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Fibration Structures and Duality in F-theory

Lara B. Anderson

Virginia Tech

Work done in collaboration with:

J. Gray

X. Gao

S. J. Lee

arXiv:1503.????

Physics and Geometry of F-theory

Max Planck Institute for Physics

February 24th, 2015

Lara Anderson (Virginia Tech) Fibration Structures and Duality Munich- Feb. 24th, ’15 1 / 30

Motivation: Towards a duality cartography for F-theory

What is F-theory? Our understanding of F-theory continues to improve

However – At present our primary

windows into the theory consist of

dualities

Can we obtain a deeper

understanding of these dualities –

what they can tell us in given cases

and how networks/chains of

F-theory dualities might be related

in the space of CY geometries?

F-theory

mM-theory/Heterotic/Type IIB


F-theory Compactification

Geometric ingredients:

An elliptically fibered Calabi-Yau

3-fold, π : Y3E−→ B2

If the fibration has a section, Y3

can be written in Weierstrass form

y 2 = x3 + f (ui )x + g(ui )

ui coords on B2, f ∈ H0(B2,K−4B2

),

g ∈ H0(B2,K−6B3

)

Degenerations of E-fiber encode

positions of 7-branes.

∆ = 4f 3 + 27g2 = 0

Divisors D ⊂ B2 ⇒ GUT

Symmetries. Intersections,

C = D ∩ D ′ ⊂ B2 ⇒ matter.


Weierstrass Form

Since the beginning of F-theory, the importance of Weierstrass models

has been evident

Strongly coupled IIB: A clear description of the physical axio-dilaton +

7-branes.

Nakayama’s Theorem : Any elliptically fibered manifold with section is

birational to a Weierstrass model

However, it is natural to ask, is there physical data in F-theory is not

readily “visible” in a Weierstrass model?

Answer at least partially yes... (e.gs: G-flux in 4-dimensions, T-branes,

etc)

Illustrate here one aspect of the theory that can be difficult to spot from

Weierstrass form, and how to incorporate it into our descriptions...


Reverse approach: A mystery manifold...

Suppose we are given a mystery CY n-fold and would like to know

whether or not it could be a good background for F-theory.

Must ask:

1 Is it genus-1 or elliptically fibered?

2 Does it have a section, multiple sections, or multi-sections?

3 If so, are they holomorphic or rational?

4 Can we manifestly put it in Weierstrass form?

5 What global “structural” features can we notice?


Observations

Consider your favorite dataset of CYs (Kreuzer-Skarke, CICYs, etc)

Immediate Observation 1: Almost all known CY 3- and 4- folds are

fibered (with elliptic, K3, CY3, fibrations, etc)

Immediate Observation 2: generic manifolds do not admit just one elliptic

fibration, they admit many (order 10s or 100s).

What can we make of this? Let’s take it one at a time...


Genericity of fibrations for known datasets

0 100 200 300 400 500h110

100

200

300

400

500h21

Figure 2: The 7524 distinct Hodge number pairs for generic elliptically fibered Calabi-Yau threefolds over

toric bases (dark/blue data points). Plot axes are Hodge numbers h11, h21. Kreuzer-Skarke Hodge pairs

are shown in background in light gray for comparison.

parameter T = k−3 is useful in characterizing the complexity of the base. The models with

T = 0, 1 are P2 and the Hirzebruch surfaces, and all lie on the left-hand side of the diagram,

ranging from F0, F1, and F2, which all have Hodge numbers (h11, h21) = (3, 243), and P2

with Hodge numbers (2, 272) to F12 with Hodge numbers (11, 491). As more points are

blown up, T increases, as does the rank of the gauge group, so h11 monotonically increases.

At the same time, h21 monotonically decreases along any blow-up sequence. The change

in h21 denotes the number of free parameters that must be tuned in the Weierstrass model

over a given base to effect a blow-up. Note that the monotonic increase in h11 and decrease

in h21 is true for any sequence of blow-up operations on the base, whether or not the base

is toric.

