of february, 2015 toric and non-toric higgs branches of f

Toric and Non-Toric Higgs Branches of F-theory

Denis Klevers

2nd Physics and Geometry of F-theory Workshop, Munich 23rd of February, 2015

arXiv:1408.4808: D.K., D. Mayorga Peña, P. Oehlmann, H. Piragua, J. ReuterarXiv:1502.nnnn: M. Cvetič, D.K., D. Mayorga Peña, P. Oehlmann, J. ReuterarXiv:1503.nnnn: M. Cvetič, D.K., H. Piragua, W. Taylor

Motivation

Why important to think about F-theory moduli space?

➡ Make statements about classes of vacua & generic features of their associated effective theories.

1) Structure of space of F-theory vacua ✤ Finiteness of F-theory landscape?✤ Connectedness/Stratification of F-theory moduli space?

2) Transition mechanisms between different branches✤ Apply transition mechanisms to predict physics: uncover residual structures, e.g.

massive U(1)’s, discrete gauge symmetries,…✤ Particularly interesting is the Higgs effect: new light states, symmetry (gauge)

enhancements (for phenomenology),…

3) Interplay Physics/Math: ✤ Hint for new CY-manifolds beyond toric geometry & CICY…➡ Enlarge set of known F-theory vacua.✤ Extend physics/geometry dictionary to incorporate new properties of vacua:

MW-group, TS-group, new non-Abelian gauge structures…

1. Construct class of CY-manifolds with all 16 toric hypersurface fibers✤ Determine intrinsic gauge groups, matter contents & Yukawas.✤ Appearance of new features: discrete gauge groups, non-toric U(1)’s, matter

with higher U(1)-charges.✤ “Toric Higgs branch”: study network of Higgsings between these models.✤ “Toric unification”: construct explicit 4D three-family particle physics models.

2. Analyze Higgs branch of rank two non-Abelian gauge groups in F-theory✤ Propose & analyze new non-toric model with U(1)2 gauge group that is needed

to describe this Higgs branch.✤ Find SU(2)xSU(2)xSU(3) gauge theory underlying general U(1)2 model.✤ Discover first concrete geometric realization of SU(3) gauge theories with

symmetric matter reps., motivated by U(1)2 model.

Goals of this talk

Zn

OutlineI. F-theory on Toric Hypersurface Fibrations

1) Constructing CY-manifolds with toric hypersurface fibers

2) The effective theories of F-theory

3) The toric Higgs branch of F-theory

4) Three family models in toric unification

II. Non-toric Higgs Branches of F-theory with rank two gauge groups

1) A more general model with U(1)2

2) The effective theory of F-theory

3) The non-toric Higgs branch & novel non-Abelian gauge structures

III. Conclusions & Outlook

I. F-theory on Toric Hypersurface FibrationsElliptic CY-manifolds with other toric fibers for F-theory: [Aldazabal,Font,Ibanez,Uranga;Klemm,Mayr,Vafa;Candelas,Font;Klemm,Lian,Roan,Yau;….…]A lot of recent activity: [Grimm,Weigand;Grimm,Hayashi;Morrison,Park;Braun,Grimm,Keitel;Borchmann,Mayrhofer,Palti,Weigand; Cvetič,Klevers,Piragua;Grimm,Kapfer,Keitel;Cvetič,Grassi,Klevers,Piragua;Küntzler,SchäferNameki; CaboBizet,Klemm,Lopes;Braun,Morrison;Morrison,Taylor;Mayrhofer,Morrison,Till,Weigand;Anderson,Haupt,Lukas;Anderson,GarciaEtxebarria,Grimm,Keitel;Mayrhofer,Palti,Till,Weigand;GarciaEtxebarria,Grimm,Keitel;Lawrie,Sacco;…]

1. Base B of X

✤ here: do not choose specific B ➡ keep analysis base-independent

2. Today: focus on torus fiber T2 of X

✤ for applications: T2 = algebraic curve of genus one

✤ here: has natural presentation as Calabi-Yau hypersurface in the 2D toric varieties associated to reflexive polyhedra.

Building blocks of torus-fibered Calabi-Yau X

C

B

T 2

X

8>>>>>><

>>>>>>:

C

Classification of B: [Morrison,Taylor;…] Application to 6D SCFTs: [Heckman,Morrison,Vafa;….]

1) Construction of toric hypersurface fibrations for F-theory

✤ Toric variety associated to 16 reflexive polytopes in 2D. ✤ Each has corresponding genus-one curve

= anti-canonical divisor in .

The toric polytopes

Jonas Reuter 8 / 28

Toric varieties from reflexive polytopes

CFi

PFiFi

PFi

PFi

✤ Combinatorics of encodes geometry of toric variety ➡ representation as generalized projective space

✤ Genus-one curve as CY-hypersurfaces in

✤ Three different types of and curves 1. cubics in = blow-ups of (14 cases)2. quartic in = (1 case)3. biquadric in = (1 case)

Toric varieties and their genus one curves

Fi

PFi

PFi

Dual Polytopes and Mirror Symmetry

the mirror dual polytopes are

F

⇤i = {q 2 M ⌦ R|hy , qi � �1, 8y 2 Fi}

constructing the dual polytope for Fi leads to F17�i , 1 i 6

Fi , 7 i 10, are selfdual

smooth toric variety corresponding to polytopes defined by

PFi =Cm+2\SR(C⇤)m

= {xk ⇠mY

a=1

�`(a)ka xk | x /2 SR ,�a 2 C⇤}

Calabi-Yau hypersurface obtained by Batyrev formula

pFi =X

q2F⇤i \M

aq

Y

k

x

hvk ,qi+1k

Jonas Reuter 9 / 28

PFi

P2

P2(1, 1, 2)

P1 ⇥ P1

PFiCFi = {pFi = 0}

PFiCFi

in

PFi

PF4

PF2

1. Ambient space: fiber over B

✤ Fibration completely determined by two divisors and on B

‣ parametrize divisor classes of the two local coordinates on the fiber.

