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Decision trees, Brill–Noether theory, and vector-like spectra in F-theorywork in progress with M. Bies, M. Cvetič, R. Donagi, M. Liu, F. Ruehle
Ling Lin
CERN Theory Department
String PhenoJune 11, 2020
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 0 / 18
Motivation
Classical problem of string pheno: find realization of (MS)SM in string landscape.
In particular: need (massless) vector-like pair(s) to accommodate the Higgs.
More generally: vector-like spectrum is charaterizing feature of 4d vacuum.
In global F-theory compactifications: difficult to control due to non-topological nature.
How can machine learning techniques help?
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 1 / 18
Motivation
Classical problem of string pheno: find realization of (MS)SM in string landscape.
In particular: need (massless) vector-like pair(s) to accommodate the Higgs.
More generally: vector-like spectrum is charaterizing feature of 4d vacuum.
In global F-theory compactifications: difficult to control due to non-topological nature.
How can machine learning techniques help?
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 1 / 18
Motivation
Classical problem of string pheno: find realization of (MS)SM in string landscape.
In particular: need (massless) vector-like pair(s) to accommodate the Higgs.
More generally: vector-like spectrum is charaterizing feature of 4d vacuum.
In global F-theory compactifications: difficult to control due to non-topological nature.
How can machine learning techniques help?
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 1 / 18
Outline
1 Computing vector-like spectrum in F-theory
2 Learning cohomology jumps with Decision Trees
3 Application to toy example
4 “Moduli” space of jumps
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 2 / 18
Computing vector-like spectrum in F-theory
Chiral matter in 4d F-theory
F-theory ∼= type IIB on B3 with (p, q)-7-branes at finite gs , by geometrizing backreaction on gsin elliptic CY-fibration π : Y4 → B3 (cf. 2017 TASI lectures by Weigand and Cvetič for recent reviews,
see also C. Long’s talk).
N = 1 gauge sector on 7-branes wrapped on (complex) surfaces S ⊂ B3 with matter localizedon curves C ⊂ S . For chiral spectrum: need to turn on gauge flux background G4 ∈ H2,2(Y4).Chiral excess is topological: can be computed via intersection theory.[Donagi/Wijnholt, 09],[Braun/Collinucci/Valandro, 11], [Marsano/Schäfer-Nameki, 11], [Krause/Mayrhofer/Weigand,
11,12], [Grimm/Hayashi, 11], [Cvetič/Grimm/Klevers, 12], [Braun/Grimm/Keitel, 13], [Cvetič/Grassi/Klevers/Piragua,
13], [Borchmann/Mayrhofer/Palti/Weigand, 13], [LL/Mayrhofer/Till/Weigand, 15]
Allows explicit construction of “quadrillions” of MSSM(-like) three-family F-theory models.[Cvetič/Klevers/Mayorga/Oehlmann/Reuter, 15], [LL/Weigand, 16], [Cvetič/LL/Liu/Oehlmann, 18],
[Cvetič/Halverson/LL/Liu/Tian, 19]
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 3 / 18
Computing vector-like spectrum in F-theory
Chiral matter in 4d F-theory
F-theory ∼= type IIB on B3 with (p, q)-7-branes at finite gs , by geometrizing backreaction on gsin elliptic CY-fibration π : Y4 → B3 (cf. 2017 TASI lectures by Weigand and Cvetič for recent reviews,
see also C. Long’s talk).
N = 1 gauge sector on 7-branes wrapped on (complex) surfaces S ⊂ B3 with matter localizedon curves C ⊂ S . For chiral spectrum: need to turn on gauge flux background G4 ∈ H2,2(Y4).
