Calabi-Yau fibrations, finiteness, and string dualities
Lara B. Anderson
Virginia TechWork done in collaboration with:(L.A, J. Gray, X. Gao, B. Hammack, S. J. Lee)- arXiv:1805.05497,1708.07907,1608.07555,1608.07554

STRINGS 2018
OIST, Okinawa
June 29th, 2018

Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’18

1 / 19Motivation
Goal: A better understanding of possible string effective theories
Within string compactifications, which EFTs? ↔ which geometries?
Even the simplest class of compactification manifolds → Calabi-Yau(complex) n-folds, unclassified for n > 2. Finite?
Even weaker question: is the topology (i.e. for CY 3-folds values ofh1,1 , h2,1 , Ch2 (TX ), etc) of Calabi-Yau n-folds bounded?
In this talk: I’ll outline one approach these questions via CY fibrations.
Quick reminder:
A fibration is a surjective morphism π : X → B such that for almost all
points b, b 0 ∈ B, π −1 (b) ' π −1 (b 0 ) (homotopy lifting)
A continuous map σ : B → X s.t. π(σ(b)) = b for all b ∈ B is called asection to the fibration.
Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’18

2 / 19

Why CY fibrations?

Fibrations play a key role in string dualities: F-theory/M-theory(E-fiber), Heterotic/F-theory (E/K3-Fiber), Heterotic/Type IIA, etc.
Fiber/base decomposition simplifies geometry, increases calculability(form of intersection numbers, metric, etc).
Theorem (Gross, Grassi, ‘93): The set of genus one fibered CY 3-folds isfinite.
(Di Cerbo + Svaldi, 2016): Finite # of bases for elliptically fibered CY 4and 5-folds.
(Terminology: If a section exists, torus/genus one fibration ⇒ “elliptic”)
Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’18

3 / 19A few results on CY fibrations:
CY n-folds can only be fibered by m-folds with c1 = 0, with 0 < m < n.
The existence of a fibration is a deformation invariant statement for CY
n-folds with n > 2: If X is a genus-1 fibered CY manifold with
H 2 (X , OX ) = 0, then every small deformation of X is also genus 1 fibered.
Motivation behind Gross’ theorem: Can finiteness of genus one fibered CY
threefolds be used to say anything about the finiteness of all CY 3-folds?

Can geometric transitions linking elliptically fibered vs. non-fibered CYs
be characterized/classified?
Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’18

4 / 19Observations

Consider your favorite dataset of CYs within the order ∼ 109 known (i.e.
CICYs (Candelas, et al ’87), toric hypersurfaces (Kreuzer/Skarke, 2000),etc)
Immediate Observation 1: Almost all known CY 3- and 4- folds arefibered (with elliptic, K3, CY3, fibrations, etc)
Immediate Observation 2: generic manifolds do not admit just one elliptic
fibration, they admit many (order 10s, 100s, 1000s...more).
What can we make of this? Let’s take it one at a time...

Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’185 / 19

Genericity of fibrations for known datasets
Toric hypersurfaces: (Rohsiepe,Braun, Morrison, Taylor, Huang...)
For the CICY3-folds 99.3%. E.g.








P2
P1

0

3

1

1

P2

3

0

19,19

CICYs 4-folds in products ofh21
500

projective spaces 99.9%

400(921, 020 out of 921, 497) (Gray,300

Haupt, Lukas ’14)
200














100

00

100

200

300

400

500

h11Figure 2: The 7524 distinct Hodge number pairs for generic elliptically fibered Calabi-Yau threefolds overP1P3

1

1000001000111P4P101110110011000P1P10010100





0

10000

1

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.

(from Taylor 1205.0952)

Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’186 / 19Genericity of fibrations for known datasets
Toric hypersurfaces: (Rohsiepe,Braun, Morrison, Taylor, Huang...)
For the CICY3-folds 99.3% E.g.





P2
P10

3

1

1P2

30

19,19






CICYs 4-folds in products ofh21
500

projective spaces 99.9%

400(921, 020 out of 921, 497) (Gray,300

Haupt, Lukas ’14)
200














100

00

100

200

300

400

500

h11Figure 2: The 7524 distinct Hodge number pairs for generic elliptically fibered Calabi-Yau threefolds overP1P3

1

100000

1000111P4P1011101

1001

1000P1P100

10100




0

100001

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.

(from Taylor 1205.0952)

Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’18

7 / 19Multiplicity of fibrations
104

Point of interest: When suchgenerically do not have just one...

Abundance

1000

manifolds have fibrations, they100

10In fact there can be many

1050

100

150

200

250

300

350

Number of fibrations per configuration

P1P21)  P1
2)











0

1

1



0

12











1

0

1

P1
P1

10

1

1

0

1

P1
P1

101



0

1

1P2
P1

10

2

0

1

1P1











0

1

1CICY4-folds(Gray,et al in1405.2073)(∼
108listfibrations)Figure 6. Distributionof elliptic
fibration
abundance
the CICY four-fold(excluding product500
manifolds). The values
lie in the range 0 - 354. We find 49,466,248 OEFs in total and on averageeach CICY four-fold configuration is elliptically fibered in 54.6 di↵erent ways.