2. Calabi-Yau hypersurface eq. of

✤ impose CY-eq. in fiber:

✤ impose CY condition on total space

➡get discrete families of Calabi-Yau manifolds

3. Derive the effective theory of F-theory for all these .

3.1 Three basic ingredients: the cubic, biquadric and quartic3.1.1 Constructing Toric Hypersurface Fibration

In this section we explain the general construction of the Calabi-Yau manifolds XFi with torichypersurface fiber CFi and base B. The following discussion applies to Calabi-Yau n-folds XFi

with a general (n � 1)-dimensional base B. The cases of most relevance for F-theory and forthis work are n = 3, 4. We refer to [37,39] for more details on the following discussion.

The starting point of the construction of the genus-one fibered Calabi-Yau manifold XFi isthe hypersurface equation (2.23) of the curve CFi . In order to obtain the equation of XFi , thecoefficients aq and the variables xi of (2.23) have to be promoted to sections of appropriate linebundles of the base B. We determine these line bundles, by first constructing a fibration of the2D toric variety PFi , which is the ambient space of CFi , over the same base B,

PFi// PB

Fi(S7,S9)

✏✏B

. (3.1)

Here PBFi(D, D) denotes the total space of this fibration. The structure of its fibration is

parametrized by two divisors in B, denoted by D and D. This can be seen by noting that allm + 2 coordinates xk on the fiber PFi are in general non-trivial sections of line bundles on B.Then, we can use the (C⇤)m-action of the toric variety PFi to set m variables to transform inthe trivial bundle of B. The divisors dual to the two remaining line bundles are precisely D, D.

Next we impose equation (2.23) in PFi(D, D). Consistency fixes the line bundles in whichthe coefficients aq have to take values in terms of the two divisors D and D. Then, we require(2.23) to be a section of the anti-canonical bundle K�1

PBFi

, which is the Calabi-Yau condition.

In addition, equation (2.23) imposed in PBFi(D, D) clearly describes a genus-one fibration over

B, since for every generic point on B, the hypersurface (2.23) describes exactly the curve CFi

in PFi . The total Calabi-Yau space resulting from the fibration of the toric hypersurface Ci isdenoted by XFi in the following. It enjoys the fibration structure

CFi// XFi

✏✏B

. (3.2)

In principle, this procedure has to be carried out for all Calabi-Yau manifolds XFi associatedto the 16 2D toric polyhedra Fi. However, we observe that all the hypersurface constraints ofthe XFi , except for XF2 and XF4 , can be obtained from the hypersurface constraint for XF1 ,after setting appropriate coefficients to zero. This is possible because if F1 is a sub-polyhedronof Fi, then the corresponding toric variety PFi is the blow-up of PF1 = P2 at a given number ofpoints, with the additional rays in Fi corresponding to the blow-up divisors. However, addingrays to the polyhedron F1 removes rays from its dual polyhedron F ⇤

1 = F16. By means of (2.23),this removes coefficients from hypersurface equation for XF1 , i.e. the hypersurface for XFi is a

17

Construction of toric hypersurface fibration . XFi

S7 S9

pFi = 0

XFi

CFi ⇢ PFi

3.1 Three basic ingredients: the cubic, biquadric and quartic3.1.1 Constructing Toric Hypersurface Fibration

In this section we explain the general construction of the Calabi-Yau manifolds XFi with torichypersurface fiber CFi and base B. The following discussion applies to Calabi-Yau n-folds XFi

with a general (n � 1)-dimensional base B. The cases of most relevance for F-theory and forthis work are n = 3, 4. We refer to [37,39] for more details on the following discussion.

The starting point of the construction of the genus-one fibered Calabi-Yau manifold XFi isthe hypersurface equation (2.23) of the curve CFi . In order to obtain the equation of XFi , thecoefficients aq and the variables xi of (2.23) have to be promoted to sections of appropriate linebundles of the base B. We determine these line bundles, by first constructing a fibration of the2D toric variety PFi , which is the ambient space of CFi , over the same base B,

CFi ⇢ PFi// XFi

✏✏B

. (3.1)

Here PBFi(D, D) denotes the total space of this fibration. The structure of its fibration is

parametrized by two divisors in B, denoted by D and D. This can be seen by noting that allm + 2 coordinates xk on the fiber PFi are in general non-trivial sections of line bundles on B.Then, we can use the (C⇤)m-action of the toric variety PFi to set m variables to transform inthe trivial bundle of B. The divisors dual to the two remaining line bundles are precisely D, D.

Next we impose equation (2.23) in PFi(D, D). Consistency fixes the line bundles in whichthe coefficients aq have to take values in terms of the two divisors D and D. Then, we require(2.23) to be a section of the anti-canonical bundle K�1

PBFi

, which is the Calabi-Yau condition.

