Web of threefold bases in F-theory and machine learning
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- Erika Burns
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1 and machine learning & with W. Taylor CTP, MIT String Data Science, Northeastern; Dec. 2th, / 33
2 Exploring a huge oriented graph 2 / 33
3 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 = x 3 + fx + g (1) We require that the elliptic fibration is generic, hence f and g are general holomorphic sections of line bundles 4K B and 6K B. The gauge groups in the 4D supergravity model are minimal(non-higgsable). The number of complex structure moduli h 3,1 (X ) is maximal. 3 / 33
4 Toric threefolds Gluing C 3 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 Z 3. For any two cones σ 1, σ 2 Σ, σ 1 σ2 is either another cone in Σ or the origin. Compactness: the total set of cones σ Σ spans the whole Z 3. Smoothness: every 3D cone is simplicial, with unit volume. z (0,0,1) 3 =0 (0,1,0) z 2 =0 (1,0,0) z 1 =0 z 4 =0 (-1,-1,-1) 4 / 33
5 Toric threefolds z (0,0,1) 3 =0 (0,1,0) z 2 =0 (1,0,0) z 1 =0 z 4 =0 (-1,-1,-1) 1D ray: v i corresponds to complex surface (divisor) D i ; z i = 0. Total number: n = h 1,1 (B) D cone: v i v j corresponds to curve z i = z j = 0. 3D cone: v i v j v k corresponds to point z i = z j = z k = 0. The convex hull of vertices v i forms a lattice polytope. Its dual polytope = {p Q 3, v i, p, v i 1} in general is not a lattice polytope. 5 / 33
6 Line bundles on toric threefolds Anti-canonical line bundle K B = i D i. Generators of holomorphic section m p of line bundle L = i a id i points p in the dual lattice Z 3 : {p Z 3, v i, p, v i a i }. (2) m p = i z p,v i +a i i (3) 6 / 33
7 Line bundles on toric threefolds Anti-canonical line bundle K B = i D i. Generators of holomorphic section m p of line bundle L = i a id i points p in the dual lattice Z 3 : {p Z 3, v i, p, v i a i }. (2) m p = i z p,v i +a i 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 Z 3, v i, p, v i 4}. (4) G = {p Z 3, v i, p, v i 6}. (5) 6 / 33
8 The edges between nodes: Blow up/down The set of toric fans Σ is infinite, because of the existence of blow up operations on Σ: (1) Blow up a point v i v j v k : add another ray ṽ = v i + v j + v k. (2) Blow up a curve v i v j : add another ray ṽ = v i + v j. 7 / 33
9 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 v i v j 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 v i, v j, v k and v = v i + v j + v k (v is a P 2 ) (2) v has 4 neighbors v i, v k, v j, v l, and either v = v i + v j or v = v k + v l. Blow up/down a smooth toric threefold will lead to another smooth toric threefold (unit volume condition). 8 / 33
10 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 D i (v i ). NO cod-1 (4,6) singularity Otherwise, the singularity x = y = z in Weierstrass model y 2 = x 3 + z 4 x + z 6 (6) cannot be resolved while keeping the Calabi-Yau condition (SUSY is broken). Lattice condition: there is at least one point p G where v i, p < 0 for each v i. Equivalent (0, 0, 0) condition: the origin (0, 0, 0) cannot lie on the boundary of G. 9 / 33
11 Constraints on Σ What if (f, g) vanishes to order (4, 6) or higher on some curves v i v j? For example: y 2 = x 3 + z 2 1 z 2 2 x + z 3 1 z 3 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. 10 / 33
12 SCFT from cod-2 (4,6) singularity An 6D example: two (-3) curves intersecting each other, SO(8) SO(8) gauge group. y 2 = x 3 + z 2 1 z 2 2 x + z 3 1 z 3 2. (8) SO(8) -3-3 z 1 =0 z 2 =0 SO(8) 11 / 33
13 SCFT from cod-2 (4,6) singularity An 6D example: two (-3) curves intersecting each other, SO(8) SO(8) gauge group. y 2 = x 3 + z 2 1 z 2 2 x + z 3 1 z 3 2. (8) SO(8) -3-3 z 1 =0 z 2 =0 SO(8) SO(8) -4 z 1 =0 Blow up the point z 1 =z 2 =0-4 z 2 =0-1 SO(8) 11 / 33
14 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 ) 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. SO(8) -3-3 SO(8) z 1 =0 z 2 =0 SO(8) -4 z 1 =0 Blow up the point z 1 =z 2 =0-4 z 2 =0-1 SO(8) Similar statement holds for 4D as well? Classify 4D N = 1 SCFT using their Higgs branch? 12 / 33
