Web of threefold bases in F-theory and machine learning 1510.04978 - - PowerPoint PPT Presentation

Web of threefold bases in F-theory and machine learning

1510.04978 & 1710.11235 with W. Taylor Yi-Nan Wang

CTP, MIT

String Data Science, Northeastern; Dec. 2th, 2017

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Exploring a huge oriented graph

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Nodes in the graph

• Physical setup: 4D F-theory compactification on an elliptic Calabi-Yau

fourfold X with complex threefold base B.

• The nodes are compact smooth toric threefold bases
• The elliptic fibration X over B is described by a Weierstrass form:

y 2 = x3 + fx + g (1) We require that the elliptic fibration is “generic”, hence f and g are general holomorphic sections of line bundles −4KB and −6KB.

• The gauge groups in the 4D supergravity model are

minimal(non-Higgsable).

• The number of complex structure moduli h3,1(X) is maximal.

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Toric threefolds

• Gluing C3 together such that there is an action of complex torus (C∗)3.
• Combinatoric description: a fan Σ which is a collection of 3D, 2D, 1D

simplicial cones in the lattice Z3.

• For any two cones σ1, σ2 ∈ Σ, σ1

σ2 is either another cone in Σ or the origin.

• Compactness: the total set of cones σ ∈ Σ spans the whole Z3.
• Smoothness: every 3D cone is simplicial, with unit volume.

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

z1=0 z2=0 z3=0 z4=0

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Toric threefolds

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

z1=0 z2=0 z3=0 z4=0

• 1D ray: vi corresponds to complex surface (divisor) Di; zi = 0. Total

number: n = h1,1(B) + 3.

• 2D cone: vivj corresponds to curve zi = zj = 0.
• 3D cone: vivjvk corresponds to point zi = zj = zk = 0.

The convex hull of vertices vi forms a lattice polytope ∆∗. Its dual polytope ∆ = {p ∈ Q3, ∀vi, p, vi ≥ −1} in general is not a lattice polytope.

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Line bundles on toric threefolds

Anti-canonical line bundle −KB =

i Di.

Generators of holomorphic section mp of line bundle L =

i aiDi ⇔

points p in the dual lattice Z3: {p ∈ Z3, ∀vi, p, vi ≥ −ai}. (2) mp =

• i

zp,vi+ai

i

(3)

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Line bundles on toric threefolds

Anti-canonical line bundle −KB =

i Di.

Generators of holomorphic section mp of line bundle L =

i aiDi ⇔

points p in the dual lattice Z3: {p ∈ Z3, ∀vi, p, vi ≥ −ai}. (2) mp =

• i

zp,vi+ai

i

(3) Hence f and g are linear combinations of monomials in set F and G, which are the lattice points of 4∆ and 6∆: F = {p ∈ Z3, ∀vi, p, vi ≥ −4}. (4) G = {p ∈ Z3, ∀vi, p, vi ≥ −6}. (5)

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The edges between nodes: Blow up/down

The set of toric fans Σ is infinite, because of the existence of blow up

• perations on Σ:

(1) Blow up a point vivjvk: add another ray ˜ v = vi + vj + vk. (2) Blow up a curve vivj: add another ray ˜ v = vi + vj.

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The edges between nodes: Blow up/down

• After the blow up, N becomes bigger, hence M, F&G are subsets of

the previous ones.

• If one blow up a curve vivj where (f , g) vanishes to order (4, 6) or

higher, F&G are unchanged.

• Blow down is the inverse process of blow up. A ray v can be removed if

and only if one of the following conditions holds: (1) v has 3 neighbors vi, vj, vk and v = vi + vj + vk (v is a P2) (2) v has 4 neighbors vi, vk, vj, vl, and either v = vi + vj or v = vk + vl.

• Blow up/down a smooth toric threefold will lead to another smooth

toric threefold (unit volume condition).

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Constraints on Σ

However, not all Σ are allowed in F-theory constructions.

• We require that (f , g) does not vanish to order (4, 6) or higher on any

divisor Di(vi).