3. Bounds

The shape of the upper bound on Hodge numbers in the “shield” configuration has been

noted in previous work, but, as far as the author of this paper knows, never explained.

– 5 –

(from Taylor 1205.0952)

Kreuzer-Skarke dataset: No. of

fibrations∼ 109 (Braun ’11)

For the CICY 3-folds 99.3%. E.g.P2

P1

P2

∣∣∣∣∣∣∣∣∣0 3

1 1

3 0

19,19

CICYs 4-folds in products of

projective spaces 99.9%

(921, 020 out of 921, 497) (Gray,

Haupt, Lukas ’14)

P1

P3

P4

P1

P1

P1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 0 0 0 0 0

1 0 0 0 1 1 1

0 1 1 1 0 1 1

0 0 1 1 0 0 0

0 0 1 0 1 0 0

0 1 0 0 0 0 1


Genericity of fibrations for known datasets

0 100 200 300 400 500h110

100

200

300

400

500h21

Figure 2: The 7524 distinct Hodge number pairs for generic elliptically fibered Calabi-Yau threefolds over

toric bases (dark/blue data points). Plot axes are Hodge numbers h11, h21. Kreuzer-Skarke Hodge pairs

are shown in background in light gray for comparison.

parameter T = k−3 is useful in characterizing the complexity of the base. The models with

T = 0, 1 are P2 and the Hirzebruch surfaces, and all lie on the left-hand side of the diagram,

ranging from F0, F1, and F2, which all have Hodge numbers (h11, h21) = (3, 243), and P2

with Hodge numbers (2, 272) to F12 with Hodge numbers (11, 491). As more points are

blown up, T increases, as does the rank of the gauge group, so h11 monotonically increases.

At the same time, h21 monotonically decreases along any blow-up sequence. The change

in h21 denotes the number of free parameters that must be tuned in the Weierstrass model

over a given base to effect a blow-up. Note that the monotonic increase in h11 and decrease

in h21 is true for any sequence of blow-up operations on the base, whether or not the base

is toric.

3. Bounds

The shape of the upper bound on Hodge numbers in the “shield” configuration has been

noted in previous work, but, as far as the author of this paper knows, never explained.

– 5 –

(from Taylor 1205.0952)

Kreuzer-Skarke dataset: No. of

fibrations∼ 109 (Braun ’11)

For the CICY 3-folds 99.3% E.g.P2

P1

P2

∣∣∣∣∣∣∣∣∣0 3

1 1

3 0

19,19

CICYs 4-folds in products of

projective spaces 99.9%

(921, 020 out of 921, 497) (Gray,

Haupt, Lukas ’14)

P1

P3

P4

P1

P1

P1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 0 0 0 0 0

1 0 0 0 1 1 1

0 1 1 1 0 1 1

0 0 1 1 0 0 0

0 0 1 0 1 0 0

0 1 0 0 0 0 1


Multiplicity of fibrations

Point of interest: When such

manifolds have fibrations, they

generically do not have just one...

In fact there can be many

E.g.

1)

P1

P2

P1

P1

P1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

0 1 1

0 1 2

1 0 1

1 0 1

1 0 1

2)

P1

P1

P2

P1

P1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 0 1

0 1 1

1 0 2

0 1 1

0 1 1

Caveat: “Obvious” fibrations

0 50 100 150 200 250 300 3501

10

100

1000

104

Number of fibrations per configuration

Abundance

Figure 6. Distribution of elliptic fibration abundance in the CICY four-fold list (excluding product

manifolds). The values lie in the range 0 - 354. We find 49,466,248 OEFs in total and on average

each CICY four-fold configuration is elliptically fibered in 54.6 di↵erent ways.

There are two di↵erent types of fibre configurations in our list. The first type is given

by block-diagonal fibre configurations, such as, for example

"1 2 0

2 0 3

#. (4.4)

This configuration describes two points in P1 times a torus [2|3]. The fibres of a total

of 2,149,222 OEFs are block-diagonal. The remaining 47,317,026 non block-diagonal fibre

configurations matrices describe irreducible tori. These fibrations can degenerate over

special loci in the base. It should be noted that some 99.4% of the fibre descriptions

contain linear constraints in the coordinates of a single projective space. However, such

linear constraints cannot be removed (by replacing the relevant Pn with Pn�1) as di↵erent

redundant descriptions of the fibre can be twisted over the base of the OEF in inequivalent

ways.

