1 S1 S which can be glued via the × × ± diffeomorphism Ξ`. For sufficiently large `, the manifold J so obtained is a G2-holonomy manifold, the twisted connected sum of X . ± Corti-Pacini-Haskins-Nordström12 G2-holonomy manifolds have two types of calibrated submanifolds: associative submanifolds, calibrated by ϕ • coassociative submanifolds, calibrated by ?ϕ •

A beautiful geometrical approach to G2-mirror symmetry is given by generalizing the SYZ argument to G2-holonomy manifolds.

Acharya 97 associative submanifolds, calibrated by ϕ • coassociative submanifolds, calibrated by ?ϕ •

A beautiful geometrical approach to G2-mirror symmetry is given by generalizing the SYZ argument to G2-holonomy manifolds.

Acharya 97

G2-holonomy manifolds have two types of calibrated submanifolds: A beautiful geometrical approach to G2-mirror symmetry is given by generalizing the SYZ argument to G2-holonomy manifolds.

Acharya 97

G2-holonomy manifolds have two types of calibrated submanifolds: associative submanifolds, calibrated by ϕ • coassociative submanifolds, calibrated by ?ϕ • associative submanifolds: obstructed • coassociative submanifolds: unobstructed • Any coassociative submanifold N J has a ⊂ + smooth moduli space of dimension b2 (N), the number of self-dual harmonic 2-forms on N.

Deformations: Any coassociative submanifold N J has a ⊂ + smooth moduli space of dimension b2 (N), the number of self-dual harmonic 2-forms on N.

Deformations: associative submanifolds: obstructed • coassociative submanifolds: unobstructed • Deformations: associative submanifolds: obstructed • coassociative submanifolds: unobstructed • Any coassociative submanifold N J has a ⊂ + smooth moduli space of dimension b2 (N), the number of self-dual harmonic 2-forms on N. Consider a compactiﬁcation of IIA on J. A D0-brane has a moduli space which equals J, which must correspond to the moduli space of a wrapped Dp-brane on J ∨. As we want a BPS conﬁguration, we must wrap a calibrated cycle. The only option left is wrapping a coassociative N J ∨ with a D4-brane. ⊂

Let (J, J ∨) denote a putative G2-mirror pair. A D0-brane has a moduli space which equals J, which must correspond to the moduli space of a wrapped Dp-brane on J ∨. As we want a BPS conﬁguration, we must wrap a calibrated cycle. The only option left is wrapping a coassociative N J ∨ with a D4-brane. ⊂

Let (J, J ∨) denote a putative G2-mirror pair. Consider a compactiﬁcation of IIA on J. As we want a BPS conﬁguration, we must wrap a calibrated cycle. The only option left is wrapping a coassociative N J ∨ with a D4-brane. ⊂

Let (J, J ∨) denote a putative G2-mirror pair. Consider a compactification of IIA on J. A D0-brane has a moduli space which equals J, which must correspond to the moduli space of a wrapped Dp-brane on J ∨. As we want a BPS configuration, we must wrap a calibrated cycle. Let (J, J ∨) denote a putative G2-mirror pair. Consider a compactification of IIA on J. A D0-brane has a moduli space which equals J, which must correspond to the moduli space of a wrapped Dp-brane on J ∨. As we want a BPS configuration, we must wrap a calibrated cycle. The only option left is wrapping a coassociative N J ∨ with a D4-brane. ⊂ For this to coincide with the D0 brane the moduli spaces must agree, in particular:

+ b1(N) + b2 (N) = 7. It is natural to conjecture that N T 4. '

The U(1) vector ﬁeld on the brane gives rise to b1(N) additional moduli, whence the physical moduli space has dimension

+ b1(N) + b2 (N). It is natural to conjecture that N T 4. '

The U(1) vector ﬁeld on the brane gives rise to b1(N) additional moduli, whence the physical moduli space has dimension

+ b1(N) + b2 (N). For this to coincide with the D0 brane the moduli spaces must agree, in particular:

+ b1(N) + b2 (N) = 7. The U(1) vector ﬁeld on the brane gives rise to b1(N) additional moduli, whence the physical moduli space has dimension

+ b1(N) + b2 (N). For this to coincide with the D0 brane the moduli spaces must agree, in particular:

+ b1(N) + b2 (N) = 7. It is natural to conjecture that N T 4. ' In the paper we present an heuristic argument to argue that this is indeed the case for the TCS G2-manifolds. The main idea is that one can glue the special lagrangian SYZ ﬁbrations of the asymptotically cylindrical building blocks X into a T 4 ± coassociative ﬁbration of J: then the G2-mirror manifold J ∨ is obtained by four T-dualities, along the lines of the original SYZ argument.