In addition, equation (2.23) imposed in PBFi(D, D) clearly describes a genus-one fibration over

B, since for every generic point on B, the hypersurface (2.23) describes exactly the curve CFi

in PFi . The total Calabi-Yau space resulting from the fibration of the toric hypersurface Ci isdenoted by XFi in the following. It enjoys the fibration structure

CFi// XFi

✏✏B

. (3.2)

In principle, this procedure has to be carried out for all Calabi-Yau manifolds XFi associatedto the 16 2D toric polyhedra Fi. However, we observe that all the hypersurface constraints ofthe XFi , except for XF2 and XF4 , can be obtained from the hypersurface constraint for XF1 ,after setting appropriate coefficients to zero. This is possible because if F1 is a sub-polyhedronof Fi, then the corresponding toric variety PFi is the blow-up of PF1 = P2 at a given number ofpoints, with the additional rays in Fi corresponding to the blow-up divisors. However, addingrays to the polyhedron F1 removes rays from its dual polyhedron F ⇤

1 = F16. By means of (2.23),this removes coefficients from hypersurface equation for XF1 , i.e. the hypersurface for XFi is acertain specialization of the hypersurface of XF1 with some aq ⌘ 0. We will be more explicitabout this in the following subsection (Section 3.1.2).

17

XFi

XFi

XFi(S7,S9)

PFi

2) The effective theories of F-theory

Gauge theory at zeros of discriminant :

✤ In resolution: Cartan matrix of G realized by s over S

✤ have codim. 1 singularities & corresp. intrinsic gauge group

➡Singularities in resolved: read off from toric polytope

➡ Points inside edges= nodes in Dynkin diagram

Non-Abelian Gauge Group� = 4f3 + 27g2

� ⇠ wn�0, n � 2

S = {w = 0}➡ gauge symm. at

Singularity

GFi

Figure 2: The 16 two dimensional reflexive polyhedra [16]. The polyhedron Fi and F17�i aredual for i = 1 . . . 6 and self-dual for i = 7 . . . 10.

Every toric variety PFi has an associated Calabi-Yau hypersurface, i.e. a genus-one curveCFi . It is defined as the generic section of its anti-canonical bundle K�1

PFi. The Calabi-Yau

hypersurface in PFi is obtained by the Batyrev construction as the following polynomial [83]

pFi =X

q2F ⇤i \M

aqY

k

xhvk,qi+1k , (2.23)

where q denotes all integral points in F ⇤i and the aq are coefficients in the field K.

We note that points vi interior to edges in Fi are usually excluded in the product (2.23)because the corresponding divisors do not intersect the hypersurface CFi . However, when con-sidering Calabi-Yau fibrations XFi of CFi as in Section 3 these divisors intersect XFi and resolvesingularities of XFi induced by singularities of its fibration, i.e. these divisors are related toCartan divisors DGI

i discussed above in Section 2.2.

3 Analysis of F-theory on Toric Hypersurface Fibrations

In this section we analyze the geometry of the Calabi-Yau manifolds XFi , that are constructedas fibration of the genus-one curves CFi over a generic base B. For each manifold we calculatethe effective field theory resulting from compactifying F-theory on it. We calculate the gaugegroup, the charged and neutral matter spectrum and the Yukawa couplings.

15

SU(3)xSU(2)⇢ GF11

e.g.

[Kodaira;Tate][Vafa;Morrison,Vafa;Bershadsky,Intriligator,Kachru,Morrison,Sadov,Vafa]

XFi

GFiXFi

P1

U(1)-symmetries Mordell-Weil group of rational sections of elliptic fibrations .

✤ rational section is map induce by rational point Q on .

Number of U(1)’s/rational sections from toric polytope:

➡ number of U(1)’s = #(vertices of )

Example:

Abelian Gauge Group

XFi

CFi

sQ : B ! XFi

Fi �3

U(1): SU(3)xSU(2)xU(1)4� 3 = 1GF11 =

Toric MW-group: [Braun,Grimm,Keitel]

[Morrison,Vafa] CFi

Figure 2: The 16 two dimensional reflexive polyhedra [16]. The polyhedron Fi and F17�i aredual for i = 1 . . . 6 and self-dual for i = 7 . . . 10.

Every toric variety PFi has an associated Calabi-Yau hypersurface, i.e. a genus-one curveCFi . It is defined as the generic section of its anti-canonical bundle K�1

PFi. The Calabi-Yau

hypersurface in PFi is obtained by the Batyrev construction as the following polynomial [83]

pFi =X

q2F ⇤i \M

aqY

k

xhvk,qi+1k , (2.23)

where q denotes all integral points in F ⇤i and the aq are coefficients in the field K.

We note that points vi interior to edges in Fi are usually excluded in the product (2.23)because the corresponding divisors do not intersect the hypersurface CFi . However, when con-sidering Calabi-Yau fibrations XFi of CFi as in Section 3 these divisors intersect XFi and resolvesingularities of XFi induced by singularities of its fibration, i.e. these divisors are related toCartan divisors DGI

i discussed above in Section 2.2.

3 Analysis of F-theory on Toric Hypersurface Fibrations

In this section we analyze the geometry of the Calabi-Yau manifolds XFi , that are constructedas fibration of the genus-one curves CFi over a generic base B. For each manifold we calculatethe effective field theory resulting from compactifying F-theory on it. We calculate the gaugegroup, the charged and neutral matter spectrum and the Yukawa couplings.

15

Intrinsic gauge group of all 16 toric hypersurface fibrations

✤ up to three U(1)’s, non-simply connected & discrete gauge groups.

✤ Key observations:

✤ intrinsic 6D matter (= 4D non-chiral) spectrum & 4D Yukawas derived➡ ideal techniques: primary decomposition, elimination ideals, etc.

✤ all theories anomaly-free. ✔

Effective theories of the 16 toric hypersurface fibrations

We start with a quick summary of some interesting results of this study. There are threepolyhedra leading to manifolds XFi without a section: F1, F2 and F4, see sections 3.2.1, 3.2.2and 3.2.3, respectively. For three polyhedra we find associated gauge groups with Mordell-Weiltorsion: F13, F15 and F16, see sections 3.6.1, 3.6.2 and 3.6.3, respectively. The analysis of thehypersurface XF3 and the corresponding effective theory of F-theory, whose spectrum containsa charge three matter, can be found in section 3.3.1.