It is also of interest to analyse the base manifolds that occur in our list. There are three

main types of base manifolds, namely products of projective spaces, almost fano complete

intersections in products of projective spaces and P1 times almost del Pezzo complete

intersections in products of projective spaces. In table 1 we further sub-divide the three

main types and present a complete classification of the base manifolds that occur in our

list. We remark that bases of the form (P1)2⇥B1 and P2⇥B1, where B1 is an almost ample

complete intersection 1-fold, such as [2|2],⇥

11

�� 11

⇤or⇥

11

�� 12

⇤, do not occur in the classification

of base manifolds. This is a consequence of the redundancy removal (more precisely, the

modding out by ine↵ective splittings and identities) that was employed in the compilation

of the CICY 4-fold list [1], since the B1 merely describe di↵erent embeddings of P1 [5].

These cases are thus already captured by (P1)3 and P1 ⇥ P2.

Of the 50,114,908 OEFs in our data set 26,088,498 satisfy the necessary condition for

admitting a section which is a generic element of a favourable divisor class as described in

– 17 –

CICY 4-folds (Gray, et al 1405.2073)(∼ 108 fibrations )

5 10 15 20 25 30 35

100

200

300

400

500

600

CICY 3-folds (∼ 78, 000 fibrations)


The physics of (multiple) fibrations

For each F-theory duality (M-/Type IIB/ Het) the fibration structure is

important:

F-theory ⇔ Type IIB → elliptic fibration

F-theory ⇔ M-theory → elliptic (/circle) fibration

F-theory ⇔ Heterotic → elliptic + K3 fibrations

Thus, each of the multiple fibrations we just saw has the potential to tell

us something about webs of dual theories

Assorted examples well known in the literature

Central Question: Can we form a map of such dualities and understand

more about which effective theories are related and how? ⇒ A duality

cartography?


Multiple Fibrations in 6-dimensions: TypeIIB and M-theory

M-theory :

Here the statement is the clear: All members of the family of fibrations

will share the same M-theory limit

However, the E/S1 that connects the 5d/6d Theories is different.

Much recent progress on determining effective F-theory action through

F-theory/M-theory uplifting (Grimm, et al)

Here have two different windows into the Coulomb branch of the theory.

TypeIIB :

Different E-fibrations correspond to different axio-dilatons/weakly

coupled theories → Sen limit or its generalizations (Collinucci/Esole,

Donagi/Katz/Wijnholt, etc). Theories are different (not usual “duality”),

but must share invariant properties....Lara Anderson (Virginia Tech) Fibration Structures and Duality Munich- Feb. 24th, ’15 12 / 30

Multiple Fibrations in 6-dimensions: Het-F duality

Heterotic on πh : XnE−→ Bn−1 ⇔ F-theory on πf : Xn+1

K3−→ Bn−1

Here B = P1. Consider a CY3 with multiple K3-fibrations:

X3

π1

E

~~E

π2 B(1)2 B(2)2

with ρ1 : B(1)2 → P1 and ρ1 : B(2)2 → P1

What kind of heterotic duals do we expect?


6D Heterotic - F-theory duality continued

In general different twists to the P1 fibrations correspond to different

topologies for holomorphic vector bundles V → K3. So duality will relate

two different heterotic gauge field backgrounds on K3.

Also, general mix of perturbative/non-perturbative dualities.

Classic e.g.s Morrison/Vafa, Duff/Minasian/Witten, Aspinwall/Gross, etc.

But in contrast: for some above only special points in moduli space are

linking the geometries (i.e. conifolds/singular points in the base). Here

the smooth moduli space is the same.

Let’s look at examples...