Chiral excess is topological: can be computed via intersection theory.[Donagi/Wijnholt, 09],[Braun/Collinucci/Valandro, 11], [Marsano/Schäfer-Nameki, 11], [Krause/Mayrhofer/Weigand,
11,12], [Grimm/Hayashi, 11], [Cvetič/Grimm/Klevers, 12], [Braun/Grimm/Keitel, 13], [Cvetič/Grassi/Klevers/Piragua,
13], [Borchmann/Mayrhofer/Palti/Weigand, 13], [LL/Mayrhofer/Till/Weigand, 15]
Allows explicit construction of “quadrillions” of MSSM(-like) three-family F-theory models.[Cvetič/Klevers/Mayorga/Oehlmann/Reuter, 15], [LL/Weigand, 16], [Cvetič/LL/Liu/Oehlmann, 18],
[Cvetič/Halverson/LL/Liu/Tian, 19]
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 3 / 18
Computing vector-like spectrum in F-theory
Chiral matter in 4d F-theory
F-theory ∼= type IIB on B3 with (p, q)-7-branes at finite gs , by geometrizing backreaction on gsin elliptic CY-fibration π : Y4 → B3 (cf. 2017 TASI lectures by Weigand and Cvetič for recent reviews,
see also C. Long’s talk).
N = 1 gauge sector on 7-branes wrapped on (complex) surfaces S ⊂ B3 with matter localizedon curves C ⊂ S . For chiral spectrum: need to turn on gauge flux background G4 ∈ H2,2(Y4).Chiral excess is topological: can be computed via intersection theory.[Donagi/Wijnholt, 09],[Braun/Collinucci/Valandro, 11], [Marsano/Schäfer-Nameki, 11], [Krause/Mayrhofer/Weigand,
11,12], [Grimm/Hayashi, 11], [Cvetič/Grimm/Klevers, 12], [Braun/Grimm/Keitel, 13], [Cvetič/Grassi/Klevers/Piragua,
13], [Borchmann/Mayrhofer/Palti/Weigand, 13], [LL/Mayrhofer/Till/Weigand, 15]
Allows explicit construction of “quadrillions” of MSSM(-like) three-family F-theory models.[Cvetič/Klevers/Mayorga/Oehlmann/Reuter, 15], [LL/Weigand, 16], [Cvetič/LL/Liu/Oehlmann, 18],
[Cvetič/Halverson/LL/Liu/Tian, 19]
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 3 / 18
Computing vector-like spectrum in F-theory
Chiral matter in 4d F-theory
F-theory ∼= type IIB on B3 with (p, q)-7-branes at finite gs , by geometrizing backreaction on gsin elliptic CY-fibration π : Y4 → B3 (cf. 2017 TASI lectures by Weigand and Cvetič for recent reviews,
see also C. Long’s talk).
N = 1 gauge sector on 7-branes wrapped on (complex) surfaces S ⊂ B3 with matter localizedon curves C ⊂ S . For chiral spectrum: need to turn on gauge flux background G4 ∈ H2,2(Y4).Chiral excess is topological: can be computed via intersection theory.[Donagi/Wijnholt, 09],[Braun/Collinucci/Valandro, 11], [Marsano/Schäfer-Nameki, 11], [Krause/Mayrhofer/Weigand,
11,12], [Grimm/Hayashi, 11], [Cvetič/Grimm/Klevers, 12], [Braun/Grimm/Keitel, 13], [Cvetič/Grassi/Klevers/Piragua,
13], [Borchmann/Mayrhofer/Palti/Weigand, 13], [LL/Mayrhofer/Till/Weigand, 15]
Allows explicit construction of “quadrillions” of MSSM(-like) three-family F-theory models.[Cvetič/Klevers/Mayorga/Oehlmann/Reuter, 15], [LL/Weigand, 16], [Cvetič/LL/Liu/Oehlmann, 18],
[Cvetič/Halverson/LL/Liu/Tian, 19]
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 3 / 18
Computing vector-like spectrum in F-theory
Zero mode counting in global modelsMassless (anti-)chiral modes in representation R on matter curves CR depend on C3 ratherthan G4 = dC3. Encoded in intermediate Jacobian of Y4 [Curio/Donagi, 98], [Donagi/Wijnholt,
12,13], [Anderson/Heckman/Katz, 13].
Can be parametrized by Chow ring CH2(Y4) [Bies/Mayrhofer(/Pehle)/Weigand, 14,17]
=⇒ computationally more feasible: given A ∈ CH2(Y4), can extract for each CR a linebundle LR such that
massless chiral modes of ←→ H0(CR,LR) ,
massless anti-chiral modes←→ H1(CR,LR) ,
χ(R) = h0 − h1 topological invariant, depends only on G4 = [A] ∈ H2,2(Y4).