We obtain the following list of all the gauge groups GFi of F-theory on the XFi :

GF1 Z3 GF7 U(1)3

GF2 U(1)⇥Z2 GF8 SU(2)2⇥U(1) GF13 (SU(4)⇥SU(2)2)/Z2

GF3 U(1) GF9 SU(2)⇥U(1)2 GF14 SU(3)⇥SU(2)2⇥U(1)GF4 (SU(2)⇥Z4)/Z2 GF10 SU(3)⇥SU(2) GF15 SU(2)4/Z2⇥U(1)GF5 U(1)2 GF11 SU(3)⇥SU(2)⇥U(1) GF16 SU(3)3/Z3

GF6 SU(2)⇥U(1) GF12 SU(2)2⇥U(1)2

From this and as a simple consequence of (2.10), we see that there is the following rule of thumbfor computing the rank of a gauge group GFi : given a polyhedron Fi with 3+ n integral pointswithout the origin, we have a gauge group with total rank n.

Let us outline the structure of this section. In the first subsection 3.1 we briefly discussthe three different representations of genus-one curves CFi realized as toric hypersurfaces: thecubic, the biquadric and the quartic. There, we define the line bundles of the base B in whichthe coefficients in these constraints have to take values in order to obtain a genus-one fiberedCalabi-Yau manifold. The functions f and g of the Weierstrass form (2.1) for the cubic, thebiquadric and the quartic can be found in Appendix B. By appropriate specializations of thecoefficients, one can obtain f , g and � = 4f 3 + 27g2 for all toric hypersurface fibration XFi .

In sections 3.2 to 3.6 we proceed to describe in detail each Calabi-Yau manifold XFi . Ineach case we first discuss the genus-one curve CFi realized as a toric hypersurface in PFi . Weproceed to construct the corresponding genus-one fibered Calabi-Yau manifold XFi and analyzeits codimension one and two singularities from which we extract the non-Abelian gauge groupand matter spectrum. If XFi has a non-trivial MW-group, we determine all its generators,their Shioda maps and the height pairing. For completeness, we also determine the Yukawacouplings from codimension three singularities. In each case we show as a consistency checkthat the necessary 6D anomalies (pure Abelian, gravitational-Abelian, pure non-Abelian, non-Abelian gravitational, non-Abelian-Abelian and purely gravitational) are canceled implyingconsistency of the considered effective theories.

We organize the Calabi-Yau manifolds XFi into five categories: those with discrete gaugesymmetries (Section 3.2), those with a gauge group of rank one and two but without discretegauge groups (Section 3.3), those with a gauge group of rank three, whose fiber polyhedrahappen to be also self-dual (Section 3.4), those with gauge groups of rank four and five withoutMW-torsion (Section 3.5) and those XFi with MW-torsion (Section 3.6). This arrangement isalmost in perfect agreement with the labeling of the polyhedra Fi in Figure 2 which facilitatesthe navigation through this section. We name the subsection containing the analysis of thespecific manifold XFi by its corresponding fiber polyhedron Fi.

16

GFi XFi

Non-simply connected groups: [Aspinwall,Morrison;Mayrhofer,Morrison,Till,Weigand]

rk(GFi) + rk(G⇤Fi) = 6 with dual to . F ⇤

i

rk(GFi) = #(pts. in Fi)� 4

Fi

[D.K.,Mayorga Peña,Oehlmann,Piragua,Reuter]

1. with : rational section exists only in Jacobian

2. with : non-toric rational section

✤ zero section is toric in

✤ tangent to zero point P in has to intersect again in rat. point Q

✤ non-toric sections has novel physics: singlet of charge q=3

✤ Matter with higher charges crucial for existence of discrete gauge symmetry: Higgs with charge q=3 matter U(1)

Non-toric properties of toric models

XF3

I2-fiber with q=3:

The toric polytopes

Jonas Reuter 8 / 28

GF3 = U(1)

XF2 U(1) ⇢ GF2 J(XF2)

XF3

cf. talk by M. Cvetič

CF3

CF3

GF1 = Z3








I2-fiber with q=3:

The toric polytopes

Jonas Reuter 8 / 28

U(1) ⇢ GF2

XF3

cf. talk by M. Cvetič

XF3

XF2 J(XF2)

GF3 = U(1)

CF3

CF3

GF1 = Z3








I2-fiber with q=3:

The toric polytopes

Jonas Reuter 8 / 28

U(1) ⇢ GF2

CF3

XF3

cf. talk by M. CvetičGF1 = Z3

CF3

XF3

XF2 J(XF2)

GF3 = U(1)

Global Models with discrete gauge groups

has no rational section, only three-section:

Spectrum of contains matter charged under :

✤ -charge is denoted by 1:

✤ all gauge invariant Yukawas respecting selection rules exist.

Similar explicit results (spectra, -selection rules) for ✤ : (discrete symmetry + non-toric U(1))

✤ : .

XF2 GF2 = U(1)⇥ Z2

Representation

Multiplicity

XF1

XF4 GF4 = (SU(2)⇥ Z4)/Z2

Zn

For Z2, related works: [Braun,Morrison; Morrison,TaylorAnderson,García-Etxebarria,Grimm,Keitel; García-Etxebarria,Grimm,Keitel;Mayrhofer,Palti,Till,Weigand]

GF1 = Z3

GF1 = Z3XF1

GF1 = Z3

Z3

� 7! exp(2⇡i/3)�

[D.K.,Mayorga Peña,Oehlmann,Piragua,Reuter]

4) The Higgs network

Higgs transitions between toric hypersurface fibrations

All toric hypersurface fibrations are connected by extremal transitions in fiber

➡ induced by blow-down in toric ambient space of fiber & subsequent complex structure deformation.