100
There are two di↵erent
types of fibre configurations in our list. The first type is given50
by block-diagonal fibre configurations, such as, for example
"
#10
1 20.
(4.4)
52 03
Abundance

E.g.
This configuration describes two points in P1 times a torus [2|3]. The fibres of a total
1of 2,149,222 OEFs are block-diagonal. The remaining 47,317,026 non block-diagonal fibre
204060fibrations80can degenerate overconfigurations matrices 0describe irreducible
tori. Theseof torus
per99.4%
configuration
special loci in the base. ItNumber
should benotedfibrations
that someof the fibre descriptions(∼ of140,000 fibrations)contain linear constraintsCICY
in the 3-folds
coordinatesa single
projective space. However, suchnlinear constraints cannot be removed (by replacing the relevant P with Pn 1 ) as di↵erent
redundant descriptions of the fibre can be twisted over the base of the OEF in inequivalentThus far: “Obvious” fibrations
Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’18

8 / 19

Multiplicity of fibrations
104

Point of interest: When suchgenerically do not have just one...

Abundance

1000

manifolds have fibrations, they100

10In fact there can be many

1050

100

150

200

250

300

350

Number of fibrations per configuration

P1P21)  P12)




0

11

0

12


1

0

1P1P1

10

1

10

1P1
P1

101



0

1

1P2P1

10

20

1

1P1








0

1

1

CICY
4-folds(Gray,et al in
1405.2073)(∼
108listfibrations)Figure 6. Distributionof elliptic
fibration
abundance
the CICY four-fold(excluding product500
manifolds). The values
lie in the range 0 - 354. We find 49,466,248 OEFs in total and on averageeach CICY four-fold configuration is elliptically fibered in 54.6 di↵erent ways.

100
There are two di↵erent
types of fibre configurations in our list. The first type is given50
by block-diagonal fibre configurations, such as, for example
"
#10
1 20.
(4.4)
5
2 03
Abundance

E.g.
This configuration describes two points in P1 times a torus [2|3]. The fibres of a total
1of 2,149,222 OEFs are block-diagonal. The remaining 47,317,026 non block-diagonal fibre
2040
60fibrations80can degenerate overconfigurations matrices 0describe irreducible
tori. Theseof torus
per99.4%
configuration
special loci in the base. ItNumber
should benotedfibrations
that someof the fibre descriptions(∼ of140,000 fibrations)contain linear constraintsCICY
in the 3-folds
coordinatesa single
projective space. However, suchnlinear constraints cannot be removed (by replacing the relevant P with Pn 1 ) as di↵erent
redundant descriptions of the fibre can be twisted over the base of the OEF in inequivalentThus far: “Obvious” fibrations
Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’18

9 / 19Result: We scanned for obvious fibrations in CICY 3-fold dataset (7, 890manifolds) and found 139, 597 genus one fibrations, 30, 974 K3-fibrations
and 208, 987 nested combinations.
But...no guarantee that obvious fibrations are the whole story. Need tobe able to characterize fibrations independent of algebraic description ofthe manifold.
It is possible to systematically characterize elliptic fibrations bycharacterizing the base: π : X → B.
Conjecture (Kollár): Let X be a Calabi-Yau n-fold. Then X is genus-1
fibered iff there exists a (1, 1)-class D in H 2 (X , Q) such that (D · C ) ≥ 0for every algebraic curve C ⊂ X , (D dim(X ) ) = 0 and D dim(X )−1 6= 0.

(proven for CY 3-folds (Wilson, Oguiso) with D effective or D · c2 (X ) 6= 0,

evidence for n > 3)(Note: the fiber class = D 2 ).
Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’18

10 / 19

Enumerating fibrations
Can all genus-one fibrations in a dataset be enumerated? Given any
dataset of CY 3-folds, can search for divisors satisfying the criteria, butneed the following:
1

Explicit control of all divisors −→ Kähler and Mori cones of X

2Intersection numbers of X

How would such a fibration count compare to “obvious” count? Onceagain, start with the simplest dataset of manifolds 7, 890 CICY 3-folds(Candelas, et al).
Developed the tools to classify fibrations in CICYs via algorithmic
replacement of CICYs with different algebraic descriptions,Gromov-Witten computations, etc.
Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’18

11 / 19

Results:
1

Scanned over all 7890 CICY 3-folds in augmented dataset (with favorablereplacements)2

For favorable CICYs in which the Kähler cone strictly descends from theambient product of projective spaces, we find (# of obvious fibrations =exhaustive list of fibrations). Otherwise, can find many more...

3> 99.4% of CICY 3-folds are genus one fibered. All E-fibered for h1,1 > 4.

4Classify 377, 559 fibrations for the set of 4, 957 Kähler favorable manifolds.

5

Many non-obvious fibrations: E.g. anti-canonical hypersurface in
dP7 × dP7 gives rise to 15, 878 genus one fibrations. Can group intofamilies based on birational equivalence.