Extending the SYZ argument to this case, leads the map h in the G2-mirror symmetry network. The main idea is that one can glue the special lagrangian SYZ ﬁbrations of the asymptotically cylindrical building blocks X into a T 4 ± coassociative ﬁbration of J: then the G2-mirror manifold J ∨ is obtained by four T-dualities, along the lines of the original SYZ argument.

Extending the SYZ argument to this case, leads the map h in the G2-mirror symmetry network. In the paper we present an heuristic argument to argue that this is indeed the case for the TCS G2-manifolds. The main idea is that one can glue the special lagrangian SYZ fibrations of the asymptotically cylindrical building blocks X into a T 4 coassociative fibration of J: ± Extending the SYZ argument to this case, leads the map h in the G2-mirror symmetry network. In the paper we present an heuristic argument to argue that this is indeed the case for the TCS G2-manifolds. The main idea is that one can glue the special lagrangian SYZ fibrations of the asymptotically cylindrical building blocks X into a T 4 ± coassociative fibration of J: then the G2-mirror manifold J ∨ is obtained by four T-dualities, along the lines of the original SYZ argument. Looking at the explicit expressions:

1 2 3,0 ?ϕ 2ω dξ Im(Ω ), ± ≡ ± − ∧ ± rotate Ω3,0 for the SYZ special lagrangians L X ± so that ± ⊂ ± 3,0 Ω = i volL± , ± L± −

in this way S1 L is a coassociative in S1 X and the compatibility× ± condition above does the× ± rest.

Such glueing is possible because:

?ϕ+ = ?(Ξ`∗ϕ ) = Ξ`∗(?ϕ ) − − by construction. rotate Ω3,0 for the SYZ special lagrangians L X ± so that ± ⊂ ± 3,0 Ω = i volL± , ± L± −

in this way S1 L is a coassociative in S1 X and the compatibility× ± condition above does the× ± rest.

Such glueing is possible because:

?ϕ+ = ?(Ξ`∗ϕ ) = Ξ`∗(?ϕ ) − − by construction. Looking at the explicit expressions:

1 2 3,0 ?ϕ 2ω dξ Im(Ω ), ± ≡ ± − ∧ ± in this way S1 L is a coassociative in S1 X and the compatibility× ± condition above does the× ± rest.

Such glueing is possible because:

?ϕ+ = ?(Ξ`∗ϕ ) = Ξ`∗(?ϕ ) − − by construction. Looking at the explicit expressions:

1 2 3,0 ?ϕ 2ω dξ Im(Ω ), ± ≡ ± − ∧ ± rotate Ω3,0 for the SYZ special lagrangians L X ± so that ± ⊂ ± 3,0 Ω = i volL± , ± L± −

Such glueing is possible because:

?ϕ+ = ?(Ξ`∗ϕ ) = Ξ`∗(?ϕ ) − − by construction. Looking at the explicit expressions:

1 2 3,0 ?ϕ 2ω dξ Im(Ω ), ± ≡ ± − ∧ ± rotate Ω3,0 for the SYZ special lagrangians L X ± so that ± ⊂ ± 3,0 Ω = i volL± , ± L± − in this way S1 L is a coassociative in S1 X and the compatibility× ± condition above does the× ± rest. Performing the four T-dualities induces a mirror map also on the asymptotic ﬁbers, and swaps the S with their mirrors (S )∨. ± ±

In particular, along the CY cylinders of X the SYZ fibers asymptote to S1 Λ , where Λ± are SYZ fibers for the K3s S . × ± ± ± In particular, along the CY cylinders of X the SYZ fibers asymptote to S1 Λ , where Λ± are SYZ fibers for the K3s S . × ± ± ± Performing the four T-dualities induces a mirror map also on the asymptotic fibers, and swaps the S with their mirrors (S )∨. ± ± It is elementary to check that

g g∨ : S+∨ S+ S S∨ → −→ − → − is a canonically deﬁned hyperk¨alerrotation!