✤ In toric polytope: Cutting corners

Corresponds to Higgsing in effective field theory

➡ worked out full network of all such Higgsings,

➡ generates subbranch of moduli space of field theory: “toric Higgs branch”.

XFi

CFi

CFi

Subtleties: SL(2,Z) transformations

after geometric transition generically the polytopes do not triviallymatch with one of the 16 polytopes, but structure agrees

find SL(2,Z) transformation which maps the polytopes to each other

thereby: the structure needs to be mapped correctly (inner points andrational sections)

amounts to redefinition of bundles

Jonas Reuter 19 / 28

✤ matched full 6D spectra (charged & uncharged).

✤ all theories obtained from maximal ones based on F13, F15, F16.

✤ all models with discrete gauge groups arise from Higgsing gauged U(1)’s:

➡ e.g. in from with Higgs of charge q=3.

➡ quantum gravity constraint: any global symmetry has to be gauged ✔

Toric Higgs branch

XF1 XF3Z3

(SU(2)⇥ Z4)/Z2

U(1)

For Z2-case: [Morrison,TaylorAnderson,García-Etxebarria,Grimm,Keitel; García-Etxebarria,Grimm,Keitel;Mayrhofer,Palti,Till,Weigand]

Arrow: extremal transitions in fiber/Higgsing

10 2 3

1

0

2

3

4

5

6

Figure 1: The network of Higgsings between all F-theory compactifications on toric hypersur-face fibrations XFi . The axes show the rank of the MW-group and the total rank of the gaugegroup of XFi . Each Calabi-Yau XFi is abbreviated by Fi and its corresponding gauge group isshown. The arrows indicate the existence of a Higgsing between two Calabi-Yau manifolds.

two and three singularities are analyzed, and the number of their complex structure moduli iscomputed. The F-theory gauge group, matter spectrum and Yukawa couplings are extractedfrom these results. Section 4 is devoted to the study of the toric Higgs branch of F-theorycompactified on the XFi . One particular Higgsing is discussed in detail in order to illustratethe relevant techniques. Here we also present the Higgsings leading to the effective theorieswith discrete gauge groups. We further elaborate on the details of the entire Higgsing chainin Appendices D and E. Our conclusions can be found in Section 5. This work contains ad-ditional Appendices on 6D anomalies (Appendix A), additional geometrical data of the XFi

(Appendix B) and the explicit Euler numbers of all Calabi-Yau threefolds XFi (Appendix C).

6

Mirror symmetric

[DK,Mayorga-Pena,Oehlmann,Reuter,Piragua]

5) Three family models in toric unification[Cvetič, DK, Mayorga-Pena, Oehlmann, Reuter]

Natural unification structure in toric Higgs branch:

Phenomenologically interesting examples

1. Standard-Model-like theory:

✤ All gauge invariant 4D Yukawas realized.

2. Pati-Salam-like theory: correct , reps & Yukawas.

3. Trinification-like theory: correct , reps & Yukawas.

Representation

Multiplicity

Representation Multiplicity Splitting Locus

(3,2)1/6 S9([K�1B ] + S7 � S9) V (I(1)) := {s3 = s9 = 0}

(1,2)�1/2([K�1

B ] + S7 � S9)

(6[K�1B ]� 2S7 � S9)

V (I(2)) := {s3 = 0

s2s25 + s1(s1s9 � s5s6) = 0}

(3,1)�2/3 S9(2[K�1B ]� S7) V (I(3)) := {s5 = s9 = 0}

(3,1)1/3 S9(5[K�1B ]� S7 � S9)

V (I(4)) := {s9 = 0

s3s25 + s6(s1s6 � s2s5) = 0}

(1,1)1(2[K�1

B ]� S7)

(3[K�1B ]� S7 � S9)

V (I(5)) := {s1 = s5 = 0}

(8,1)0 1 + S9S9�[K�1

B ]

2 Figure 19 s9 = 0

(1,3)01 + S7�S9

2

⇥([K�1B ] + S7 � S9)

Figure 19 s3 = 0

Table 20: Charged matter representations under SU(3) ⇥ SU(2) ⇥ U(1) and correspondingcodimension two fibers of XF11 . The adjoint matter is included for completeness.

3.5.2 Polyhedron F12: GF12 = SU(2)2 ⇥ U(1)2

In this section, we analyze the elliptically fibered Calabi-Yau manifold XF12 with base B andgeneral elliptic fiber given by the elliptic curve E in PF12 . The toric data of PF12 can beextracted from Figure 20, where the fiber polyhedron F12 together with a choice of homogeneous

69


(3,2)1/6 S9([K�1B ] + S7 � S9) V (I(1)) := {s3 = s9 = 0}

(1,2)�1/2([K�1

B ] + S7 � S9)

(6[K�1B ]� 2S7 � S9)

V (I(2)) := {s3 = 0

s2s25 + s1(s1s9 � s5s6) = 0}

(3,1)�2/3 S9(2[K�1B ]� S7) V (I(3)) := {s5 = s9 = 0}

(3,1)1/3 S9(5[K�1B ]� S7 � S9)

V (I(4)) := {s9 = 0

s3s25 + s6(s1s6 � s2s5) = 0}

(1,1)1(2[K�1

B ]� S7)