6

For one manifold we find a parametrically infinite set of distinct elliptic
fibrations ⇒ (h1,1 , h2,1 ) = (19, 19) manifold (Oguiso, Aspinwall/Gross).

7

These results extend to CYs with π1 (X ) 6= 0, gCICYs, etc.

Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’18

12 / 19

What about the physics of multiple elliptic fibrations?
E.g. in 6D: M-theory on Y3 ' F-theory on Y3 × S 1
Different fibrations of Y3 will lead to different F-theory vacua, all with thesame M-theory limit (i.e. Coulomb branch)
Example with h1,1 = 7, h2,1 = 26:
P1 1 1 0
 1
 P 0 0 1
 2
 P 0 0 1

X3 =  P2 0 0 0
 2
 P 0 1 0

 2
 P 0 0 0P2 1 0 0
Lara Anderson (Virginia Tech)

00000100000110010001010010011010010

Calabi-Yau fibrations, finiteness, and string dualities0
0 
0 
1 
0 
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0STRINGS 2018, June 29th, ’18

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E1 : U(1)5
V = 5, T = 0, H = 278
E9 : SU(2) × U(1)4

E2 : SU(2) × U(1)3

V = 7, T = 0, H = 280

V = 6, T = 1, H = 250

E8 : SU(2) × U(1)45D M-theory Limitwith

V = 7, T = 0, H = 280

E3 : SU(2) × U(1)3

V = 6, T = 1, H = 250

V M = 6, H M = 27
E7 : SU(2) × U(1)4

E4 : U(1)4

V = 7, T = 0, H = 280

V = 4, T = 1, H = 248

E6 : SU(2)2 × U(1)3

E5 : SU(2) × U(1)3

V = 9, T = 0, H = 282

V = 6, T = 1, H = 250

Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’18

14 / 19

Heterotic/F-theory Duality
Based on 8-dimensional correspondence (Vafa):
K3

E

Heterotic on πh : Xn −→ Bn−1 ⇔ F-theory on πf : Yn+1 −→ Bn−1
Leads to nested fibrations (w/ sections)
Yn+1K3↓Bn−1

E−→=←→Bn↓

P1

Bn−1

Heterotic geometry also includes Mumford (slope) stable bundles V → Xn
Let’s explore the possibilities for multiple K 3 fibrations within the theF-theory geometry, Yn+1 , and what it means for the heterotic duals...
Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’18

15 / 19

Case 1: Multiple K 3-fibrations with distinct elliptic fibrations → Samestory as M-/F- correspondence. Duality in 1D lower.
Case 2: Multiple K 3-fibrations with the same elliptic fibration
BnP1

P1
ρ1}

ρ2!

Bn−1

0
Bn−1

→ identical effective heterotic theories on different geometries:

0πh : Xn → Bn−1and πh0 : Xn0 → Bn−1
Case 3: One K 3-fibration with multiple elliptic fibrations:
Yn+1
E

BnE
π2π1}

P1
ρ1"B0 nP1!

|ρ2

Bn−1→ distinct heterotic theories on the same CY geometry: πh : X → Bn−1
Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’18

16 / 19

Examples“Case 3” Can describe E8 × E8 /SO(32) duality in 8D and in lower
dimensions (e.g. P1 (SO(32), E8 × E8 duality → πf : Y3 → F4 )Simple example of “Case 2”: πf : Y3 → F0 = P1 × P1
Morrison+Vafa: These multiple fibrations lead to 6D heterotic/heteroticduality of Duff, Minasian, Witten (nT = 1)
When nT > 1 much more is possible, by inspecting multiple fibrations, wefind many generalizations of DMW in 6D (e.g. geometry given by conic
bundles not P1 -bundles).

Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’18

17 / 19

Novel dualities in 4-dimensions
Consider multiple elliptic fibrations within
The 4-dimensional heterotic theory is independent of which way “up” it’s
elliptic fiber is oriented. Suppose
X3
EB2



Eπ2π1B20

Leads to F-theory on two distinct 4-folds:
K3

ρ : Y4 −→ B2 ,,00

K30

ρ : Y4 −→ B2

with the same effective physics!
Can shed light on G-flux.
Systematic study of F-theory duals can lead to important bounds on thestructure of MX (c(V )) (Mod. space of stable sheaves on X ) (LA + Taylor)
Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’18

18 / 19

Conclusions and Future Directions

We have completed the first complete classification/enumeration ofgenus-one fibrations in a set of CICY 3-folds.
Evidence is accumulating that as h1,1 increases, all known CY manifolds
are elliptically fibered → generically with multiple fibrations.
Can this be established for all CY 3-folds? If so, may provide a tractableapproach to finiteness...
Multiple fibrations can impact a huge array of dualities and shed light on
new aspects of string/F-theory.

Lara Anderson (Virginia Tech)Calabi-Yau fibrations, finiteness, and string dualities

STRINGS 2018, June 29th, ’18

19 / 19