Examples: Double K3-fibrations

1)

P1

P2

P1

P1

P1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

0 1 1

0 1 2

2 0 0

1 1 0

1 0 1

h1,1(B) = 6,χ(B) = 8

2)

P1

P1

P2

P1

P1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 0

1 0 1

1 2 0

0 0 2

0 1 1

B = P1 × P1

Heterotic dual:

pert./non-pert.

duality

1)

P1

P2

P2

P1

P1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 0 0

1 0 0 2

0 1 1 1

1 0 1 0

1 0 1 0

B = P1 × P1

2)

P1

P2

P2

P1

P1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 0 1 0

1 0 0 2

0 1 1 1

1 1 0 0

1 0 1 0

B = P1 × P1

Pert./Pert. heterotic

duality

1)

P1

P2

P3

P1

∣∣∣∣∣∣∣∣∣∣∣

0 0 1 1

0 0 2 1

1 1 1 1

1 1 0 0

B = F2

2)

P2

P3

P1

P1

∣∣∣∣∣∣∣∣∣∣∣

0 0 2 1

1 1 1 1

0 0 1 1

1 1 0 0

B = F0

Same bases as classic

DMW examples

(P1,1,2,8,12[24]), but over

whole moduli space


Multiple Fibrations in 4-dimensions

M-theory/Type IIB : Same ideas as in 6D. Also can add in Flux G4.

(Note: “horizontal/vertical” components can change. Braun’s talk).

Heterotic a) Consider a multiply fibered CY 3-fold as a base in a

heterotic compactification.

Here the weakly coupled heterotic theory doesn’t care at all which way is

“up”. Distinct F-theory 4-folds will give identical theories.

Heterotic b) Consider the heterotic duals of a multiply K3-fibered CY

4-fold.

The F-theory duals theories (with different bases to the K3-fibration)

should be dual to heterotic theories on different base geometries.

This idea was introduced many years ago in the context of (0, 2) mirror

symmetry (Distler, Greene, Blumenhagen, etc).Lara Anderson (Virginia Tech) Fibration Structures and Duality Munich- Feb. 24th, ’15 16 / 30

Multiple fibrations in 8-dimensions

Point of clarification: Although

similar structure is possible for

K3-surfaces in 8D, it is also quite

different...

Unlike for CY 3- and 4-folds, for K3

surfaces the existence of a

fibration/section varies in moduli

space

Only two log semi-stable

degenerations possible: 1) E8 × E8

and 2) SO(32) heterotic

(Clingher/Morgan)

x

x

x

3

2

1

Figure 1: The reflexive polyhedron that contains both the Weierstrass and the new SO(32)

triangle.

and the ‘bottom’ correspond to extended Dynkin diagrams of E8, the extension points being

the points (−2, −3, 1) and (−2, −3, −1) above and below the ‘section point’, respectively.

In the second case our triangle lies in the plane x1 = 0 and is given as the convex hull of

the points

( 0, 1, 0), ( 0, −1, −2), and ( 0, −1, 2). (4)

This triangle is dual to the Newton polyhedron of IP1,1,2[4]. Now the K3 polyhedron is split

in an asymmetric way: On one side we have just a single point, the corresponding divisor

being a smooth fiber, whereas on the other side we have 17 relevant points forming the

extended Dynkin diagram of SO(32). Here we have two different sections, determined by

4

(Candelas et al hep-th/9706226)

Relationship to log stable

degenerations more globally?

(relate/constrain:

Donagi/Wijnholt/Katz,

Heckman/Yau, etc)


Scans in progress...

Having identified the intriguing possibilities above, it remains to explore

what can really happen....

Several concrete goals:

1 Develop tools to identify/analyze multiple fibrations in any given CY n-fold

2 Carry out a systematic survey for some dataset of manifolds.

3 Chose to begin with the CICY 3-folds (7890 manifolds) → smallest and

technically easier to control in some ways. (Also, interesting/different

features with complete intersection fibers.)

4 Scans currently underway...

So today, I will highlight the first aspect...


Let’s get started...

Having identified a family of fibrations, how to read off the dual theories?

Shioda-Tate-Wazir Thm: If X3 is elliptically fibered with section then

every divisor must fall into one of the following three categories:

π∗(Dbase) or (zero section) or MW (X3) or blow-up in the fiber

So we are lead to Question no. 1: Do the fibrations have sections?