(3[K�1B ]� S7 � S9)

V (I(5)) := {s1 = s5 = 0}

(8,1)0 1 + S9S9�[K�1

B ]

2 Figure 19 s9 = 0

(1,3)01 + S7�S9

2

⇥([K�1B ] + S7 � S9)

Figure 19 s3 = 0




69


(3,2)1/6 S9([K�1B ] + S7 � S9) V (I(1)) := {s3 = s9 = 0}

(1,2)�1/2([K�1

B ] + S7 � S9)

(6[K�1B ]� 2S7 � S9)

V (I(2)) := {s3 = 0

s2s25 + s1(s1s9 � s5s6) = 0}

(3,1)�2/3 S9(2[K�1B ]� S7) V (I(3)) := {s5 = s9 = 0}

(3,1)1/3 S9(5[K�1B ]� S7 � S9)

V (I(4)) := {s9 = 0

s3s25 + s6(s1s6 � s2s5) = 0}

(1,1)1(2[K�1

B ]� S7)

(3[K�1B ]� S7 � S9)

V (I(5)) := {s1 = s5 = 0}

(8,1)0 1 + S9S9�[K�1

B ]

2 Figure 19 s9 = 0

(1,3)01 + S7�S9

2

⇥([K�1B ] + S7 � S9)

Figure 19 s3 = 0




69


(3,2)1/6 S9([K�1B ] + S7 � S9) V (I(1)) := {s3 = s9 = 0}

(1,2)�1/2([K�1

B ] + S7 � S9)

(6[K�1B ]� 2S7 � S9)

V (I(2)) := {s3 = 0

s2s25 + s1(s1s9 � s5s6) = 0}

(3,1)�2/3 S9(2[K�1B ]� S7) V (I(3)) := {s5 = s9 = 0}

(3,1)1/3 S9(5[K�1B ]� S7 � S9)

V (I(4)) := {s9 = 0

s3s25 + s6(s1s6 � s2s5) = 0}

(1,1)1(2[K�1

B ]� S7)

(3[K�1B ]� S7 � S9)

V (I(5)) := {s1 = s5 = 0}

(8,1)0 1 + S9S9�[K�1

B ]

2 Figure 19 s9 = 0

(1,3)01 + S7�S9

2

⇥([K�1B ] + S7 � S9)

Figure 19 s3 = 0




69

⇥


(3,2)1/6 S9([K�1B ] + S7 � S9) V (I(1)) := {s3 = s9 = 0}

(1,2)�1/2([K�1

B ] + S7 � S9)

(6[K�1B ]� 2S7 � S9)

V (I(2)) := {s3 = 0

s2s25 + s1(s1s9 � s5s6) = 0}

(3,1)�2/3 S9(2[K�1B ]� S7) V (I(3)) := {s5 = s9 = 0}

(3,1)1/3 S9(5[K�1B ]� S7 � S9)

V (I(4)) := {s9 = 0

s3s25 + s6(s1s6 � s2s5) = 0}

(1,1)1(2[K�1

B ]� S7)

(3[K�1B ]� S7 � S9)

V (I(5)) := {s1 = s5 = 0}

(8,1)0 1 + S9S9�[K�1

B ]

2 Figure 19 s9 = 0

(1,3)01 + S7�S9

2

⇥([K�1B ] + S7 � S9)

Figure 19 s3 = 0




69


(3,2)1/6 S9([K�1B ] + S7 � S9) V (I(1)) := {s3 = s9 = 0}

(1,2)�1/2([K�1

B ] + S7 � S9)

(6[K�1B ]� 2S7 � S9)

V (I(2)) := {s3 = 0

s2s25 + s1(s1s9 � s5s6) = 0}

(3,1)�2/3 S9(2[K�1B ]� S7) V (I(3)) := {s5 = s9 = 0}

(3,1)1/3 S9(5[K�1B ]� S7 � S9)

V (I(4)) := {s9 = 0

s3s25 + s6(s1s6 � s2s5) = 0}

(1,1)1(2[K�1

B ]� S7)

(3[K�1B ]� S7 � S9)

V (I(5)) := {s1 = s5 = 0}

(8,1)0 1 + S9S9�[K�1

B ]

2 Figure 19 s9 = 0

(1,3)01 + S7�S9

2

⇥([K�1B ] + S7 � S9)

Figure 19 s3 = 0




69


(3,2)1/6 S9([K�1B ] + S7 � S9) V (I(1)) := {s3 = s9 = 0}

(1,2)�1/2([K�1

B ] + S7 � S9)

(6[K�1B ]� 2S7 � S9)

V (I(2)) := {s3 = 0

s2s25 + s1(s1s9 � s5s6) = 0}

(3,1)�2/3 S9(2[K�1B ]� S7) V (I(3)) := {s5 = s9 = 0}

(3,1)1/3 S9(5[K�1B ]� S7 � S9)

V (I(4)) := {s9 = 0

s3s25 + s6(s1s6 � s2s5) = 0}

(1,1)1(2[K�1

B ]� S7)

(3[K�1B ]� S7 � S9)

V (I(5)) := {s1 = s5 = 0}

(8,1)0 1 + S9S9�[K�1

B ]

2 Figure 19 s9 = 0

(1,3)01 + S7�S9

2

⇥([K�1B ] + S7 � S9)

Figure 19 s3 = 0




69


(3,2)1/6 S9([K�1B ] + S7 � S9) V (I(1)) := {s3 = s9 = 0}

(1,2)�1/2([K�1

B ] + S7 � S9)

(6[K�1B ]� 2S7 � S9)

