Exceptional gauge groups in F-theory models without section Yusuke Kimura

arXiv: 1511.06912 1603.03212 1607.02978 1608.07219

Talk Plan F-theory models with section, without section, physical implications Models without section –constructions: hypersurfaces, double covers Non-Abelian Gauge Symmetries on 7-branes, enhancement to Exceptional Gauge Groups Jacobian Fibrations, U(1) Gauge Field and Mordell-Weil Groups

1. F-theory models with section, without section, physical implications

Brief review of F-theory F-theory: non-perturbative extension of type IIB superstring

Brief review of F-theory F-theory: non-perturbative extension of type IIB superstring F-theory is compactified on genus-one fibered Calabi-Yau n-fold

Brief review of F-theory F-theory: non-perturbative extension of type IIB superstring F-theory is compactified on genus-one fibered Calabi-Yau n-fold genus-one fiber is usually considered auxiliary

Brief review of F-theory F-theory: non-perturbative extension of type IIB superstring F-theory is compactified on genus-one fibered Calabi-Yau n-fold genus-one fiber is usually considered auxiliary The complex structure of a genus-one fiber is identified with axio-dilation

7-branes, non-Abelian gauge groups Genus-one fibered CY n-fold E 𝑌 𝑛 𝐵 𝑛−1

7-branes, non-Abelian gauge groups Genus-one fibered CY n-fold E 𝑌 𝑛 𝐵 𝑛−1 Genus-one fibers degenerate over codim-1 locus in base 𝐵 𝑛−1

Such a locus is called the discriminant locus

Such a locus is called the discriminant locus 7-branes are wrapped on each component of the discriminant locus

Such a locus is called the discriminant locus 7-branes are wrapped on each component of the discriminant locus Non-Abelian gauge groups on the 7-branes are in correspondence with degenerate fibers Types of singular fibers: Kodaira-Néron classification

Section to Elliptic Fibration

Section to Elliptic Fibration pick one point in each fiber

Section to Elliptic Fibration move point throughout the base = section to the fibration

Section to Elliptic Fibration Thus, a section meets a fiber in one point

Section to Elliptic Fibration Thus, a section meet a fiber in one point section fiber

General genus-one fibration General genus-one fibrations do not have a global section

General genus-one fibration General genus-one fibrations do not have a global section But they always admit a multisection to the fibration

multisection Multisection of degree n is often referred to as an ``n-section’’

multisection Multisection of degree n is often referred to as an ``n-section’’ n-section intersects with a fiber in n points multisection fiber

Physical meaning moduli of genus-one fibrations

Physical meaning elliptic fibrations with a section moduli of genus-one fibrations

Physical meaning Elliptic fibration with a section has the Mordell-Weil group MW group = group of sections rk MW = # U(1) Morrison-Vafa ‘96 e.g. when MW ≅ 𝑍 2 , there is 𝑈 1 2

Physical meaning Morrison-Taylor ‘14 argued that moving from an elliptic fibration with a section to a genus-one fibration without a section can be viewed as a Higgsing process

Physical meaning Morrison-Taylor ‘14 argued that moving from an elliptic fibration with a section to a genus-one fibration without a section can be viewed as a Higgsing process in which U(1) symmetry breaks and only a discrete gauge symmetry remains

Physical meaning Morrison-Taylor ‘14 argued that moving from an elliptic fibration with a section to a genus-one fibration without a section can be viewed as a Higgsing process in which U(1) symmetry breaks and only a discrete gauge symmetry remains For example, for F-theory model with an n-section, 𝑈 1 𝑛−1 breaks to a discrete 𝑍 𝑛 symmetry

2.Models without section

Two constructions We construct CY genus-one fibrations without a section

Two constructions We construct CY genus-one fibrations without a section We consider two constructions

Two constructions We construct CY genus-one fibrations without a section We consider two constructions -hypersurfaces in product of projective spaces -double covers of product of projective spaces

Hypersurface Construction We consider hypersurfaces in products of projective spaces to construct CY spaces

Hypersurface Construction We consider hypersurfaces in products of projective spaces to construct CY spaces K3 surface: (3,2) hypersurface in 𝑃 2 × 𝑃 1 CY 4-fold: (3,2,2,2) hypersurface in 𝑃 2 × 𝑃 1 × 𝑃 1 × 𝑃 1

Hypersurface Construction (3,2) hypersurface S in 𝑃 2 × 𝑃 1 has natural projection onto 𝑃 1