If not, what is the minimal order of multi-sections?

And a section exists, how many? (i.e. rk(MW (X )))

How difficult is it to find the correct blow-down sequence to hit

Weierstrass?


An algorithmic search for sections

First caveat: If the divisors do not descend from the ambient space no

tools in general to compute cohomology/intersection no.s, etc.

Next, consider topology first:

Oguiso Criteria: Over each point p ∈ B, S · Fp = 1

Intersection critera: S2 · Dα = −[c1(B)] · S · Dα for all divisors Dα in B2.

Does a section S (birationally) define the base B ⊂ X? (i.e. topology

match)

Holomorphic vs. rational section? Recall, elements of MW (X ) are in

general rational sections → can non-trivially “wrap” blow-ups in the fiber

in the resolved limit.


An illustration for K3:

Consider the following description of K3:

P1xP2yP1z

∣∣∣∣∣∣∣∣∣1 1

1 2

1 1

Putative sections: A divisor S ⊂ X will be called a “putative section” if

a) over each point p ∈ B, S intersects the fiber in a single point.

b) Topology match w/ base (i.e. χ(S) = χ(B), . . .)

Let S = O(αx , αy , αz) then need

2αx + 3αy = 1

−6αxαy − 2α2y − 4αxαz − 6αyαz = 2

Leads to soln: αx = 12 (1− 3αy ), αz = 1

2 (−2− 3αy + yαy2) (αy odd,

integer).

E.g. S = O(−1, 1, 1), H∗(K3,O(−1, 1, 1)) = (1, 0, 0).

Problem: How to describe without good coordinates?


The divisor D = −Dx + Dy + Dz naturally splits into two parts:

D = Dzero + Dpole

Dzero = Dy + Dz , Dpole = Dx with F · Dzero = 3 and F · D = 2

On each point p ∈ B, the two points of Fp ∩ Dpole match with two of the

three points of Fp ∩ Dzero . With the sections (szero) and (spole) chosen this

way, the remaining point Fp ∩ Dzero gives the single physical intersection

point Fp ∩ S .

Practically: for a chosen complex structure

The two criteria for a putative section then lead to

2↵x + 3↵y = 1 , (7)

�6↵x↵y � 2↵2y � 4↵x↵z � 6↵y↵z = 2 , (8)

whose general solution can be parametrised as

↵x =1

2(1 � 3↵y) , ↵z =

1

2(�2 � 3↵y + 7↵2

y) , (9)

for an arbitrary odd integer ↵y.

1.2 Parametric Expression for a Section of the Elliptic Fibration

Let us focus on LD = OX(�1, 1, 1) by taking a solution (↵x,↵y,↵z) = (�1, 1, 1) from the family, Eq. (9),of line bundles. The divisor class [D] = �Jx + Jy + Jz naturally splits into two parts by the e↵ectivedivisors, Dzero and Dpoles such that

[Dzero] = Jy + Jz , [Dpole] = Jx . (10)

With [Fp] · [Dzero]z=p = 3 and [Fp] · [Dpole]z=p = 2, the section D of the elliptic fibration can be constructedby appropriately choosing the two sections,

szero 2 H0(X, LDzero) , spole 2 H0(X, LDpole) , (11)

of Lzero = OX(0, 1, 1) and Lpole = OX(1, 0, 0) so that on each p 2 B the two points of Fp \ Dpole matchwith two of the three points of Fp \Dzero. With the sections szero and spole chosen as such, the remainingone point of Fp \Dzero gives the single intersection point of Fp \D. In the following, we provide how theprocedure works at a practical level.