This suggests a canonical way of extending the hyperk¨ahlerrotation to include the B-ﬁeld

g∗B = B+. − − This suggests a canonical way of extending the hyperk¨ahlerrotation to include the B-ﬁeld

g∗B = B+. − − It is elementary to check that

g g∨ : S+∨ S+ S S∨ → −→ − → − is a canonically deﬁned hyperk¨alerrotation! Moreover, the X∨ asymptotic CY cylinders must have the structure±

1 X∨ R+ (S )∨ (S )∨, ± ∼ × × ± 1 where (S )∨ is the T-dual circle and (S )∨ are mirrors of the S as K3 surfaces. ± ±

We ﬁnd that the mirror G2-manifold J ∨ so obtained is a TCS of the mirrors of the asymptotically cylindrical building blocks X∨. ± 1 where (S )∨ is the T-dual circle and (S )∨ are mirrors of the S as K3 surfaces. ± ±

We ﬁnd that the mirror G2-manifold J ∨ so obtained is a TCS of the mirrors of the asymptotically cylindrical building blocks X∨. ±

Moreover, the X∨ asymptotic CY cylinders must have the structure±

1 X∨ R+ (S )∨ (S )∨, ± ∼ × × ± We ﬁnd that the mirror G2-manifold J ∨ so obtained is a TCS of the mirrors of the asymptotically cylindrical building blocks X∨. ±

Moreover, the X∨ asymptotic CY cylinders must have the structure±

1 X∨ R+ (S )∨ (S )∨, ± ∼ × × ± 1 where (S )∨ is the T-dual circle and (S )∨ are mirrors of the S as K3 surfaces. ± ± but this argument is meant to be no more than a motivation to look for examples of TCS G2-mirror pairs (J, J ∨) with this a structure.

Remarkably, this structure naturally emerges for G2-manifolds that are twisted connected sums of asymptotically cylindrical Calabi-Yau threefolds constructed from dual tops.

Of course, there are lots of subtleties we are not addressing here (which are in part related with the subtleties in the original SYZ argument and also go beyond), Remarkably, this structure naturally emerges for G2-manifolds that are twisted connected sums of asymptotically cylindrical Calabi-Yau threefolds constructed from dual tops.

Of course, there are lots of subtleties we are not addressing here (which are in part related with the subtleties in the original SYZ argument and also go beyond), but this argument is meant to be no more than a motivation to look for examples of TCS G2-mirror pairs (J, J ∨) with this a structure. Of course, there are lots of subtleties we are not addressing here (which are in part related with the subtleties in the original SYZ argument and also go beyond), but this argument is meant to be no more than a motivation to look for examples of TCS G2-mirror pairs (J, J ∨) with this a structure.

Remarkably, this structure naturally emerges for G2-manifolds that are twisted connected sums of asymptotically cylindrical Calabi-Yau threefolds constructed from dual tops. The construction rests on the polar duality between reﬂexive polytopes.

In the CY case, the largest class of examples of Calabi-Yau manifolds for which a mirror is readily constructed (and in fact the largest class of examples of CY manifolds) is given by CY hypersurfaces and complete intersections in toric varieties.

Batyrev 94, Batyrev-Borisov 94 In the CY case, the largest class of examples of Calabi-Yau manifolds for which a mirror is readily constructed (and in fact the largest class of examples of CY manifolds) is given by CY hypersurfaces and complete intersections in toric varieties.

Batyrev 94, Batyrev-Borisov 94

The construction rests on the polar duality between reﬂexive polytopes. This paper is dedicated to the memory of Max Kreuzer.

1. Introduction To date, the largest class of Calabi-Yau threefolds that has been constructed explicitly, consists of hypersurfaces in toric varieties which are associated to reﬂexive polytopes via the Batyrev construction [1]. Kreuzer and the third author have given a complete list of 473,800,776 such polytopes [2,3]. The Hodge numbers h1,1 and h1,2 play an important role in the classiﬁcation of Calabi-Yau manifolds and in applications of these manifolds to string theory. There are combinatorial formulas for these numbers in terms of the polytopes, that are given in [1]. By computing the Hodge numbers associated to the polytopes in the list, one obtains a list of 30,108 distinct pairs of values for (h1,1,h2,1).

960 720 480 240 0 240 480 720 960

500 500

400 400

300 300

200 200

100 100

0 0 960 720 480 240 0 240 480 720 960

Figure 1: The Hodge plot for the list or reﬂexive 4-polytopes. The Euler number =2 h1,1 h1,2 is plotted against the height y = h1,1 + h1,2. The oblique axes correspond to h1,1 =0and h1,2 =0.

Figure from1 Candelas-Constantin-Skarke 12 the G2-mirror pairs are canonically obtained by switching the roles of the dual tops used in the construction.