V (I(2)) := {s3 = 0

s2s25 + s1(s1s9 � s5s6) = 0}

(3,1)�2/3 S9(2[K�1B ]� S7) V (I(3)) := {s5 = s9 = 0}

(3,1)1/3 S9(5[K�1B ]� S7 � S9)

V (I(4)) := {s9 = 0

s3s25 + s6(s1s6 � s2s5) = 0}

(1,1)1(2[K�1

B ]� S7)

(3[K�1B ]� S7 � S9)

V (I(5)) := {s1 = s5 = 0}

(8,1)0 1 + S9S9�[K�1

B ]

2 Figure 19 s9 = 0

(1,3)01 + S7�S9

2

⇥([K�1B ] + S7 � S9)

Figure 19 s3 = 0




69


(3,2)�1/6 S9([K�1B ] + S7 � S9) V (I(1)) := {s3 = s9 = 0}

(1,2)1/2([K�1

B ] + S7 � S9)

(6[K�1B ]� 2S7 � S9)

V (I(2)) := {s3 = 0

s2s25 + s1(s1s9 � s5s6) = 0}

(3,1)�2/3 S9(2[K�1B ]� S7) V (I(3)) := {s5 = s9 = 0}

(3,1)1/3 S9(5[K�1B ]� S7 � S9)

V (I(4)) := {s9 = 0

s3s25 + s6(s1s6 � s2s5) = 0}

(1,1)�1(2[K�1

B ]� S7)

(3[K�1B ]� S7 � S9)

V (I(5)) := {s1 = s5 = 0}

(8,1)0 1 + S9S9�[K�1

B ]

2 Figure 19 s9 = 0

(1,3)01 + S7�S9

2

⇥([K�1B ] + S7 � S9)

Figure 19 s3 = 0




69


(3,2)�1/6 S9([K�1B ] + S7 � S9) V (I(1)) := {s3 = s9 = 0}

(1,2)1/2([K�1

B ] + S7 � S9)

(6[K�1B ]� 2S7 � S9)

V (I(2)) := {s3 = 0

s2s25 + s1(s1s9 � s5s6) = 0}

(3,1)�2/3 S9(2[K�1B ]� S7) V (I(3)) := {s5 = s9 = 0}

(3,1)1/3 S9(5[K�1B ]� S7 � S9)

V (I(4)) := {s9 = 0

s3s25 + s6(s1s6 � s2s5) = 0}

(1,1)�1(2[K�1

B ]� S7)

(3[K�1B ]� S7 � S9)

V (I(5)) := {s1 = s5 = 0}

(8,1)0 1 + S9S9�[K�1

B ]

2 Figure 19 s9 = 0

(1,3)01 + S7�S9

2

⇥([K�1B ] + S7 � S9)

Figure 19 s3 = 0




69

⇥

XF16 GF16

XF13 GF14

Standard Model

Trinification

Pati-Salam

XF11

SM via tops of dP2: [Lin,Weigand]

[DK,Mayorga-Pena,Oehlmann,Reuter,Piragua]

Construction of three family models in 4D

Model Building Strategy:

✤ Construct G4-flux by computing for (SM), (PS), (Trinif) following .

✤ Solve F-theory constraints on G4-flux, formulated in terms of 3D CS-terms.

✤ Require massless UY(1) in Standard Model.

✤ Compute minimal number of families so that nD3 is integral & positive and 3D CS-terms are integral G4-flux quantization.

✤ Use toric Higgs branch to describe unification in Pati-Salam and/or Trinification.

➡Explicit results for concrete fourfolds with base .

XF13XF11 XF16

B = P3

[Cvetič,DK,Mayorga-Pena,Oehlmann,Reuter]

H(2,2)V (XFi)

[Cvetič,Grassi,DK,Piragua]

[Dasgupta,Rajesh,Sethi;Marsano,Schäfer-Nameki;GrimmHayashi;Cvetič,Grassi,DK,Piragua]CS-terms for 6D F-theory: [Grimm,Kapfer Keitel]

Construction of three family models in 4D

Standard Model: (#(families),nD3) Pati-Salam: (#(families),nD3)

Trinification: (#(families),nD3)

✤ All models admit three families➡positive & integral nD3 ➡quantized CS-term/G4-flux.

✤ Unification with three families possible.

see poster by Paul-Konstantin Oehlmann and Jonas Reuter

[Cvetič,DK,Mayorga-Pena,Oehlmann,Reuter]

Divisor parameters ( , ) (n7,n9)S7 S9

II. Non-toric Higgs Branches of F-theory with rank two groups[Cvetič, DK, Piragua, Taylor]

Question: What are the geometries needed to describe elementary Higgsings G U(1)rk(G)?➡ Elliptic fibrations with higher rank MW-group crucial for F-theory moduli space.

Rank one case understood: 6D F-theory with SU(2) Higgses to model with U(1) described by quartic in .

Answer similar question for rank two gauge groups SU(2)xSU(2), SU(3) ✤ Higgsings to U(1)2 not seen in toric models➡ need more general, non-toric U(1)2 models.Spin-off: In return, general U(1)2 models enhance to SU(2)xSU(2)xSU(3) gauge groups with novel SU(3) gauge group realization.

SU(2) on Riemann surface U(1)

rk(MW)=0 rk(MW)=1 tuning moduli

adjoint VEV ⌃g

The general question

P2(1, 1, 2)

1) A more general model withU(1)2

Any elliptic fibration X with MW-rank two is cubic.✤ Keep three rat. points in elliptic fiber at general positions.

✤ Three rational points: X has rk(MW)=2

In an elliptic fibration, the coefficients ai, bi are non-trivial polynomials➡ cannot perform variable transformation to put P, Q to toric points.