LR given as collection of points in B3 =⇒ can be modeled as coherent sheaf on B3.
H i (CR,LR) computed via Ext groups; algorithm implemented in computer algebra systemCAP [Bies, 17], [Bies/Posur, 19].
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 4 / 18
Computing vector-like spectrum in F-theory
Zero mode counting in global modelsMassless (anti-)chiral modes in representation R on matter curves CR depend on C3 ratherthan G4 = dC3. Encoded in intermediate Jacobian of Y4 [Curio/Donagi, 98], [Donagi/Wijnholt,
12,13], [Anderson/Heckman/Katz, 13].
Can be parametrized by Chow ring CH2(Y4) [Bies/Mayrhofer(/Pehle)/Weigand, 14,17]
=⇒ computationally more feasible: given A ∈ CH2(Y4), can extract for each CR a linebundle LR such that
massless chiral modes of ←→ H0(CR,LR) ,
massless anti-chiral modes←→ H1(CR,LR) ,
χ(R) = h0 − h1 topological invariant, depends only on G4 = [A] ∈ H2,2(Y4).
LR given as collection of points in B3 =⇒ can be modeled as coherent sheaf on B3.
H i (CR,LR) computed via Ext groups; algorithm implemented in computer algebra systemCAP [Bies, 17], [Bies/Posur, 19].
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 4 / 18
Computing vector-like spectrum in F-theory
Computational challenges
Sheaf description very general, e.g., CR need not be smooth. However, implementation oncomputer extremely resource intensive, fails, e.g., if genus(CR) too large (& 10).
I Typical F-theory models have curves with g > 20 (oftentimes the would-be Higgs!).
Cohomologies depend on complex structure of Y4, of which there are in general O(100).I Determine (even just part of) complex structure dependence of vector-like spectrum tricky.
In practice, sheaf description lacks type IIB-ish intuitions about localized matter.I Difficult to compute Yukawa couplings in global models [Cvetič/LL/Liu/Zhang/Zoccarato, 19].
First order questions: What is the “generic” value of h0? When does it “jump”?
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 5 / 18
Learning cohomology jumps with Decision Trees
Machine Learning line bundle cohomology
Surge of recent interest to study H i (X ,L) with machine learning [Ruehle, 17],
[Kläwer/Schlechter, 18], [Larfors/Schneider, 19,20], [Brodie/Constantin/Deen/Lukas, 19], mostly suited forheterotic compactifications: X is smooth with known Pic(X ) 3 L.
Qualitatively different in F-theory: CR and LR given by polynomials on B3.I CR can be singular.I Pic(CR) in general not known.I LR specified by sum of points
∑i λipi , where pi ∈ B3 lie on CR.
=⇒ which pi are rationally equivalent? Even non-trivial if LR is pull-back bundle!I Both CR and LR can simultaneously change with complex structure.
Instead of training algorithm with optimal predictive power, we want to be able tointerpret its strategy geometrically =⇒ use binary decision trees.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 6 / 18
Learning cohomology jumps with Decision Trees
Machine Learning line bundle cohomology
Surge of recent interest to study H i (X ,L) with machine learning [Ruehle, 17],
[Kläwer/Schlechter, 18], [Larfors/Schneider, 19,20], [Brodie/Constantin/Deen/Lukas, 19], mostly suited forheterotic compactifications: X is smooth with known Pic(X ) 3 L.
Qualitatively different in F-theory: CR and LR given by polynomials on B3.I CR can be singular.I Pic(CR) in general not known.I LR specified by sum of points
∑i λipi , where pi ∈ B3 lie on CR.
=⇒ which pi are rationally equivalent? Even non-trivial if LR is pull-back bundle!I Both CR and LR can simultaneously change with complex structure.
Instead of training algorithm with optimal predictive power, we want to be able tointerpret its strategy geometrically =⇒ use binary decision trees.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 6 / 18
Learning cohomology jumps with Decision Trees
Machine Learning line bundle cohomology
Surge of recent interest to study H i (X ,L) with machine learning [Ruehle, 17],
[Kläwer/Schlechter, 18], [Larfors/Schneider, 19,20], [Brodie/Constantin/Deen/Lukas, 19], mostly suited forheterotic compactifications: X is smooth with known Pic(X ) 3 L.