Our main result is that a structure analogous to that of the Batyrev mirror map is in place for G2-manifolds that are twisted connected sums of asymptotically cylindrical CY threefolds, whenever these are built from dual pairs of tops: Our main result is that a structure analogous to that of the Batyrev mirror map is in place for G2-manifolds that are twisted connected sums of asymptotically cylindrical CY threefolds, whenever these are built from dual pairs of tops: the G2-mirror pairs are canonically obtained by switching the roles of the dual tops used in the construction. The J and J ∨ so obtained are twisted connected sums of X ± and (X )∨ respectively and the asymptotic CY cylinders± are K3 ﬁbered by mirror pairs of K3s S and (S )∨ respectively. ± ±

In particular, we find that by switching the dual tops in the construction, we obtain G2-mirror pairs that have precisely the structure predicted by our heuristic SYZ-like argument. The J and J ∨ so obtained are twisted connected sums of X ± and (X )∨ respectively ± In particular, we find that by switching the dual tops in the construction, we obtain G2-mirror pairs that have precisely the structure predicted by our heuristic SYZ-like argument. The J and J ∨ so obtained are twisted connected sums of X ± and (X )∨ respectively and the asymptotic CY cylinders± are K3 fibered by mirror pairs of K3s S and (S )∨ respectively. ± ± Our method allows to construct several millions of examples of such pairs, some of which are explicitly discussed in the paper.

Moreover, we are able to show that the pairs of G2 holonomy manifolds so obtained indeed satisfy the Shatashvili-Vafa relation

b2(J) + b3(J) = b2(J ∨) + b3(J ∨). some of which are explicitly discussed in the paper.

Moreover, we are able to show that the pairs of G2 holonomy manifolds so obtained indeed satisfy the Shatashvili-Vafa relation

b2(J) + b3(J) = b2(J ∨) + b3(J ∨).

Our method allows to construct several millions of examples of such pairs, Moreover, we are able to show that the pairs of G2 holonomy manifolds so obtained indeed satisfy the Shatashvili-Vafa relation

b2(J) + b3(J) = b2(J ∨) + b3(J ∨).

Our method allows to construct several millions of examples of such pairs, some of which are explicitly discussed in the paper. It is natural to expect an analogous phenomenon in the context of G2-mirrors. Our construction directly yields

2 4 2 4 H (J, Z) H (J, Z) H (J ∨, Z) H (J ∨, Z) ⊕ ' ⊕

for a G2 mirror pair.

The case of H2(J, Z) H3(J, Z) is in progress. ⊕

CY-mirror symmetry entails the isomorphism of even odd the lattices H (X, Z) and H (X∨, Z). Our construction directly yields

2 4 2 4 H (J, Z) H (J, Z) H (J ∨, Z) H (J ∨, Z) ⊕ ' ⊕

for a G2 mirror pair.

The case of H2(J, Z) H3(J, Z) is in progress. ⊕

CY-mirror symmetry entails the isomorphism of even odd the lattices H (X, Z) and H (X∨, Z). It is natural to expect an analogous phenomenon in the context of G2-mirrors. The case of H2(J, Z) H3(J, Z) is in progress. ⊕

CY-mirror symmetry entails the isomorphism of even odd the lattices H (X, Z) and H (X∨, Z). It is natural to expect an analogous phenomenon in the context of G2-mirrors. Our construction directly yields

2 4 2 4 H (J, Z) H (J, Z) H (J ∨, Z) H (J ∨, Z) ⊕ ' ⊕ for a G2 mirror pair. CY-mirror symmetry entails the isomorphism of even odd the lattices H (X, Z) and H (X∨, Z). It is natural to expect an analogous phenomenon in the context of G2-mirrors. Our construction directly yields

2 4 2 4 H (J, Z) H (J, Z) H (J ∨, Z) H (J ∨, Z) ⊕ ' ⊕ for a G2 mirror pair.

The case of H2(J, Z) H3(J, Z) is in progress. ⊕ It is going to be rather technical – I am sorry about this.

In the rest of the talk I am going to give an overview of the methods we have used to establish the result. In the rest of the talk I am going to give an overview of the methods we have used to establish the result.

It is going to be rather technical – I am sorry about this. A building block Z is ﬁbration

1 π : Z P → whose generic ﬁbre

1 π− (p) S ≡ p is a non-singular K3 surface) that satisﬁes some further technical assumptions.