15 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 E 8 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. 13 / 33
16 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 E 8 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. 13 / 33
17 Hodge numbers of elliptic CY4 For good toric bases B, we can compute (string theoretic) Hodge numbers h 1,1 and h 3,1 of a generic elliptic CY4 X over B: h 1,1 (X ) = h 1,1 (B) + N(blp) + rk(g) + 1, (9) h 3,1 (X ) = h 3,1 (X ) = F + G l (Θ) 4 + Θ,dim Θ=2 Θ i,θ i,dim(θ i )=dim(θ i )=1 l (Θ i ) l (Θ i ). (10) l (Θ) is the number of interior points on a facet Θ. 14 / 33
18 Approach 1 Random walk on the toric threefold landscape ( w/ Taylor) Start from P 3, do a random sequence of 100,000 blow up/downs. Never pass through bases with cod-1 or cod-2 (4,6) singularities (excluding E 8 gauge group). In total 100 runs. 15 / 33
19 Approach 1 Random walk on the toric threefold landscape ( w/ Taylor) Start from P 3, do a random sequence of 100,000 blow up/downs. Never pass through bases with cod-1 or cod-2 (4,6) singularities (excluding E 8 gauge group). In total 100 runs. SU(2) SU(3) G 2 SO(7) SO(8) F 4 E 6 E Average number of non-higgsable gauge group on a base. 76% of bases have SU(3) SU(2) non-higgsable cluster. Total number max(h 1,1 (B)) / 33
20 Approach 2 Combinatorially generate toric threefold bases by blowing up Fano bases ( 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 bases. 16 / 33
21 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. 17 / 33
22 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. 17 / 33
23 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. P 3, 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 h 1,1 (B) 10 and the end point are only resolvable. 17 / 33
24 New One-way Monte Carlo approach Thousands of steps Good bases Resolvable bases Thousands of steps 18 / 33
25 Estimate the number of nodes on each layer For each path p = a 1 a 2 a 3... a k, we assign a dynamic weight factor k 1 N out (a i ) D(p) = (11) N in (a i+1 ) i=1 Then the number of nodes on layer k equals to the average of D(p): N nodes (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. 19 / 33
26 Estimate the number of nodes on each layer For each path p = a 1 a 2 a 3... a k, we assign a dynamic weight factor k 1 N out (a i ) D(p) = (11) N in (a i+1 ) i=1 Then the number of nodes on layer k equals to the average of D(p): N nodes (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 4 Layer 3 Layer 2 Layer 1 e.g. for a bipartite tree, one trivially get N nodes (k) = 2 k 1. This formula can be easily proved for any oriented graph Webwith of athreefold single root. bases in F-theory 19 / 33
27 Results In total, we generated 2,000 random blow up sequences starting from P / 33
28 Results In total, we generated 2,000 random blow up sequences starting from P 3. The end points are concentrated at certain layers. For example, 15% percent of branches end on layer 2249 and 15% percent of branches end on layer But there s nothing between them. End points are highly non-random. 20 / 33
29 Results In total, we generated 2,000 random blow up sequences starting from P 3. The end points are concentrated at certain layers. For example, 15% percent of branches end on layer 2249 and 15% percent of branches end on layer But there s nothing between them. End points are highly non-random. The gauge groups on end point bases are SU(2) a G2 b F 4 c E 8 d H, where [ a h 1,1 ] [ (B) + 1 =, b 6 h 1,1 ] [ (B) + 1 =, c 9 h 1,1 ] [ (B) + 1 =, d 24 h 1,1 ] (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. 20 / 33
30 Results After computing h 1,1 (X ), h 3,1 (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 h 1,1 (B) = 2303, h 1,1 (X ) = 3878, h 3,1 (X ) = 2: mirror of generic elliptic CY4 over P 3. (2) For the bases with h 1,1 (B) = 2591, h 1,1 (X ) = 4358, h 3,1 (X ) = 3: mirror of generic elliptic CY4 over generalized Hirzebruch threefold F 3. 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 h1,1 (X ) 10 O(103) , / 33
31 Results The distribution of resolvable bases is centralized at very large h 1,1 5, 000. The total number of resolvable bases from blowing up P ,964, bigger than the number in The height h can be as high as / 33