Step 1: Fix the complex structure of X From the configuration matrix, Eq. (1), we choose a genericpair of defining equations:

P (x,y, z) = 2x0y0z0 + 12x0y1z0 + 14x1y1z0 + 6x0y2z0 � 6x1y2z0 + 18x0y0z1 + 19x1y0z1 � 16x0y1z1

�14x1y1z1 � 20x0y2z1 + 4x1y2z1 ,

Q(x,y, z) = 12x0y20z0 + 17x1y

20z0 � 19x0y0y1z0 � 16x1y0y1z0 � 17x0y

21z0 + 4x1y

21z0 � 15x0y0y2z0

�10x1y0y2z0 � x0y1y2z0 � 5x1y1y2z0 + 17x0y22z0 � 11x1y

22z0 + 10x0y

20z1 � 12x1y

20z1

+7x0y0y1z1 + x1y0y1z1 + 15x0y21z1 + 2x1y

21z1 + x0y0y2z1 � 12x1y0y2z1 � 13x0y1y2z1

�19x1y1y2z1 + 6x0y22z1 + 17x1y

22z1 .

Step 2: Demand Fp \ Dpole ⇢ Fp \ Dzero over each p 2 B Starting from a generic section ofLDpole

= OX(1, 0, 0),spole(x,y, z) = x0 + 10x1 , (12)

we shall look for an appropriate section of LDzero = OX(0, 1, 1),

szero(x,y, z) = a1y0z0 + a2y0z1 + a3y1z0 + a4y1z1 + a5y2z0 + a6y2z1 , (13)

2


Considering the tuning of Dzero/Dpole allows for an analytic solution for

the section:

for which Fp \ Dpole ⇢ Fp \ Dzero. Demoting the base coordinates z = [z0 : z1] to parameters and solvingthe system,

P (x,y, z) = 0 , (14)

Q(x,y, z) = 0 , (15)

spole(x,y, z) = 0 , (16)

for x and y, one obtains two solutions (x(i), y(i)) = (x(i)(z), y(i)(z)), for i = 1, 2. We then substituteeach of these to Eq. (13) and demand that szero(x

(i),y(i)) ⌘ 0 as a function of z, for i = 1, 2. This turnsout to fix the section szero uniquely (up to scaling) as

(a1, a2, a3, a4, a5, a6) = � (20, 161, 106,�146, 66,�204) , with � 2 C⇤ . (17)

Step 3: Obtain the parametric expression for D Having specified the two divisors, Dzero and Dpole

of X as the vanishing loci of szero and spole, respectively, we can now explicitly parametrise the section,D = Dzero \ Dpole, of the elliptic fibration in terms of the base coordinates z. As a result we obtain thefollowing parametric expression for D,

x0 = A0(z) , x1 = A1(z) , y0 = B0(z) , y1 = B1(z) , y2 = B2(z) , (18)

with the quintic polynomials Ai=0,1(z) and the quadratic polynomials Bi=0,1,2(z) given by

A0(z) = 241226z50 � 2444409z4

0z1 + 6970327z30z

21 � 4889388z2

0z31 � 2858859z0z

41 + 992331z5

1 ,

A1(z) = 152844z50 � 1296506z4

0z1 + 3553577z30z

21 � 8289055z2

0z31 + 11322255z0z

41 � 5290227z5

1 ,

B0(z) = 77z20 � 447z0z1 + 144z2

1 , (19)

B1(z) = �82z20 � 12z0z1 + 306z2

1 ,

B2(z) = �99z20 + 428z0z1 � 561z2

1 .

2 Blowing Down the Fibre

Once a section is shown to exist, the elliptic fibration needs to be put in the Weierstrass form so that thesingularity structure can be studied in more detail. Systematic algorithm for obtaining a Weierstrass formhas not been understood for a general elliptic fibration given in a CICY form (cf. Ref. []). Our strategyis to first blow down the fibre to a simpler elliptic curve, desirably to a cubic hypersurface in P2, in whichcase, the Jacobian of the elliptic fibration can be obtained by using the procedure given in Ref. [].

In this section, we illustrate the procedure by the following example,

F =

266664

P1x1

1 1 0 0 0 0

P1x2

1 0 1 0 0 0

P2x3

0 2 0 0 0 1

P3x4

0 0 1 1 1 1

377775

�! X =

26666666664

P1x1

1 1 0 0 0 0

P1x2

1 0 1 0 0 0

P2x3

0 2 0 0 0 1

P3x4

0 0 1 1 1 1

P1x5

1 0 0 1 0 0

P1x6

0 1 0 0 1 0

37777777775

, (20)

3

All other divisors blow-ups of this one. Can use it to explicitly put K3 in

Weierstrass form.