Building blocks are threefolds which give a remarkably elegant way of producing the asymptotically cylindrical CYs needed in the TCS construction of G2-manifolds. Building blocks are threefolds which give a remarkably elegant way of producing the asymptotically cylindrical CYs needed in the TCS construction of G2-manifolds. A building block Z is ﬁbration

1 π : Z P → whose generic ﬁbre

1 π− (p) S ≡ p is a non-singular K3 surface) that satisfies some further technical assumptions. (i.e. there is no line bundle L such that n L⊗ = [KZ] for any n > 1) and equal to the class of the generic fibre, S:[ K ] = [S] − Z ii.) we may pick a smooth and irreducible fibre S0, such that there is no monodromy upon orbiting around S0, i.e. the fibration is trivial in the vicinity of S0. There is a natural restriction map

2 2 ρ : H (Z, Z) H (S0, Z) →

i.) the anticanonical class of Z is primitive and equal to the class of the generic fibre, S:[ K ] = [S] − Z ii.) we may pick a smooth and irreducible fibre S0, such that there is no monodromy upon orbiting around S0, i.e. the fibration is trivial in the vicinity of S0. There is a natural restriction map

2 2 ρ : H (Z, Z) H (S0, Z) →

i.) the anticanonical class of Z is primitive (i.e. there is no line bundle L such that n L⊗ = [KZ] for any n > 1) ii.) we may pick a smooth and irreducible ﬁbre S0, such that there is no monodromy upon orbiting around S0, i.e. the ﬁbration is trivial in the vicinity of S0. There is a natural restriction map

2 2 ρ : H (Z, Z) H (S0, Z) →

i.) the anticanonical class of Z is primitive (i.e. there is no line bundle L such that n L⊗ = [KZ] for any n > 1) and equal to the class of the generic ﬁbre, S:[ K ] = [S] − Z There is a natural restriction map

2 2 ρ : H (Z, Z) H (S0, Z) →

i.) the anticanonical class of Z is primitive (i.e. there is no line bundle L such that n L⊗ = [KZ] for any n > 1) and equal to the class of the generic fibre, S:[ K ] = [S] − Z ii.) we may pick a smooth and irreducible fibre S0, such that there is no monodromy upon orbiting around S0, i.e. the fibration is trivial in the vicinity of S0. i.) the anticanonical class of Z is primitive (i.e. there is no line bundle L such that n L⊗ = [KZ] for any n > 1) and equal to the class of the generic fibre, S:[ K ] = [S] − Z ii.) we may pick a smooth and irreducible fibre S0, such that there is no monodromy upon orbiting around S0, i.e. the fibration is trivial in the vicinity of S0. There is a natural restriction map

2 2 ρ : H (Z, Z) H (S0, Z) → we demand that the quotient Γ3,19/N is torsion free, i.e. the embedding N, Γ3,19 is primitive → iv.) H3(Z, Z) has no torsion. Under these assumptions, it follows that Z is simply connected and the Hodge numbers H1,0(Z) and H2,0(Z) vanish. As Z is a K3 ﬁbration over P1, the normal bundle of the ﬁbre, and in particular of S0, is trivial.

iii.) Denoting the image of ρ by N, i.e. the embedding N, Γ3,19 is primitive → iv.) H3(Z, Z) has no torsion. Under these assumptions, it follows that Z is simply connected and the Hodge numbers H1,0(Z) and H2,0(Z) vanish. As Z is a K3 ﬁbration over P1, the normal bundle of the ﬁbre, and in particular of S0, is trivial.

iii.) Denoting the image of ρ by N, we demand that the quotient Γ3,19/N is torsion free, iv.) H3(Z, Z) has no torsion. Under these assumptions, it follows that Z is simply connected and the Hodge numbers H1,0(Z) and H2,0(Z) vanish. As Z is a K3 ﬁbration over P1, the normal bundle of the ﬁbre, and in particular of S0, is trivial.

iii.) Denoting the image of ρ by N, we demand that the quotient Γ3,19/N is torsion free, i.e. the embedding N, Γ3,19 is primitive → Under these assumptions, it follows that Z is simply connected and the Hodge numbers H1,0(Z) and H2,0(Z) vanish. As Z is a K3 ﬁbration over P1, the normal bundle of the ﬁbre, and in particular of S0, is trivial.

iii.) Denoting the image of ρ by N, we demand that the quotient Γ3,19/N is torsion free, i.e. the embedding N, Γ3,19 is primitive → iv.) H3(Z, Z) has no torsion. As Z is a K3 ﬁbration over P1, the normal bundle of the ﬁbre, and in particular of S0, is trivial.