• NO cod-1 (4,6) singularity

Otherwise, the singularity x = y = z in Weierstrass model y 2 = x3 + z4x + z6 (6) cannot be resolved while keeping the Calabi-Yau condition (SUSY is broken).

• Lattice condition: there is at least one point p ∈ G where vi, p < 0

for each vi.

• Equivalent (0, 0, 0) condition: the origin (0, 0, 0) cannot lie on the

boundary of G.

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Constraints on Σ

• What if (f , g) vanishes to order (4, 6) or higher on some curves vivj?

For example: y 2 = x3 + z2

1z2 2x + z3 1z3 2.

(7)

• When cod-2 (4,6) singularity appears, after the resolution process, the

elliptic fibration is non-flat (fiber components with complex dimension higher than 1 appears)(Katz/Morrison/Schafer-Nameki/Sully 11’,

Lawrie/Schafer-Nameki 12’).

• In the physics language, there are tensionless string in the low energy

effective theory/ SCFT coupled to the supergravity theory.

• Tensionless string: M5 brane wrapping the real 4-dimensional fiber

component in the M-theory dual picture. In the F-theory limit, these fiber components shrink to zero size and the string become tensionless.

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SCFT from cod-2 (4,6) singularity

• An 6D example: two (-3) curves intersecting each other, SO(8)×SO(8)

gauge group. y 2 = x3 + z2

1z2 2x + z3 1z3 2.

(8)

• 3
• 3

SO(8) SO(8) z1=0 z2=0

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SCFT from cod-2 (4,6) singularity

• An 6D example: two (-3) curves intersecting each other, SO(8)×SO(8)

gauge group. y 2 = x3 + z2

1z2 2x + z3 1z3 2.

(8)

• 3
• 3

SO(8) SO(8) z1=0 z2=0

• 4
• 4

SO(8) SO(8) z1=0 z2=0

• 1

Blow up the point z1=z2=0

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SCFT from cod-2 (4,6) singularity

• 6D (1,0) theories can be classified by their tensor branches using

F-theory tools (Heckman/Morrison/Rudelius/Vafa 13’ 15’)

• If we take this part of geometry (-4/-1/-4) out, then the tensor branch

is “[SO(8)] 1 [SO(8)]”

• If we shrink the (−1)-curve to zero size, the v.e.v. of scalar in tensor

multiplet vanishes and we get an SCFT.

• 3
• 3

SO(8) SO(8) z1=0 z2=0

• 4
• 4

SO(8) SO(8) z1=0 z2=0

• 1

Blow up the point z1=z2=0

• Similar statement holds for 4D as well? Classify 4D N = 1 SCFT using

their Higgs branch?

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Constraints on Σ (continued)

• In our scanning of the oriented graph, cod-2 (4,6) singularities are

generally allowed.

• The nodes are separated into two classes: good bases and resolvable

bases.

• Good toric base: no toric cod-2 (4,6) singularity; there may be some

(4,6) curves on a divisor carrying an E8 but they can be easily blown up to resolve the problem. After these additional blow ups, the 4D low energy theory is a gauge theory coupled to gravity.

• Resolvable base: has toric cod-2 (4,6) singularities but satisfy (0, 0, 0)

condition; non-Lagrangian.

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Constraints on Σ (continued)

• In our scanning of the oriented graph, cod-2 (4,6) singularities are

generally allowed.

• The nodes are separated into two classes: good bases and resolvable

bases.

• Good toric base: no toric cod-2 (4,6) singularity; there may be some

(4,6) curves on a divisor carrying an E8 but they can be easily blown up to resolve the problem. After these additional blow ups, the 4D low energy theory is a gauge theory coupled to gravity.

• Resolvable base: has toric cod-2 (4,6) singularities but satisfy (0, 0, 0)

condition; non-Lagrangian.

• Other issues: cod-3 (4,6), terminal singularities (Arras/Grassi/Weigand

16’); generally allowed.

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Hodge numbers of elliptic CY4

For good toric bases B, we can compute (string theoretic) Hodge numbers h1,1 and h3,1 of a generic elliptic CY4 X over B: h1,1(X) = h1,1(B) + N(blp) + rk(G) + 1, (9) h3,1(X) ∼ = ˜ h3,1(X) = |F| + |G| −

• Θ∈∆,dim Θ=2

l′(Θ) − 4 +

• Θi∈∆,Θ∗

i ∈∆∗,dim(Θi)=dim(Θ∗ i )=1

l′(Θi) · l′(Θ∗

i ) .