Hypersurface Construction (3,2) hypersurface S in 𝑃 2 × 𝑃 1 has natural projection onto 𝑃 1 Fiber of the projection is cubic hypersurface in 𝑃 2 which is a genus-one curve

Hypersurface Construction (3,2) hypersurface S in 𝑃 2 × 𝑃 1 has natural projection onto 𝑃 1 Fiber of the projection is cubic hypersurface in 𝑃 2 which is a genus-one curve Thus, (3,2) hypersurface admits genus-one fibration

Hypersurface Construction Generic (3,2) hypersurface has Néron-Severi lattice 0 3 3 2 Thus, it has a 3-section, but it lacks a global section

Hypersurface Construction (3,2,2,2) hypersurface in 𝑃 2 × 𝑃 1 × 𝑃 1 × 𝑃 1 admits genus-one fibration in a similar fashion Base 3-fold is 𝑃 1 × 𝑃 1 × 𝑃 1

Hypersurface Construction (3,2,2,2) hypersurface in 𝑃 2 × 𝑃 1 × 𝑃 1 × 𝑃 1 admits genus-one fibration in a similar fashion Base 3-fold is 𝑃 1 × 𝑃 1 × 𝑃 1 Projection onto 𝑃 1 × 𝑃 1 also gives a K3 fibration

Hypersurface Construction We particularly consider specific form of equations, which we call `Fermat type’

Hypersurface Construction We particularly consider specific form of equations, which we call `Fermat type’ Fermat type K3 surface 𝑡− 𝛼 1 𝑡− 𝛼 2 𝑋 3 + 𝑡− 𝛼 3 𝑡− 𝛼 4 𝑌 3 + 𝑡− 𝛼 5 𝑡− 𝛼 6 𝑍 3 =0

Hypersurface Construction We particularly consider specific form of equations, which we call `Fermat type’ Fermat type K3 surface 𝑡− 𝛼 1 𝑡− 𝛼 2 𝑋 3 + 𝑡− 𝛼 3 𝑡− 𝛼 4 𝑌 3 + 𝑡− 𝛼 5 𝑡− 𝛼 6 𝑍 3 =0 [X:Y:Z] coord. on 𝑃 2 t coord. on 𝑃 1

Hypersurface Construction (3,2,2,2) CY hypersurface of Fermat type 𝑡− 𝛼 1 𝑡− 𝛼 2 𝑓 𝑋 3 + 𝑡− 𝛼 3 𝑡− 𝛼 4 𝑔 𝑌 3 + 𝑡− 𝛼 5 𝑡− 𝛼 6 ℎ 𝑍 3 =0

Hypersurface Construction (3,2,2,2) CY hypersurface of Fermat type 𝑡− 𝛼 1 𝑡− 𝛼 2 𝑓 𝑋 3 + 𝑡− 𝛼 3 𝑡− 𝛼 4 𝑔 𝑌 3 + 𝑡− 𝛼 5 𝑡− 𝛼 6 ℎ 𝑍 3 =0 [X:Y:Z] coord. on 𝑃 2 t coord. on 𝑃 1 f,g,h : bidegree (2,2) polynomials on 𝑃 1 × 𝑃 1 𝑃 2 × 𝑃 1 × 𝑃 1 × 𝑃 1

Hypersurface Construction (3,2,2,2) CY hypersurface of Fermat type 𝑡− 𝛼 1 𝑡− 𝛼 2 𝑓 𝑋 3 + 𝑡− 𝛼 3 𝑡− 𝛼 4 𝑔 𝑌 3 + 𝑡− 𝛼 5 𝑡− 𝛼 6 ℎ 𝑍 3 =0 K3 fiber is Fermat type (3,2) hypersurface, which lacks a global section, therefore (3,2,2,2) hypersurface of Fermat type does not have a rational section

Double Cover Construction We consider double covers of products of projective spaces to construct CY spaces

Double Cover Construction We consider double covers of products of projective spaces to construct CY spaces K3 surface: double covers of 𝑃 1 × 𝑃 1 branched along a (4,4) curve CY 4-fold: double covers of 𝑃 1 × 𝑃 1 × 𝑃 1 × 𝑃 1 branched along a (4,4,4,4) 3-fold

Double Cover Construction Double cover of 𝑃 1 × 𝑃 1 branched along a (4,4) curve has a natural projection onto 𝑃 1 Fiber of this projection is a double cover of 𝑃 1 branched over 4 points, which is a genus-one curve Thus, the projection gives a genus-one fibration

Double Cover Construction Generic double cover of 𝑃 1 × 𝑃 1 branched along a (4,4) curve has Néron-Severi lattice 0 2 2 0 Therefore, it has a bisection, but it lacks a global section.