iii.) Denoting the image of ρ by N, we demand that the quotient Γ3,19/N is torsion free, i.e. the embedding N, Γ3,19 is primitive → iv.) H3(Z, Z) has no torsion. Under these assumptions, it follows that Z is simply connected and the Hodge numbers H1,0(Z) and H2,0(Z) vanish. iii.) Denoting the image of ρ by N, we demand that the quotient Γ3,19/N is torsion free, i.e. the embedding N, Γ3,19 is primitive → iv.) H3(Z, Z) has no torsion. Under these assumptions, it follows that Z is simply connected and the Hodge numbers H1,0(Z) and H2,0(Z) vanish. As Z is a K3 fibration over P1, the normal bundle of the fibre, and in particular of S0, is trivial. By excising a fibre, we may form the open space

X Z S . ≡ \ 0 The manifold X so obtained is an asymptotically cylindrical CY threefold.

The lattice N naturally embeds into the Picard lattice of S0 and we can think of the ﬁbres as being elements of a family of lattice polarized K3 surfaces with polarizing lattice N. The manifold X so obtained is an asymptotically cylindrical CY threefold.

The lattice N naturally embeds into the Picard lattice of S0 and we can think of the ﬁbres as being elements of a family of lattice polarized K3 surfaces with polarizing lattice N. By excising a ﬁbre, we may form the open space

X Z S . ≡ \ 0 The lattice N naturally embeds into the Picard lattice of S0 and we can think of the ﬁbres as being elements of a family of lattice polarized K3 surfaces with polarizing lattice N. By excising a ﬁbre, we may form the open space

X Z S . ≡ \ 0 The manifold X so obtained is an asymptotically cylindrical CY threefold. The geometric data encoding the glueing of X translates into combinatorial conditions about± the embedding of the lattices N of the two 2 ± building blocks Z within H (S0, Z). ± In particular Hk(J, Z) are completely determined.

Building blocks are an extremely useful technical tool to form TCS G2-manifolds J. In particular Hk(J, Z) are completely determined.

Building blocks are an extremely useful technical tool to form TCS G2-manifolds J. The geometric data encoding the glueing of X translates into combinatorial conditions about± the embedding of the lattices N of the two 2 ± building blocks Z within H (S0, Z). ± Building blocks are an extremely useful technical tool to form TCS G2-manifolds J. The geometric data encoding the glueing of X translates into combinatorial conditions about± the embedding of the lattices N of the two 2 ± building blocks Z within H (S0, Z). ± In particular Hk(J, Z) are completely determined. 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 ⊕ ⊕ ⊕ H (J, Z) = H (S) (T+ T ) − 3⊕,19 ∩ 3,19 Γ /(N + T+) Γ /(N+ + T ) − − ⊕ 3 3 ⊕ H (Z+) H (Z ) K+∗ K∗ ⊕ ⊕ − ⊕ ⊕ − Here, T are the complements of N in 2 ± ± H (S0, Z), the group K is defined as K ker(ρ)/[S ] and K∗ is its dual. ≡ 0 In this construction, the polytope ∆ is the Newton polytope giving rise to all of the monomials of the defining equation and the polytope ∆◦, after an appropriate triangulation, defines the toric ambient space.

A pair of lattice polytopes (∆, ∆◦) satisfying

∆, ∆◦ 1 h i ≥ − under the canonical pairing on Rn are called reﬂexive and deﬁne a Calabi-Yau manifold X(∆,∆◦) embedded as a hypersurfaces in a toric variety. A pair of lattice polytopes (∆, ∆◦) satisfying

∆, ∆◦ 1 h i ≥ − under the canonical pairing on Rn are called reflexive and define a Calabi-Yau manifold X(∆,∆◦) embedded as a hypersurfaces in a toric variety. In this construction, the polytope ∆ is the Newton polytope giving rise to all of the monomials of the defining equation and the polytope ∆◦, after an appropriate triangulation, defines the toric ambient space. A top ♦◦ is defined as a bounded lattice polytope (w.r.t. a lattice N) defined by relations

m , ◦ 1 m , ◦ 0 h i ♦ i ≥ − h 0 ♦ i ≥

for a set of (primitive) lattice points mi and m0, all sitting in the dual lattice M. The last relation deﬁnes a hyperplane F and ◦ F must be a ♦ ∩ reﬂexive polytope ∆F◦ .

