the boundaries of the 4d f-theory landscape · 2020. 6. 15. · ux choices 10272 ;000(10224 with...

The Boundaries of the 4D F-theory Landscape

Yi-Nan Wang

University of Oxford

StringPheno 2020, Online

June 12th, 2020

Yi-Nan Wang The Boundaries of the 4D F-theory Landscape 1 / 35

Papers Mentioned

1 YNW, On the Elliptic Calabi-Yau Fourfold with Maximal h1,1,

(2001.07258)

2 Taylor, YNW, Scanning the skeleton of the 4D F-theory landscape,

(1710.11235)

3 Taylor, YNW, The F-theory geometry with most flux vacua,

(1511.03209)


Landscape vs Swampland

String Landscape

Swampland

Inconsistent theories

Boundary of String Landscape

SwamplandBounds

• Enlarge the boundary of known string landscape

• Refine the swampland bounds



String Landscape

Swampland



SwamplandBounds

String Landscape


Boundary of Quantum Gravity

Assuming String Universality



String Landscape

Swampland



SwamplandBounds

• Enlarge the boundary of known string landscape

• Refine the swampland bounds


Questions on the Boundary of Landscape

• Fix a space-time dimension and N supersymmetry

• In the low energy effective theory, what is the upper limit of a certain

type of matter fields coupled to Einstein gravity in the same dimension?

1 Vector multiplet; The total rank of gauge group G

2 Axion scalars

3 Tensor multiplet (in 6d)

• Quick answers in the known landscape:

(1) 4d N = 1: max(rk(G )) ∼ 105, max(Naxion) ∼ 105

(2) 6d (1,0): max(rk(G )) ∼ 102, max(T ) ∼ 102

(3) 5d N = 1: max(rk(G )) ∼ 102


4d N = 1 Supergravity

• Supergravity multiplet, Vector multiplet, Chiral multiplet

• String realizations:

1 F-theory on elliptic CY4

2 Heterotic string on CY3

3 M-theory on G2 manifold

4 Intersecting brane models, e. g. IIA with D6 branes

• F-theory landscape provides the known upper limits to the previous

questions


4d N = 1 Supergravity

• Supergravity multiplet, Vector multiplet, Chiral multiplet

• String realizations:

1 F-theory on elliptic CY4

2 Heterotic string on CY3

3 M-theory on G2 manifold

4 Intersection brane models, e. g. IIA with D6 branes

• F-theory landscape provides the known upper limits to the previous

questions


4d F-theory

• (Singular) elliptic CY4 X4 fibered over a smooth base threefold B3:

y2 = x3 + f (zi )x + g(zi ) . (1)

• f (zi ) ∈ O(−4KB3 ), g(zi ) ∈ O(−6KB3 )

∆(zi ) = 4f (zi )3 + 27g(zi )

2 . (2)

• Discriminant locus ∆ = 0: the elliptic fiber is singular.

Codimension Sub-manifold Physical field

One divisor D ⊂ B3 4d non-Abelian gauge group

Two curve C ⊂ D weakly coupled matter fields

Three point p ⊂ D Yukawa coupling

• flat fibration: ordD/C/p(f , g ,∆) lower than (4, 6, 12)

• The crepant resolution of X4: X4 with at most terminal singularity


Non-minimal Loci in 4d F-theory

ord(f , g ,∆) cod-1 cod-2 cod-3

≥ (4, 6, 12) bad Strongly coupled matter Higher Yukawa-coupling

≥ (8, 12, 24) bad bad Strongly coupled sector

≥ (12, 18, 36) bad bad bad

• Cod-1 (4, 6), cod-2 (8, 12), cod-3 (12, 18): the singularity of X4 is

worse than canonical, strictly forbidden. (No SUSY vacua)


Non-minimal Loci in 4d F-theory

• Cod-2 (4, 6)

1 Blow up C : ρ : B ′3 → B3, X ′

4 does not have cod-2 (4, 6) locus.

X ′4 ∼ X4, has the same Hodge numbers.