Double Cover Construction Double cover of 𝑃 1 × 𝑃 1 × 𝑃 1 × 𝑃 1 branched along a (4,4,4,4) 3-fold admits a genus-one fibration in a similar fashion Base 3-fold is 𝑃 1 × 𝑃 1 × 𝑃 1 Projection onto 𝑃 1 × 𝑃 1 also gives a K3 fibration

Double Cover Construction We particularly consider double covers given by specific form of equations

Double Cover Construction We consider K3 constructions as double covers given by the following form of equations: 𝜏 2 =𝑓(𝑡) 𝑥 4 +𝑔(𝑡) x, t are inhomogeneous coordinates on 𝑃 1 ′ 𝑠 f(t), g(t) are polynomials in t of degree 4

Double Cover Construction By splitting polynomials f(t), g(t) into linear factors, the equation can be rewritten as 𝜏 2 = 𝑖=1 4 (𝑡− 𝛼 𝑖 ) 𝑥 4 + 𝑗=5 8 (𝑡− 𝛼 𝑗 )

Double Cover Construction We similarly consider CY 4-fold constructions as double covers given by the following form of equations 𝜏 2 =𝑓 𝑖=1 4 (𝑡− 𝛼 𝑖 ) 𝑥 4 +𝑔 𝑗=5 8 (𝑡− 𝛼 𝑗 )

Double Cover Construction We similarly consider CY 4-fold constructions as double covers given by the following form of equations 𝜏 2 =𝑓 𝑖=1 4 (𝑡− 𝛼 𝑖 ) 𝑥 4 +𝑔 𝑗=5 8 (𝑡− 𝛼 𝑗 ) t coord. on 𝑃 1 x coord. on 𝑃 1 f,g bidegree (4,4) polynomials on 𝑃 1 × 𝑃 1 𝑃 1 × 𝑃 1 × 𝑃 1 × 𝑃 1

Double Cover Construction We similarly consider CY 4-fold constructions as double covers given by the following form of equations 𝜏 2 =𝑓 𝑖=1 4 (𝑡− 𝛼 𝑖 ) 𝑥 4 +𝑔 𝑗=5 8 (𝑡− 𝛼 𝑗 ) K3 fiber is double cover of 𝑃 1 × 𝑃 1 ramified along a (4,4) curve double cover of 𝑃 1 × 𝑃 1 × 𝑃 1 × 𝑃 1 branched along a (4,4,4,4) 3-fold, given by the above equation, does not admit a rational section

3. Non-Abelian Gauge Symmetries on 7-branes, enhancement to Exceptional Gauge Groups

Fermat type K3 hypersurface Singular fibers of Fermat type K3 hypersurface 𝑡− 𝛼 1 𝑡− 𝛼 2 𝑋 3 + 𝑡− 𝛼 3 𝑡− 𝛼 4 𝑌 3 + 𝑡− 𝛼 5 𝑡− 𝛼 6 𝑍 3 =0 are six type IV fibers at 𝑡= 𝛼 𝑖 (𝑖=1,2,…,6)

Fermat type K3 hypersurface Singular fibers of Fermat type K3 hypersurface 𝑡− 𝛼 1 𝑡− 𝛼 2 𝑋 3 + 𝑡− 𝛼 3 𝑡− 𝛼 4 𝑌 3 + 𝑡− 𝛼 5 𝑡− 𝛼 6 𝑍 3 =0 are six type IV fibers at 𝑡= 𝛼 𝑖 (𝑖=1,2,…,6) Therefore, non-Abelian gauge symmetries are 𝑆𝑈(3)

Fermat type K3 hypersurface When the multiplicity of 𝛼 𝑖 is 2, e.g. 𝛼 1 = 𝛼 2 , two type IV fibers collide and they are enhanced to type 𝐼 𝑉 ∗ fiber Corresponding gauge symmetry on the 7-branes is exceptional 𝐸 6 gauge group

Fermat type CY hypersurface We next consider (3,2,2,2) CY hypersurface of Fermat type