(10) l′(Θ) is the number of interior points on a facet Θ.

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Approach 1

Random walk on the toric threefold landscape (1510.04978 w/ Taylor)

• Start from P3, do a random sequence of 100,000 blow up/downs.
• Never pass through bases with cod-1 or cod-2 (4,6) singularities

(excluding E8 gauge group).

• In total 100 runs.

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Approach 1

Random walk on the toric threefold landscape (1510.04978 w/ Taylor)

• Start from P3, do a random sequence of 100,000 blow up/downs.
• Never pass through bases with cod-1 or cod-2 (4,6) singularities

(excluding E8 gauge group).

• In total 100 runs.

SU(2) SU(3) G2 SO(7) 13.6 2.0 9.7 4 × 10−6 SO(8) F4 E6 E7 1.0 2.8 0.3 0.2

Average number of non-Higgsable gauge group on a base.

• 76% of bases have SU(3) × SU(2) non-Higgsable cluster.
• Total number ∼ 1048. max(h1,1(B)) ∼ 120.

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Approach 2

Combinatorially generate toric threefold bases by blowing up Fano bases (1706.02299 Halverson/Long/Sung)

• Put additional “height constraint” during the blow up process: h ≤ 6.
• Blow ups of points before blow ups of curves.
• Generally allow cod-2 (4,6) singularities.
• Rigorously proved that N ≥ 4

3 × 2.96 × 10755 bases.

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New One-way Monte Carlo approach

• We want to include all the resolvable bases in our oriented graph and

draw all the edges between them. We also want to generate some good bases in this process.

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New One-way Monte Carlo approach

• We want to include all the resolvable bases in our oriented graph and

draw all the edges between them. We also want to generate some good bases in this process.

• In this approach, we cannot perform a random walk, because the good

bases are extremely rare among resolvable bases.

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New One-way Monte Carlo approach

• We want to include all the resolvable bases in our oriented graph and

draw all the edges between them. We also want to generate some good bases in this process.

• In this approach, we cannot perform a random walk, because the good

bases are extremely rare among resolvable bases.

• We do a random sequence of blow ups starting from a single base, e.g.

P3, until we hit the end point where any blow up will break the (0, 0, 0)

• condition. At each step, the possibility of choosing each outgoing path is

equal.

• According to the definition, the end point is always good. But most of

the bases between h1,1(B) ∼ 10 and the end point are only resolvable.

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New One-way Monte Carlo approach

Thousands of steps Thousands of steps Good bases Resolvable bases

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Estimate the number of nodes on each layer

For each path p = a1 → a2 → a3 . . . ak, we assign a “dynamic weight factor” D(p) =

k−1

• i=1

Nout(ai) Nin(ai+1) (11) Then the number of nodes on layer k equals to the average of D(p): Nnodes(k) =

• p

P(p)D(p) (12) where the sum is over all the paths and P(p) is the probability of one goes along this path.

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Estimate the number of nodes on each layer

For each path p = a1 → a2 → a3 . . . ak, we assign a “dynamic weight factor” D(p) =

k−1

• i=1

Nout(ai) Nin(ai+1) (11) Then the number of nodes on layer k equals to the average of D(p): Nnodes(k) =

• p

P(p)D(p) (12) where the sum is over all the paths and P(p) is the probability of one goes along this path.

Layer 1 Layer 2 Layer 3 Layer 4

e.g. for a bipartite tree, one trivially get Nnodes(k) = 2k−1. This formula can be easily proved for any oriented graph with a single root.

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Results

In total, we generated 2,000 random blow up sequences starting from P3.

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Results

In total, we generated 2,000 random blow up sequences starting from P3.

• The end points are concentrated at certain layers. For example, 15%

percent of branches end on layer 2249 and 15% percent of branches end

• n layer 2303. But there’s nothing between them.
• End points are highly non-random.