The building blocks used in the construction of G2 manifolds as twisted connected sums can be obtained in a similar way from a pair of four-dimensional projecting tops ♦, ♦◦. The last relation deﬁnes a hyperplane F and ◦ F must be a ♦ ∩ reﬂexive polytope ∆F◦ .

The building blocks used in the construction of G2 manifolds as twisted connected sums can be obtained in a similar way from a pair of four-dimensional projecting tops ♦, ♦◦.

A top ♦◦ is deﬁned as a bounded lattice polytope (w.r.t. a lattice N) deﬁned by relations

m , ◦ 1 m , ◦ 0 h i ♦ i ≥ − h 0 ♦ i ≥ for a set of (primitive) lattice points mi and m0, all sitting in the dual lattice M. The building blocks used in the construction of G2 manifolds as twisted connected sums can be obtained in a similar way from a pair of four-dimensional projecting tops ♦, ♦◦.

A top ♦◦ is deﬁned as a bounded lattice polytope (w.r.t. a lattice N) deﬁned by relations

m , ◦ 1 m , ◦ 0 h i ♦ i ≥ − h 0 ♦ i ≥ for a set of (primitive) lattice points mi and m0, all sitting in the dual lattice M. The last relation defines a hyperplane F and ◦ F must be a ♦ ∩ reflexive polytope ∆F◦ . Let us specialize to our case of interest, in which N and M are four-dimensional. We may always exploit SL(4, Z) to fix m0 = (0, 0, 0, 1), and a top with this choice of m0 is called projecting if the projection π4 forgetting the fourth coordinate maps π ( ◦) ∆◦ . 4 ♦ ⊇ F

Tops appear naturally as halves of reflexive polytopes defining Calabi-Yau hypersurfaces which are fibred by a Calabi-Yau hypersurface of one dimension lower, which is in turn defined by the reflexive pair (∆F , ∆F◦ ).

Avram-Kreuzer-Mandelberg 97, Candelas-Font 98 We may always exploit SL(4, Z) to ﬁx m0 = (0, 0, 0, 1), and a top with this choice of m0 is called projecting if the projection π4 forgetting the fourth coordinate maps π ( ◦) ∆◦ . 4 ♦ ⊇ F

Avram-Kreuzer-Mandelberg 97, Candelas-Font 98

Let us specialize to our case of interest, in which N and M are four-dimensional. Tops appear naturally as halves of reflexive polytopes defining Calabi-Yau hypersurfaces which are fibred by a Calabi-Yau hypersurface of one dimension lower, which is in turn defined by the reflexive pair (∆F , ∆F◦ ).

Avram-Kreuzer-Mandelberg 97, Candelas-Font 98

Let us specialize to our case of interest, in which N and M are four-dimensional. We may always exploit SL(4, Z) to ﬁx m0 = (0, 0, 0, 1), and a top with this choice of m0 is called projecting if the projection π4 forgetting the fourth coordinate maps π ( ◦) ∆◦ . 4 ♦ ⊇ F Here, our notation ◦ is meant to indicate ‘dual’ in the sense of the above relation rather than ‘polar dual’.

For any projecting top ◦ with ◦ F = ∆◦ , ♦ ♦ ∩ F F = m0⊥, there is a dual top ♦ satisfying:

, ◦ 1 , ν 0 h♦ ♦ i ≥ − h♦ 0i ≥ with ν = (0, 0, 0, 1). 0 − For any projecting top ◦ with ◦ F = ∆◦ , ♦ ♦ ∩ F F = m0⊥, there is a dual top ♦ satisfying:

, ◦ 1 , ν 0 h♦ ♦ i ≥ − h♦ 0i ≥ with ν0 = (0, 0, 0, 1). Here, our notation ◦ is meant to indicate− ‘dual’ in the sense of the above relation rather than ‘polar dual’. A generic section of (D ) deﬁnes a smooth O ♦ hypersurface Z(♦,♦◦), given by

m,ν0 m,νi +1 ◦ h i h i Z(♦,♦ ) : 0 = z0 zi m νi X∈♦ Y

For a projecting top, ∆◦ = ◦ F and F ♦ ∩ ∆F = ♦ F are a reflexive pair, thus this hypersurface∩ is fibred by a K3 surface which is defined by the reflexive pair (∆F , ∆F◦ ).

As a convex lattice polytope, ♦ defines a toric variety T , as well as a line bundle (D ) on T . ♦ O ♦ ♦ For a projecting top, ∆◦ = ◦ F and F ♦ ∩ ∆F = ♦ F are a reflexive pair, thus this hypersurface∩ is fibred by a K3 surface which is defined by the reflexive pair (∆F , ∆F◦ ).