Fermat type CY hypersurface We next consider (3,2,2,2) CY hypersurface of Fermat type 𝑡− 𝛼 1 𝑡− 𝛼 2 𝑓 𝑋 3 + 𝑡− 𝛼 3 𝑡− 𝛼 4 𝑔 𝑌 3 + 𝑡− 𝛼 5 𝑡− 𝛼 6 ℎ 𝑍 3 =0

Fermat type CY hypersurface We next consider (3,2,2,2) CY hypersurface of Fermat type 𝑡− 𝛼 1 𝑡− 𝛼 2 𝑓 𝑋 3 + 𝑡− 𝛼 3 𝑡− 𝛼 4 𝑔 𝑌 3 + 𝑡− 𝛼 5 𝑡− 𝛼 6 ℎ 𝑍 3 =0 Discriminant components are 𝑡= 𝛼 𝑖 𝑖=1,2,…,6 𝑓=0, 𝑔=0, ℎ=0 For generic equations, singular fibers on these components are type IV

Fermat type CY hypersurface We next consider (3,2,2,2) CY hypersurface of Fermat type 𝑡− 𝛼 1 𝑡− 𝛼 2 𝑓 𝑋 3 + 𝑡− 𝛼 3 𝑡− 𝛼 4 𝑔 𝑌 3 + 𝑡− 𝛼 5 𝑡− 𝛼 6 ℎ 𝑍 3 =0 Discriminant components are 𝑡= 𝛼 𝑖 𝑖=1,2,…,6 𝑓=0, 𝑔=0, ℎ=0 For generic equations, singular fibers on these components are type IV Therefore, gauge symmetries are 𝑆𝑈(3)

Fermat type CY hypersurface When the multiplicity of 𝛼 𝑖 is 2, e.g. 𝛼 1 = 𝛼 2 , components t= 𝛼 1 and t= 𝛼 2 become coincident, and fiber type on the component is enhanced to type 𝐼 𝑉 ∗ Corresponding gauge symmetry on the 7-branes is exceptional 𝐸 6 gauge group

Double Covers Singular fibers of K3 surface constructed as double cover 𝜏 2 = 𝑖=1 4 (𝑡− 𝛼 𝑖 ) 𝑥 4 + 𝑗=5 8 (𝑡− 𝛼 𝑗 ) are at 𝑡= 𝛼 𝑖 (𝑖=1,2,…,8)

Double Covers Singular fibers of K3 surface constructed as double cover 𝜏 2 = 𝑖=1 4 (𝑡− 𝛼 𝑖 ) 𝑥 4 + 𝑗=5 8 (𝑡− 𝛼 𝑗 ) are at 𝑡= 𝛼 𝑖 (𝑖=1,2,…,8) Fibers are type III, therefore gauge symmetries are 𝑆𝑈(2)

Double Covers When three type III fibers collide, e.g. when 𝛼 1 = 𝛼 2 = 𝛼 3 , fiber type is enhanced to 𝐼𝐼 𝐼 ∗ Corresponding gauge symmetry is exceptional 𝐸 7 gauge group

Double Covers We next consider CY 4-fold constructed as double cover of 𝑃 1 × 𝑃 1 × 𝑃 1 × 𝑃 1 𝜏 2 =𝑓 𝑖=1 4 (𝑡− 𝛼 𝑖 ) 𝑥 4 +𝑔 𝑗=5 8 (𝑡− 𝛼 𝑗 )

Double Covers We next consider CY 4-fold constructed as double cover of 𝑃 1 × 𝑃 1 × 𝑃 1 × 𝑃 1 𝜏 2 =𝑓 𝑖=1 4 (𝑡− 𝛼 𝑖 ) 𝑥 4 +𝑔 𝑗=5 8 (𝑡− 𝛼 𝑗 ) Discriminant components are 𝑡= 𝛼 𝑖 𝑖=1,2,…,8 𝑓=0, 𝑔=0

Double Covers Singular fibers on discriminant components are type III fibers Therefore, 𝑆𝑈(2) gauge symmetries arise on the 7-branes

Double Covers When three type III fibers collide, e.g. 𝛼 1 = 𝛼 2 = 𝛼 3 , three discriminant components 𝑡= 𝛼 1 , 𝑡= 𝛼 2 , 𝑡= 𝛼 3 become coincident, and fiber type is enhanced to type 𝐼𝐼 𝐼 ∗ Corresponding gauge symmetry is exceptional 𝐸 7 gauge group

4. Jacobian Fibrations, U(1) Gauge Field and Mordell-Weil Groups

Jacobian fibration and U(1) Number of U(1) factors is equal to the rank of MW group But, for F-theory model without a section, compactification space does not have the Mordell-Weil group