Qualitatively different in F-theory: CR and LR given by polynomials on B3.I CR can be singular.I Pic(CR) in general not known.I LR specified by sum of points
∑i λipi , where pi ∈ B3 lie on CR.
=⇒ which pi are rationally equivalent? Even non-trivial if LR is pull-back bundle!I Both CR and LR can simultaneously change with complex structure.
Instead of training algorithm with optimal predictive power, we want to be able tointerpret its strategy geometrically =⇒ use binary decision trees.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 6 / 18
Learning cohomology jumps with Decision Trees
Decision trees
A decision tree is a directed, connected graph with unique root vertex/node.Binary tree: each node has either 0 or 2 sub-nodes. Nodes with no sub-nodes are “leaves”.
Data organized by numeric features ~x . Decision tree “classifies” input with splitting criteriaat each node n:if xj ≤ κ
(n)j , then input assigned to one sub-node, otherwise to the other sub-node.
At the leaves, all assigned inputs ideally of same class (for us: h0 “generic” or jumps).However, in general not possible; failure measured by Gini impurity (∼how many differentclasses are assigned to node).
For training: minimize Gini impurity for given training data.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 7 / 18
Learning cohomology jumps with Decision Trees
The data set
Data collected from curves with genus 1 ≤ g ≤ 6 on S = dP3.I Each curve class [C ]←→ {P :=
∑k akmk = 0} ⊂ S , with homogeneous monomials mk .
I For each class [C ], we consider up to 13 line bundles L ∈ Pic(S).I For each pair ([C ], L) (fixed χ), we compute h0(Ca, L|Ca) via CAP, where Ca is curve of class
[C ], determined by a choice ak ∈ {0, 1} for the coefficients ak of P.
For genus 1 only 127 data points per pair ([C ], L), while for genus 6 we have ∼ 260 000.Lowest value h0 considered generic, anything above classified as a “jump”.
Consider different features of data: coefficients ak , “split-type” (topology of Ca),“intersection” (Γl · L, where Γl is component of Ca), "intersection + split-type".
Based on feature, train tree to classify whether input is generic or has a jump.Training-testing data ratio: 90:10.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 8 / 18
Learning cohomology jumps with Decision Trees
The data set
Data collected from curves with genus 1 ≤ g ≤ 6 on S = dP3.I Each curve class [C ]←→ {P :=
∑k akmk = 0} ⊂ S , with homogeneous monomials mk .
I For each class [C ], we consider up to 13 line bundles L ∈ Pic(S).I For each pair ([C ], L) (fixed χ), we compute h0(Ca, L|Ca) via CAP, where Ca is curve of class
[C ], determined by a choice ak ∈ {0, 1} for the coefficients ak of P.
For genus 1 only 127 data points per pair ([C ], L), while for genus 6 we have ∼ 260 000.Lowest value h0 considered generic, anything above classified as a “jump”.
Consider different features of data: coefficients ak , “split-type” (topology of Ca),“intersection” (Γl · L, where Γl is component of Ca), "intersection + split-type".
Based on feature, train tree to classify whether input is generic or has a jump.Training-testing data ratio: 90:10.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 8 / 18
Learning cohomology jumps with Decision Trees
The data set
Data collected from curves with genus 1 ≤ g ≤ 6 on S = dP3.I Each curve class [C ]←→ {P :=
∑k akmk = 0} ⊂ S , with homogeneous monomials mk .
I For each class [C ], we consider up to 13 line bundles L ∈ Pic(S).I For each pair ([C ], L) (fixed χ), we compute h0(Ca, L|Ca) via CAP, where Ca is curve of class
[C ], determined by a choice ak ∈ {0, 1} for the coefficients ak of P.
For genus 1 only 127 data points per pair ([C ], L), while for genus 6 we have ∼ 260 000.Lowest value h0 considered generic, anything above classified as a “jump”.
Consider different features of data: coefficients ak , “split-type” (topology of Ca),“intersection” (Γl · L, where Γl is component of Ca), "intersection + split-type".