As a convex lattice polytope, ♦ deﬁnes a toric variety T♦, as well as a line bundle (D♦) on T♦. A generic section of (D ) deﬁnesO a smooth O ♦ hypersurface Z(♦,♦◦), given by

m,ν0 m,νi +1 ◦ h i h i Z(♦,♦ ) : 0 = z0 zi m νi X∈♦ Y thus this hypersurface is fibred by a K3 surface which is defined by the reflexive pair (∆F , ∆F◦ ).

As a convex lattice polytope, ♦ deﬁnes a toric variety T♦, as well as a line bundle (D♦) on T♦. A generic section of (D ) deﬁnesO a smooth O ♦ hypersurface Z(♦,♦◦), given by

m,ν0 m,νi +1 ◦ h i h i Z(♦,♦ ) : 0 = z0 zi m νi X∈♦ Y

For a projecting top, ∆F◦ = ♦◦ F and ∆ = F are a reflexive pair,∩ F ♦ ∩ As a convex lattice polytope, ♦ defines a toric variety T♦, as well as a line bundle (D♦) on T♦. A generic section of (D ) definesO a smooth O ♦ hypersurface Z(♦,♦◦), given by

m,ν0 m,νi +1 ◦ h i h i Z(♦,♦ ) : 0 = z0 zi m νi X∈♦ Y

For a projecting top, ∆◦ = ◦ F and F ♦ ∩ ∆F = ♦ F are a reflexive pair, thus this hypersurface∩ is fibred by a K3 surface which is defined by the reflexive pair (∆F , ∆F◦ ). Consider a G2-manifold J which is a twisted connected sum of the asymptotically cylindrical CY three-folds

X Z( , ◦ ) S ± ≡ ♦± ♦± \ ±

The G2-mirror of J is a TCS of the asymptotically cylindrical CY three-folds

(X )∨ Z( ◦ , ) (S )∨ ± ≡ ♦± ♦± \ ±

The CY-mirror symmetry for hypersurfaces within toric varieties is realized by

X(∆,∆◦) = (X(∆◦,∆))∨. The G2-mirror of J is a TCS of the asymptotically cylindrical CY three-folds

(X )∨ Z( ◦ , ) (S )∨ ± ≡ ♦± ♦± \ ±

The CY-mirror symmetry for hypersurfaces within toric varieties is realized by

X(∆,∆◦) = (X(∆◦,∆))∨.

Consider a G2-manifold J which is a twisted connected sum of the asymptotically cylindrical CY three-folds

X Z( , ◦ ) S ± ≡ ♦± ♦± \ ± The CY-mirror symmetry for hypersurfaces within toric varieties is realized by

X(∆,∆◦) = (X(∆◦,∆))∨.

Consider a G2-manifold J which is a twisted connected sum of the asymptotically cylindrical CY three-folds

X Z( , ◦ ) S ± ≡ ♦± ♦± \ ±

The G2-mirror of J is a TCS of the asymptotically cylindrical CY three-folds

(X )∨ Z( ◦ , ) (S )∨ ± ≡ ♦± ♦± \ ± Notice that swapping the role of the two tops, one obtains automatically asymptotic CY cylinders which are fibered by the mirror K3 in the asymptotic region. In particular, the fibres of the mirror building blocks Z(♦,♦◦) and Z(♦◦,♦) are from algebraic mirror families of K3 surfaces. With the methods discussed above we have constructed several millions of novel examples of G2-mirror manifolds. There is lots of work to be done: sigma-models, topological G2-strings, branes, supersymmetric field theory... Stay tuned!

Thank you for you attention.

Our work brings evidence for the G2-mirror symmetry conjecture. There is lots of work to be done: sigma-models, topological G2-strings, branes, supersymmetric ﬁeld theory... Stay tuned!

Thank you for you attention.

Our work brings evidence for the G2-mirror symmetry conjecture. With the methods discussed above we have constructed several millions of novel examples of G2-mirror manifolds. There is lots of work to be done: sigma-models, topological G2-strings, branes, supersymmetric ﬁeld theory... Stay tuned! Our work brings evidence for the G2-mirror symmetry conjecture. With the methods discussed above we have constructed several millions of novel examples of G2-mirror manifolds. There is lots of work to be done: sigma-models, topological G2-strings, branes, supersymmetric ﬁeld theory... Stay tuned!

Thank you for you attention.