Jacobian fibration and U(1) Number of U(1) factors is equal to the rank of MW group But, for F-theory model without a section, compactification space does not have the Mordell-Weil group Braun-Morrison ‘14 For F-theory model without a section, # U(1) = rk MW(J(Y)) where J(Y) is the Jacobian fibration of the compactification space Y

Jacobian fibration and U(1) For some specific models, we can determine the Mordell-Weil groups of the Jacobian fibrations Thus, we can determine U(1) factors in gauge symmetries in those models without a section by computing the Mordell-Weil groups of the Jacobian fibrations

Fermat type CY hypersurface We consider Fermat type (3,2,2,2) hypersurface 𝑡− 𝛼 1 2 𝑓 𝑋 3 + 𝑡− 𝛼 2 2 𝑔 𝑌 3 + 𝑡− 𝛼 3 2 ℎ 𝑍 3 =0

Fermat type CY hypersurface We consider Fermat type (3,2,2,2) hypersurface 𝑡− 𝛼 1 2 𝑓 𝑋 3 + 𝑡− 𝛼 2 2 𝑔 𝑌 3 + 𝑡− 𝛼 3 2 ℎ 𝑍 3 =0 Gauge symmetry is 𝐸 6 3 ×𝑆𝑈 3 3

Fermat type CY hypersurface We consider Fermat type (3,2,2,2) hypersurface 𝑡− 𝛼 1 2 𝑓 𝑋 3 + 𝑡− 𝛼 2 2 𝑔 𝑌 3 + 𝑡− 𝛼 3 2 ℎ 𝑍 3 =0 Gauge symmetry is 𝐸 6 3 ×𝑆𝑈 3 3 Jacobian fibration is given by 𝑦 2 = 𝑥 3 − 2 4 3 3 𝑖=1 3 𝑡− 𝛼 𝑖 4 𝑓 2 𝑔 2 ℎ 2

Fermat type CY hypersurface The Mordell-Weil group of the Jacobian fibration is 𝑍 3 (The global structure of the non-Abelian gauge symmetry is therefore 𝐸 6 3 ×𝑆𝑈 3 3 / 𝑍 3 ) rk MW(J(Y)) = 0, thus this F-theory model does not have a U(1) gauge field

CY 4-fold constructed as Double Cover Next, we consider CY 4-fold constructed as double cover of 𝑃 1 × 𝑃 1 × 𝑃 1 × 𝑃 1 ramified over a (4,4,4,4) 3-fold 𝜏 2 =𝑓 𝑡− 𝛼 1 3 (𝑡− 𝛼 2 ) 𝑥 4 +𝑔 𝑡− 𝛼 2 𝑡− 𝛼 3 3

CY 4-fold constructed as Double Cover Next, we consider CY 4-fold constructed as double cover of 𝑃 1 × 𝑃 1 × 𝑃 1 × 𝑃 1 ramified over a (4,4,4,4) 3-fold 𝜏 2 =𝑓 𝑡− 𝛼 1 3 (𝑡− 𝛼 2 ) 𝑥 4 +𝑔 𝑡− 𝛼 2 𝑡− 𝛼 3 3 Gauge symmetry is 𝐸 7 2 ×𝑆𝑂 7 ×𝑆𝑈 2 2

CY 4-fold constructed as Double Cover Next, we consider CY 4-fold constructed as double cover of 𝑃 1 × 𝑃 1 × 𝑃 1 × 𝑃 1 ramified over a (4,4,4,4) 3-fold 𝜏 2 =𝑓 𝑡− 𝛼 1 3 (𝑡− 𝛼 2 ) 𝑥 4 +𝑔 𝑡− 𝛼 2 𝑡− 𝛼 3 3 Gauge symmetry is 𝐸 7 2 ×𝑆𝑂 7 ×𝑆𝑈 2 2 Jacobian fibration is given by 𝑦 2 = 1 4 𝑥 3 − 𝑡− 𝛼 1 3 𝑡− 𝛼 2 2 𝑡− 𝛼 3 3 𝑓𝑔⋅𝑥

CY 4-fold constructed as Double Cover MW group of the Jacobian fibration is 𝑍 2 (The global structure of the gauge symmetry is 𝐸 7 2 ×𝑆𝑂 7 ×𝑆𝑈 2 2 / 𝑍 2 ) rk MW(J(Y)) = 0, thus this F-theory model does not have a U(1) gauge field