2 Directly study (non-flat) X4 and X4 over B3: strongly coupled

matter localized on C , e. g. 6d (1,0) SCFT reduced on C . (Kim,

Razamat, Vafa, Zafrir 17’). . .

• Cod-3 (8, 12)

1 Blow up p: ρ : B ′3 → B3, X ′

4 does not have cod-3 (8, 12) loci.

X ′4 ∼ X4, has the same Hodge numbers.

2 Directly study X4 and X4 over B3: strongly coupled sector localized

at p (Apruzzi, Heckman, Morrison, Tizzano 18’)


Non-minimal Loci of 4d F-theory

• “Good” fibration: X4 without cod-1 (4, 6), cod-2 (4, 6) or cod-3

(8, 12) loci. A weakly coupled description of low energy theory exists.

Allowed:

• Cod-3 (4, 6): if ordp(f , g ,∆) ≥ (4, 6, 12) but not as high as (8, 12, 24).

Higher order yukawa coupling (Achmed-Zade, Garcıa-Etxebarria, Mayrhofer 18’)

• Terminal singularities in X4.

• The Tate-Shioda-Wazir formula can be applied:

h1,1(X4) = h1,1(B3) + rk(G ) + 1 . (3)


Classification of compact 4d F-theory models

1 Classify the topology of compact base threefolds B3

2 Classify different X4 over each B3 (birationally inequivalent X4)

3 Fix the geometry, enumerate the G4 flux choices

G4 +1

2c2(X4) ∈ H4(X4,Z) . (4)

• G4 is crucial for a chiral spectrum, moduli stabilization etc.

• The number of G4 choices is bounded:

ND3 =χ(X4)

24− 1

2

∫X4

G4 ∧ G4 ≥ 0 . (5)


Classification of CY4

• CY4 as hypersurface in ambient toric space: (Kreuzer, Skarke

97’)(Candelas, Perevalov, Rajesh 97’)(Klemm, Lian, Roan, Yau 97’). . .


Classification of elliptic CY4

• The number of elliptic CY4 up to isomorphism in cod-1 is finite (Di

Cerbo, Svaldi 16’)

• Number of possible Hodge number combinations and upper bound on

Hodge numbers: finite

• Topology of B3: one of the following or a blow-up of:

1 Fano threefold

2 B2 over P1

3 P1 over B2


Explore the set of smooth toric B3

• B3 = P1 over B2 (Halverson,Taylor 15’), N > 105

• Random flops starting from P3 (Taylor, YNW 15’), N > 1048

• Blow up weak-Fano toric threefolds in a combinatoric way (Halverson,

Long, Sung 17’), N > 10755; allowing non-minimal loci.• Random blow-up from P3 (Taylor, YNW 17’)

1 Bases that support a good fibration: N > 10250

2 Bases that supports a non-minimal fibration: N > 103000


Random toric blow-up from P3(Taylor, YNW 17’)

• In the random blow-up process, allow non-minimal cod-2 (4,6) and

cod-3 (8,12) loci

• The end point (any further blow-up leads to a bad base) always

support a good fibration

• 15% of the random blow-up sequences ends up at bases with

h1,1(B3) = 2303!

• Question: what is the structure of end point bases?Yi-Nan Wang The Boundaries of the 4D F-theory Landscape 17 / 35

Classification of elliptic CY4

• Boundary of 4d F-theory landscape

• CY4 with the largest h3,1 and G4 flux choices (Taylor, YNW 15’)

• CY4 with the largest h1,1, gauge rank and axions (YNW 20’)


Elliptic CY4 with the largest h3,1(Taylor, YNW 15’)

• (h1,1, h2,1, h3,1) = (252, 0, 303148)

• The base B3 is a B2 fibration over P1, where B2 is the base surface for

the self-mirror elliptic CY3 with (h1,1, h2,1) = (251, 251).

G = E 98 × F 8

4 × (G2 × SU(2))16 (6)

• Typical non-Higgsable gauge group in 4d F-theory landscape (Halverson,

Long, Sung, Taylor, YNW...)