Based on feature, train tree to classify whether input is generic or has a jump.Training-testing data ratio: 90:10.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 8 / 18
Learning cohomology jumps with Decision Trees
Example of tree trained on split-type (g = 3, d = 3)split type <= 5.5
gini = 0.492samples = 4095
value = [1791, 2304]class = no jump
split type <= 4.5gini = 0.25
samples = 1710value = [250, 1460]
class = no jump
True
split type <= 11.5gini = 0.457
samples = 2385value = [1541, 844]
class = jump
False
split type <= 0.5gini = 0.198
samples = 1440value = [160, 1280]
class = no jump
gini = 0.444samples = 270
value = [90, 180]class = no jump
gini = 0.192samples = 1363
value = [147, 1216]class = no jump
split type <= 1.5gini = 0.281
samples = 77value = [13, 64]class = no jump
gini = 0.324samples = 54
value = [11, 43]class = no jump
split type <= 2.5gini = 0.159
samples = 23value = [2, 21]
class = no jump
gini = 0.198samples = 18value = [2, 16]
class = no jump
gini = 0.0samples = 5value = [0, 5]
class = no jump
split type <= 6.5gini = 0.497
samples = 1080value = [500, 580]class = no jump
split type <= 25.5gini = 0.323
samples = 1305value = [1041, 264]
class = jump
gini = 0.499samples = 912
value = [434, 478]class = no jump
split type <= 8.5gini = 0.477
samples = 168value = [66, 102]class = no jump
split type <= 7.5gini = 0.444
samples = 87value = [29, 58]class = no jump
split type <= 9.5gini = 0.496
samples = 81value = [37, 44]class = no jump
gini = 0.444samples = 69
value = [23, 46]class = no jump
gini = 0.444samples = 18value = [6, 12]
class = no jump
gini = 0.5samples = 42
value = [21, 21]class = jump
split type <= 10.5gini = 0.484
samples = 39value = [16, 23]class = no jump
gini = 0.444samples = 15value = [5, 10]
class = no jump
gini = 0.497samples = 24
value = [11, 13]class = no jump
split type <= 17.5gini = 0.407
samples = 885value = [633, 252]
class = jump
split type <= 34.5gini = 0.056
samples = 420value = [408, 12]
class = jump
split type <= 12.5gini = 0.431
samples = 674value = [462, 212]
class = jump
split type <= 18.5gini = 0.307
samples = 211value = [171, 40]
class = jump
gini = 0.371samples = 366
value = [276, 90]class = jump
split type <= 13.5gini = 0.478
samples = 308value = [186, 122]
class = jump
gini = 0.444samples = 69
value = [23, 46]class = no jump
split type <= 14.5gini = 0.434
samples = 239value = [163, 76]
class = jump
gini = 0.371samples = 183
value = [138, 45]class = jump
split type <= 16.5gini = 0.494
samples = 56value = [25, 31]class = no jump
split type <= 15.5gini = 0.48
samples = 30value = [12, 18]class = no jump
gini = 0.5samples = 26
value = [13, 13]class = jump
gini = 0.494samples = 18value = [8, 10]
class = no jump
gini = 0.444samples = 12value = [4, 8]
class = no jump
gini = 0.0samples = 52value = [52, 0]class = jump
split type <= 21.5gini = 0.377
samples = 159value = [119, 40]
class = jump
split type <= 20.5gini = 0.496
samples = 33value = [18, 15]
class = jump
split type <= 22.5gini = 0.318
samples = 126value = [101, 25]
class = jump
split type <= 19.5gini = 0.444samples = 9value = [3, 6]
class = no jump
gini = 0.469samples = 24value = [15, 9]class = jump
gini = 0.444samples = 3value = [1, 2]
class = no jump
gini = 0.444samples = 6value = [2, 4]
class = no jump
gini = 0.165samples = 33value = [30, 3]class = jump
split type <= 23.5gini = 0.361
samples = 93value = [71, 22]
class = jump
gini = 0.444samples = 12value = [4, 8]
class = no jump
split type <= 24.5gini = 0.286
samples = 81value = [67, 14]
class = jump
gini = 0.444samples = 3value = [2, 1]class = jump
gini = 0.278samples = 78
value = [65, 13]class = jump
split type <= 33.5gini = 0.091
samples = 188value = [179, 9]