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Results

In total, we generated 2,000 random blow up sequences starting from P3.

• The end points are concentrated at certain layers. For example, 15%

percent of branches end on layer 2249 and 15% percent of branches end

• n layer 2303. But there’s nothing between them.
• End points are highly non-random.
• The gauge groups on end point bases are SU(2)a × G b

2 × F c 4 × E d 8 × H,

where a ∼ = h1,1(B) + 1 6

• , b ∼

= h1,1(B) + 1 9

• , c ∼

= h1,1(B) + 1 24

• , d ∼

= h1,1(B) 68

• .

(13) H is some other gauge group that rarely appears. For example, if the end point is on layer 2999, then H =SU(3).

• For different end point bases on the same layer, they have same

non-Higgsable gauge groups but the their adjacency are different.

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Results

• After computing h1,1(X), h3,1(X) of generic elliptic CY4 X over the

end point bases B, we found that they resemble the mirror of simple elliptic CY4s over simple bases. (1) For the bases with h1,1(B) = 2303, h1,1(X) = 3878, h3,1(X) = 2: mirror of generic elliptic CY4 over P3. (2) For the bases with h1,1(B) = 2591, h1,1(X) = 4358, h3,1(X) = 3: mirror of generic elliptic CY4 over generalized Hirzebruch threefold ˜ F3.

• Maybe easy to compute the Gukov-Vafa-Witten potential on these

geometry because they have simple mirror CY4s.

• The number of flux vacua on each of these manifolds

∼ 100.8×h1,1(X) ∼ 10O(103) ≪ 10272,000

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Results

• The distribution of resolvable bases is centralized at very large

h1,1 ∼ 5, 000.

• The total number of resolvable bases from blowing up

P3 ∼ 3.5 × 101,964, bigger than the number 10755 in 1706.02299.

• The height h can be as high as 331.

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New results

• The distribution of good bases is centralized at the end points.
• The total number of good bases ∼ 9.1 × 10253, which is much smaller

than the total number of resolvable bases. Indeed the good bases form a tiny fraction of the whole set.

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New results

We also tried other starting points such as P1 × P1 × P1 and generalized Hirzebruch threefold ˜ F2.

• The qualitative feature are the same, while the total number of

resolvable bases from blowing up ˜ F2 is much bigger ∼ 103046.

• The total number of good bases does not differ by much, h1,1(B) of

end points are the same.

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New results

• These numbers are underestimated, since we get extremely small

numbers on the layer k 10, 000: 10−2000. Due to the existence of multiple roots.

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New results

• To get a feeling of the abundance of roots, we try to randomly blow

down an end point base (reverse the orientation of edge). Can we get P3?

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New results

• To get a feeling of the abundance of roots, we try to randomly blow

down an end point base (reverse the orientation of edge). Can we get P3?

• It turns out that we will be stuck at some “exotic starting point” base

with 50-100 toric rays which cannot be further blown down to get another smooth toric base.

• Consistent with Mori theory.
• Estimate the number of these exotic starting points? Allowing singular

bases?

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Questions for machine learning

• We have generated a huge complicated network with 102,000 ∼ 103,000

nodes, where a typical node has degree ∼ 103. But a typical node in the network also contains a lot of data!

• Two classes of big data/machine learning questions:

(1) “Local” geometric information on one base. Deriving non-Higgsable gauge group with local geometric data? Finding the structure of non-Higgsable clusters? Criteria for cod-1/cod-2 (4,6) singularity? (2) Studying the structure of the network. Navigating problem: how to get a particular base such as the one with largest number of rays?

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Derive non-Higgsable gauge group

• In 6D, the gauge groups can be easily read out with the intersection

numbers of curves.

• e.g. an isolated (-3/-4/-5/-6/-8/-12) curve will give SU(3), SO(8), F4,

E6, E7, E8 gauge groups.

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Derive non-Higgsable gauge group

• In 6D, the gauge groups can be easily read out with the intersection

numbers of curves.

• e.g. an isolated (-3/-4/-5/-6/-8/-12) curve will give SU(3), SO(8), F4,

E6, E7, E8 gauge groups.