Elliptic CY4 with the largest h3,1

• Rough estimation of G4 flux choices:

1

2

∫G4 ∧ G4 ≤

χ(X4)

24. (7)

• Flux(self-dual flux): count the number of lattice points in a

b4(b4/2)-dimensional ball with radius√χ/12

• # of flux choices 10272,000 (10224,000 with self-duality condition).

• Both b4 and χ of this geometry are the largest → largest # of flux

• Can be over-counting due to non-trivial metric of the flux space (Cheng,

Moore, Paquette 19’)Yi-Nan Wang The Boundaries of the 4D F-theory Landscape 20 / 35

Elliptic CY4 with the largest h1,1(YNW 20’)

• (h1,1, h2,1, h3,1) = (303148, 0, 252)

• G = E 25618 × F 7576

4 × G 201682 × SU(2)30200 (Candelas, Perevalov, Rajesh 97’)

• Largest gauge group in the known 4d N = 1 landscape

• Tate-Shioda-Wazir formula

h1,1(X4) = h1,1(B3) + rk(G ) + 1 (8)

• h1,1(B3) = 181819, largest number of axions (Grimm, Taylor 12’)

Naxion = h1,1(B3) + 1 = 181820 (9)

• What is the structure of B3?


Construction of B3

• Step 1: construct a non-compact toric threefold BE8 with the following

3d polytope:

• After triangulation, the toric fan has 5016 3d cones, 7576 2d cones and

2561 rays

• Tune an E8 on each of these rays!

• Step 2: add two more rays into BE8 to make it compact

• Step 3: blow up the 5016 (E8,E8,E8) non-minimal loci and 7576

(E8,E8) non-minimal loci.


Construction of B3

• Blow-up of (E8,E8,E8) non-minimal loci (Apruzzi, Heckman, Morrison,

Tizzano 18’)

• Blow-up of (E8,E8) non-minimal loci: similar to the tensor branch of

(E8,E8) conformal matter in 6d (Morrison, Taylor 12’)(Heckman, Morrison, Vafa

13’)(Del Zotto, Heckman, Tomasiello, Vafa 14’)

E8

E8

E8


Construction of B3

• Blow-up of (E8,E8,E8) non-minimal loci (Apruzzi, Heckman, Morrison,

Tizzano 18’)

• Blow-up of (E8,E8) non-minimal loci: similar to the tensor branch of

(E8,E8) conformal matter in 6d (Morrison, Taylor 12’)(Heckman, Morrison, Vafa

13’)(Del Zotto, Heckman, Tomasiello, Vafa 14’)

100

010

001

501

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510

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221211

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E8

E8

E8

F4

F4

F4

G2

G2

G2

G2

G2

G2

G2

SU(2)

SU(2)

SU(2)

SU(2)

SU(2)

SU(2)

SU(2)

SU(2)

SU(2)

II II

II

II

II

II

IIII

II

II

II

II

II

II

II

II

II

II

II


Construction of B3

• Remarks:

(1) G2 − SU(2)− II collision: cod-3 (4,6) non-minimal loci

(2) II − II collision: terminal singularity in X4

• Step 4: there are 619 E8 divisors with non-toric (4, 6)-curves, need to

blow them up.

• Finally get B3 with h1,1(B3) = 181819! X4 is a good fibration.


Construction of B3

Comments:

• B3 is an end point, cannot further blow it up

• G = E 25618 × F 7576

4 × G 201682 × SU(2)30200 is the non-Higgsable gauge

group on B3. Cannot tune bigger gauge groups

• Number of self-dual G4 flux choices ∼ 10194000, smaller than the CY4

with largest h3,1

• Exists a number of D3-branes depending on G4 → Additional 4d gauge

groups

ND3 ≤χ

24= 75852 (10)


Counting flop phases

• Question: how many B3 has the same h1,1(B3) = 181819?

• There exists many flip/flop phases within a single (E8,E8,E8) triangle!