32 New results The distribution of good bases is centralized at the end points. The total number of good bases , which is much smaller than the total number of resolvable bases. Indeed the good bases form a tiny fraction of the whole set. 23 / 33
33 New results We also tried other starting points such as P 1 P 1 P 1 and generalized Hirzebruch threefold F 2. The qualitative feature are the same, while the total number of resolvable bases from blowing up F 2 is much bigger The total number of good bases does not differ by much, h 1,1 (B) of end points are the same. 24 / 33
34 New results These numbers are underestimated, since we get extremely small numbers on the layer k 10, 000: Due to the existence of multiple roots. 25 / 33
35 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 P 3? 26 / 33
36 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 P 3? It turns out that we will be stuck at some exotic starting point base with 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? 26 / 33
37 Questions for machine learning We have generated a huge complicated network with 10 2, ,000 nodes, where a typical node has degree 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? 27 / 33
38 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), F 4, E 6, E 7, E 8 gauge groups. 28 / 33
39 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), F 4, E 6, E 7, E 8 gauge groups. How to read out the non-higgsable gauge groups in 4D? Formula using canonical class, normal bundle and intersection relations of divisors (Morrison Taylor 14 ). No formula with triple intersection numbers as input; no classification of 4D NHC. 28 / 33
40 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), F 4, E 6, E 7, E 8 gauge groups. How to read out the non-higgsable gauge groups in 4D? Formula using canonical class, normal bundle and intersection relations of divisors (Morrison Taylor 14 ). No formula with triple intersection numbers as input; no classification of 4D NHC. A natural play ground for supervised learning. Input: set of local triple intersection numbers. Output: gauge group. 28 / 33
41 Derive non-higgsable gauge group 29 / 33
42 Derive non-higgsable gauge group Generalized Hirzebruch threefold F n. n 1 v n 0 n v n v 4 n 2 -n n n 2 v 3 0 v 5 30 / 33
43 Derive non-higgsable gauge group If we want to derive the gauge group on a P 2 divisor, we take three different input sets with different vector sizes: small, medium, large. 1 n 1 n n 2 n 1 31 / 33
44 Derive non-higgsable gauge group Input set: 20,000 P 2 divisors on good base from the one-way Monte Carlo program. Training method Accuracy (V s ) Accuracy (V m ) Accuracy (V l ) LogisticRegression Markov NearestNeighbors NeuralNetwork RandomForest SupportVectorMachine / 33
45 Conclusion We probed the most complete connected set of toric threefold bases. A lower limit of the total number nodes is estimated as 10 3, / 33
46 Conclusion We probed the most complete connected set of toric threefold bases. A lower limit of the total number nodes is estimated as 10 3,000. Bases without cod-2 (4,6) are rare, but they are well organized. The gauge contents follow a universal pattern. Intermediate bases? 33 / 33
47 Conclusion We probed the most complete connected set of toric threefold bases. A lower limit of the total number nodes is estimated as 10 3,000. 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? 33 / 33
48 Conclusion We probed the most complete connected set of toric threefold bases. A lower limit of the total number nodes is estimated as 10 3,000. 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. 33 / 33
49 Conclusion We probed the most complete connected set of toric threefold bases. A lower limit of the total number nodes is estimated as 10 3,000. 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? 33 / 33
50 Conclusion We probed the most complete connected set of toric threefold bases. A lower limit of the total number nodes is estimated as 10 3,000. 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! 33 / 33
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