• How to read out the non-Higgsable gauge groups in 4D?
• Formula using canonical class, normal bundle and intersection relations
• f divisors (Morrison Taylor 14’).
• No formula with triple intersection numbers as input; no classification
• f 4D NHC.

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Derive non-Higgsable gauge group

• In 6D, the gauge groups can be easily read out with the intersection

numbers of curves.

• e.g. an isolated (-3/-4/-5/-6/-8/-12) curve will give SU(3), SO(8), F4,

E6, E7, E8 gauge groups.

• How to read out the non-Higgsable gauge groups in 4D?
• Formula using canonical class, normal bundle and intersection relations
• f divisors (Morrison Taylor 14’).
• No formula with triple intersection numbers as input; no classification
• f 4D NHC.
• A natural play ground for supervised learning.

Input: set of local triple intersection numbers. Output: gauge group.

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Derive non-Higgsable gauge group

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Derive non-Higgsable gauge group

Generalized Hirzebruch threefold ˜ Fn.

1 1 1 1 1 1 n n n n2 n2

• n
• n
• n

v4 v2 v1 v3 v5

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Derive non-Higgsable gauge group

If we want to derive the gauge group on a P2 divisor, we take three different input sets with different vector sizes: small, medium, large.

1 1 1 n n2 n n

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Derive non-Higgsable gauge group

Input set: 20,000 P2 divisors on good base from the one-way Monte Carlo program. Training method Accuracy (Vs) Accuracy (Vm) Accuracy (Vl) LogisticRegression 0.846808 0.925754 0.929904 Markov 0.928204 0.9962 0.99745 NearestNeighbors 0.948503 0.976201 0.980251 NeuralNetwork 0.931603 0.979351 0.99445 RandomForest 0.957402 0.99955 0.99955 SupportVectorMachine 0.957902 0.9991 0.9964

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Conclusion

• We probed the most complete connected set of toric threefold bases. A

lower limit of the total number nodes is estimated as ∼ 103,000.

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Conclusion

• We probed the most complete connected set of toric threefold bases. A

lower limit of the total number nodes is estimated as ∼ 103,000.

• Bases without cod-2 (4,6) are rare, but they are well organized. The

gauge contents follow a universal pattern. Intermediate bases?

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Conclusion

• We probed the most complete connected set of toric threefold bases. A

lower limit of the total number nodes is estimated as ∼ 103,000.

• Bases without cod-2 (4,6) are rare, but they are well organized. The

gauge contents follow a universal pattern. Intermediate bases?

• The global structure of the graph is still not well understood. There

seem to be many roots in the oriented graph (which are resolvable bases). Total number? Structure?

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Conclusion

• We probed the most complete connected set of toric threefold bases. A

lower limit of the total number nodes is estimated as ∼ 103,000.

• Bases without cod-2 (4,6) are rare, but they are well organized. The

gauge contents follow a universal pattern. Intermediate bases?

• The global structure of the graph is still not well understood. There

seem to be many roots in the oriented graph (which are resolvable bases). Total number? Structure?

• There are questions on the dataset well suited for machine learning.

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Conclusion

• We probed the most complete connected set of toric threefold bases. A

lower limit of the total number nodes is estimated as ∼ 103,000.

• Bases without cod-2 (4,6) are rare, but they are well organized. The

gauge contents follow a universal pattern. Intermediate bases?

• The global structure of the graph is still not well understood. There

seem to be many roots in the oriented graph (which are resolvable bases). Total number? Structure?

• There are questions on the dataset well suited for machine learning.
• Cosmological implications? Understand 4D N = 1 SCFT? Flux?

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Conclusion

• We probed the most complete connected set of toric threefold bases. A

lower limit of the total number nodes is estimated as ∼ 103,000.

• Bases without cod-2 (4,6) are rare, but they are well organized. The

gauge contents follow a universal pattern. Intermediate bases?

• The global structure of the graph is still not well understood. There

seem to be many roots in the oriented graph (which are resolvable bases). Total number? Structure?

• There are questions on the dataset well suited for machine learning.
• Cosmological implications? Understand 4D N = 1 SCFT? Flux?

Thank you!

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