100

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023012

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140130

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320210

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221 211

311

411

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E8

E8

E8

F4

F4

F4

G2

G2

G2

G2

G2

G2

G2

SU(2)

SU(2)

SU(2)

SU(2)

SU(2)

SU(2)

SU(2)

SU(2)

SU(2)

II II

II

II

II

II

IIII

II

II

II

II

II

II

II

II

II

II

II



• There exists many flip/flop phases within a single (E8,E8,E8) triangle!

100

010

001

501

401

301

201

101

302

203

102

103

104

105

051

041031

021

011

032

023012

013

014

015

150

140130

120

110

230

320210

310

410

510

111121131141

221211

311

411

112

113

114

122

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132

231

321

312

213

123

E8

E8

E8

F4

F4

F4

G2

G2

G2

G2

G2

G2

G2

SU(2)

SU(2)

SU(2)

SU(2)

SU(2)

SU(2)

SU(2)

SU(2)

SU(2)

II II

II

II

II

II

II

II

II

II

II

II

II

II

II

II

II

II

• No cod-3 (4, 6)-locus, no terminal singularity from II − II collision

• X4 is smooth and flatYi-Nan Wang The Boundaries of the 4D F-theory Landscape 28 / 35


• At least 11003 different flip/flop phases in a single (E8,E8,E8) triangle

• In total 5016 of these triangles:

Nflp > 110015048

≈ 7.5× 1045766 .(11)

• The number of self-dual G4 flux choices 10194000 multiplied by Nflp,

bigger than the number 10224000 on the CY4 with largest h3,1!


Pheno Implications

• N-inflation with large number (105) of axions

• Large amount (104) of dark gauge sectors

• Standard model sector still hard to construct, more likely from

D3-brane sectors


Solving the End Point Puzzle

• The end point bases in the random blow-up program with

h1,1(B3) = 2303 can be constructed in a similar way

• Step 1: take a polytope that is the dual of P3 polytope. Triangulate it

and get BE8 .

• Step 2: tune an E8 on every divisor of BE8 , blow up all the non-minimal

loci

• Step 3: blow up non-toric (4, 6)-curves

• Get B3 that supports X4 with (h1,1, h2,1, h3,1) = (3878, 0, 2): mirror of

the generic elliptic CY4 over P3!


Other Dimensions

(1) 6d (1,0): tensor multiplet T , vector multiplet V , hypermultiplet H

• Largest number of tensor multiplets and vector multiplets: F-theory on

elliptic CY3 X3 with (h1,1, h2,1) = (491, 11) (Morrison, Taylor 12’)(Taylor 12’)

• T = 193, V = 296, G = E 178 × F 16

4 × (G2 × SU(2))32.

• Base geometry:

(−12//− 11//(−12//)13,−11//− 12, 0) . (12)

// ≡ −1,−2,−2,−3,−1,−5,−1,−3,−2,−2,−1 . (13)

• Gravity coupled to the tensor branch of non-minimal (E8,E8) conformal

matter with 17 E8s


Other Dimensions

(2) 5d N = 1: vector multiplet V , hypermultiplet H

• Largest number of vector multiplets: M-theory on the resolved CY3 X3

with (h1,1, h2,1) = (491, 11)

• Non-Abelian quiver gauge theory (Ohmori, Shimizu, Tachikawa, Yonekura

12’)(Apruzzi, Schafer-Nameki, YNW 19’):

SU(48)

|SU(16)− SU(32)− SU(48)− SU(64)− SU(80)− SU(96) −SU(64)− SU(32) .

(14)

• SU(96) is the largest SU(N) gauge group coupled to 5d N = 1

supergravity? (Katz, Kim, Tarazi, Vafa 20’)

(3) 2d (0, 2): F-theory on elliptic CY5 (ongoing work with Tian)


Further Questions

• Prove the bounds with swampland constraints?

• Challenge the bounds in other parts of string landscape? (For example

M-theory on G2)

• Phenomenological Implications


Further Questions

• Prove the bounds with swampland constraints?

• Challenge the bounds in other parts of string landscape? (For example

M-theory on G2)

• Phenomenological Implications

• Thank you!


the boundaries of the 4d f-theory landscape · 2020. 6. 15. · ux choices 10272 ;000(10224 with...

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