Important: points P, Q and R can freely collide with each other.

Construction of non-toric model with U(1)2

uf2(u, v, w) +3Y

i=1

(aiv + biw) = 0

f2 = s1u2 + s2uv + s3v

2 + s5uw + s6vw + s8w2

P = [0 : �b1 : a1] Q = [0 : �b2 : a2] R = [0 : �b3 : a3]

C[Deligne;Borchmann,Mayrhofer,Palti,Weigand;Cvetič,DK,Piragua]

l1u� l2(aiv + biw) = m1f2 �m2(ajv + bjw) = l2m2 + (akv + bkw)l1m1 = 0

i 6= j 6= k(l1 : l2 : m1 : m2) 2 P1 ⇥ P1

Elliptic fibrations X with non-toric curve over any base B specified by four divisor classes: [a1] , [a2], [a3], [s8].

Can construct global CY-resolution of X as complete intersection in higher dimensional ambient space:

Construction of non-toric model with U(1)2

with and for every

[Cvetič, DK, Piragua, Taylor]C

2) The effective theory of F-theory

Full 6D spectrum over general base B

Charges Multiplicity(2,0)(0,2)

(-2,-2)(-1,1)(-2,-1)(-1,-2)(1,1)(1,0)(0,1)

U(1)xU(1) charge lattice

✤ rich particle spectrum with new types of representations✤ checked cancellation of all 6D anomalies. ✔

4[b31b32s

33] · ([a1b2]� [KB ])

4[b31b33s

33] · ([a1b3]� [KB ])

[Cvetič, DK, Piragua, Taylor]

3) The non-toric Higgs branch & novel non-Abelian gauge structures

General strategy to reduce Mordell-Weil rank of X✤ tune moduli of X to place rational points on top of each other➡ for X with rk(MW)=2:

Geometry of Higgs transitions in F-theory

uf2(u, v, w) +3Y

i=1

(aiv + biw) = 0



✤ rk(MW)=2 1: 0

PQ

PR

uf2(u, v, w) + �1(a1v + b1w)2(a3v + b3w) = 0


Tuning induces codimension 1 singularities in X:1. SU(2)xSU(2) at ,2. SU(3) singularity at

In toric models: Placing rational points on top of each other not possible➡ by construction fixed at different toric points (e.g. u=v=0 and u=w=0)


�1 = 0 �2 = 0

f2(0,�b1, a1) = a21s8 � a1b1s6 + b21s3 = 0

✤ rk(MW)=2 1: 0✤ rk(MW)=1 0: 0

PQ

PR

uf2(u, v, w) + �1�2(a1v + b1w)3 = 0

Different strata in moduli space of X yield tuned models with1. G=SU(2)xSU(2) and adjoints2. G=SU(3) and adjoints➡ Reproduce exactly their most general Weierstrass models. ✔➡ Full spectra match. ✔

General situations: have to have full SU(2)xSU(2)xSU(3) ➡ needed to incorporate all matter reps. of general U(1)2 theory.

Enhancements to SU(2)2 and SU(3)

1. SU(2)xSU(2): 2. SU(3):

Tuned model has I3-singularity over divisor with ordinary double point singularity at a1=b1=0:

✤ Weierstrass model looks like I3ns: monodromy cover is irreducible

✤ Interplay between structure of T and fiberation: double point of T not deformable ➡ first explicit example with symmetric + antisymmetric matter at a1=b1=0

Generic situation: Z2-monodromy exchanges nodes in fiber around I3ns yielding G=SU(2)

Novel realization of SU(3) with symmetric matter representations

T

BD

D \ T

T = {a21s8 � a1b1s6 + b21s3 = 0}

2 + (s26 � 4s3s8) = 0D := {s26 � 4s3s8 = 0}


[Morrison, Taylor]

Tuned model has I3-singularity over divisor with ordinary double point singularity at a1=b1=0:

✤ Weierstrass model looks like I3ns: monodromy cover is irreducible

✤ Interplay between structure of T and fiberation: double point of T not deformable ➡ first explicit example with symmetric + antisymmetric matter at a1=b1=0

Here: D = discriminant locus of T ➡ intersection points in

pair up: no monodromy➡ tangential intersection

locally: D={x2+O(x3)} at➡ I3 is split: I3s yielding SU(3)

T

BD

D \ T

D \ T

Novel realization of SU(3) with symmetric matter representations

T = {a21s8 � a1b1s6 + b21s3 = 0}

2 + (s26 � 4s3s8) = 0D := {s26 � 4s3s8 = 0}


[Morrison, Taylor]

III. Conclusions & Outlook

Summary1. Constructed & analyzed all genus-one fibrations with fiber in toric varieties

associated to 16 2D reflexive polytopes Fi.✤ Full effective theory in 6D (= non-chiral 4D) determined➡ intrinsic gauge groups & matter content: discrete gauge groups, singlets with

charge q=3, SM, Pati-Salam, Trinification,… .✤ Construction of explicit three family models exhibiting toric unification.✤ Network of Higgsings relating all effective theories studied.

2. Construction of (most?) general, non-toric CY-elliptic fibration with U(1)2

✤ Determination of full effective theory in 6D (= non-chiral 4D).✤ All F-theory models with rank two non-Abelian gauge groups embedded via

tuning/Higgsings.✤ Novel explicit construction of SU(3) gauge theory with two-times symmetric

representation in global F-theory.

Outlook✤ Explore further phenomenology of all toric hypersurface fibrations.✤ Constructing of Higgs branch of larger non-Abelian gauge groups: SU(5)…

CFi

of february, 2015 toric and non-toric higgs branches of f

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