For CY 3- and 4-folds, sections obtained as above are generically rational

Technical obstacle: Care must be taken with blow-downs. Generally

many “paths”, not all compatible with a given rational section.


Summary of approach

Find all elliptic/K3 fibrations using toric projection or other methods

Scan for putative sections

Analyze blow-downs and sections

Write down Weierstrass models for each relevant fibration

Compare “duals”

Note: Generic e.g.s with complete intersection fibers, multiple sections,

multi-sections, etc. Timely application of new tools!

First database scan underway for CICY 3-folds and to be available soon...

(same techniques can be applied to toric datasets)


Implications for heterotic moduli spaces in 4-dimensions

In recent work with W. Taylor, we have used 4D Het/F-theory to obtain

systematically study the general properties/constraints of EFT for this

class of string compactifications

Develop a general formalism: For smooth X3, possible B2 classified

(generalized del Pezzo). Build an algorithm to construct all B3 that are

non-degenerate P1 fibrations over any B2.

In the context of today’s talk: What happens in this context if heteroticX3 has multiple fibrations?

X3

π1

E

~~E

π2 B(1)2

B(2)2

Leads to multiple F-theory duals Y(1)4 , Y

(2)4 , . . .


η: Building bundles and B3

Idea: Choose topology of bundles (V1,V2) ⇔ Build ρ : B3P1

−→ B2

Heterotic:

Can expand:

c2(Vi ) = ηi ∧ ω0 + ζi ,

w/ ηi (resp. ζi ) {1, 1} (resp.

{2, 2}) forms on B2 and ω0 dual

to the zero section.

Anomaly Cancellation ⇒

η1,2 = 6c1(B2)± t

Can build B3 over B2 by

“twisting” the P1 fibration

(analog of Fn surfaces in 6D)

B3 = P(O ⊕ L)

c1(B3) = c1(B2) + 2Σ + t

where Σ is dual to the

zero-section of the P1-fibration

In Het/F-dual pairs, two t’s are the same (FMW), (Grimm + Taylor)

Next: Bounds on twists ⇒ finite # B3 sol’ns/enumeration


Bounds on the structure group, H

“Generic” symmetries on Y4 provide

rank(V )-dependent vanishing criteria

for M(c(V )). (First studied by

Rajesh and Berglund & Myer)

Also constraints on which symmetries

can be enhanced

non-Higgsable SU(2),SU(3) 6→ SU(5)

Can be pinned at exactly one

symmetry (or a sparse set)

Intriguing for string pheno...

H η ≥ Nc1(B2)

N =

SU(n) n (n ≥ 2)

SO(7) 4

SO(m) m2 (m ≥ 8)

Sp(k) 2k (k ≥ 2)

F4133

G272

E692

E7143

E8 5


Multiple windows into the moduli space

In the context of multiple fibrations, we can see that if the heterotic base

X3 admits multiple fibrations:

Several F-theory duals Y4 (possibly over different bases) for the same

bundle

Consistency conditions: Y4 can be resolved into a smooth Calabi-Yau 4-fold

Need vanishing degrees of (f , g ,∆) ≤ (4, 6, 12) on every divisor in B3

f , g cannot vanish to orders 4, 6 on any curve.

Each of these Y4 can constrain different components (η, η′, . . .) of c2(V )

Could potentially lead to more complete control of the heterotic moduli

space


Conclusions and Future Directions

Most known CY manifolds are elliptically fibered → generically with

multiple fibrations

Multiple fibrations can impact a huge array of dualities and shed light on

different aspects of string/F-theory

On multiple fronts, the technical tools are just now becoming available to

analyze these relationships in detail

We have provided an algorithmic toolkit to make it possible/feasible to

analyze these dualities systematically

Goal: A map of dualities

Scans underway, stay tuned...


The End


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