class = jump
split type <= 46.5gini = 0.026
samples = 232value = [229, 3]
class = jump
split type <= 26.5gini = 0.053
samples = 182value = [177, 5]
class = jump
gini = 0.444samples = 6value = [2, 4]
class = no jump
gini = 0.0samples = 81value = [81, 0]class = jump
split type <= 27.5gini = 0.094
samples = 101value = [96, 5]class = jump
gini = 0.444samples = 3value = [1, 2]
class = no jump
split type <= 31.5gini = 0.059
samples = 98value = [95, 3]class = jump
split type <= 29.5gini = 0.027
samples = 74value = [73, 1]class = jump
split type <= 32.5gini = 0.153
samples = 24value = [22, 2]class = jump
split type <= 28.5gini = 0.035
samples = 56value = [55, 1]class = jump
gini = 0.0samples = 18value = [18, 0]class = jump
gini = 0.0samples = 8value = [8, 0]class = jump
gini = 0.041samples = 48value = [47, 1]class = jump
gini = 0.278samples = 12value = [10, 2]class = jump
gini = 0.0samples = 12value = [12, 0]class = jump
split type <= 45.5gini = 0.043
samples = 135value = [132, 3]
class = jump
gini = 0.0samples = 97value = [97, 0]class = jump
split type <= 44.5gini = 0.016
samples = 123value = [122, 1]
class = jump
gini = 0.278samples = 12value = [10, 2]class = jump
gini = 0.0samples = 36value = [36, 0]class = jump
gini = 0.023samples = 87value = [86, 1]class = jump
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 9 / 18
Learning cohomology jumps with Decision Trees
Average accuracy
1 2 3 4 5 6Genus
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Acc
urac
y
Average accuracy vs genus for different features
coefficients split types intersections intersections+split types
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 10 / 18
Learning cohomology jumps with Decision Trees
Interpretation of result
Training on coefficients reach near perfect performance.Expected since coefficients entirely specifies setup, but no “intuitive” understanding.
Topological criteria work surprisingly well (combining split-type and intersection numbersaround and above 95% accuracy). Better suited for “extrapolation” to higher genus!
Based on our data (without any further algebraic geometry considerations): h0(Ca, L|Ca)more likely to jump if Ca = C̃a ∪ P1.
Small fraction of failure of topological criteria =⇒ other sources of jumps in cohomology.These are likely to be under-represented due to bias in our data set.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 11 / 18
Application to toy example
A toy F-theory model
Compact fourfold Y4 → B3 with B3 a hypersurface in toric space. F-theory on Y4 hasSU(5)× U(1) gauge symmetry, with SU(5) supported on S ∼= dP3 ⊂ B3 [Bies, 17].
In a “U(1)-flux” gauge background [Grimm/Weigand, 10], we have chiral spectrum:χ(101) = 3 , χ(5−2) = −18 , χ(53) = 15.
Focus on C53 ≡ C , with g = 24, polynomial has 44 coefficients; deg(L53) = 38.
Finding splitting C → C̃ ∪ P1 easy in this case (dP3 has 6 rigid divisors).I Only two of them (E1,2) lead to jump when split off. They have L · E1,2 < −1.I Splitting off E1 and E2 in combination, can get h0 = 17, 18, ..., 20.
I Cannot get h0 = 16 this way!
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 12 / 18
Application to toy example
A toy F-theory model
Compact fourfold Y4 → B3 with B3 a hypersurface in toric space. F-theory on Y4 hasSU(5)× U(1) gauge symmetry, with SU(5) supported on S ∼= dP3 ⊂ B3 [Bies, 17].
In a “U(1)-flux” gauge background [Grimm/Weigand, 10], we have chiral spectrum:χ(101) = 3 , χ(5−2) = −18 , χ(53) = 15.
Focus on C53 ≡ C , with g = 24, polynomial has 44 coefficients; deg(L53) = 38.
Finding splitting C → C̃ ∪ P1 easy in this case (dP3 has 6 rigid divisors).I Only two of them (E1,2) lead to jump when split off. They have L · E1,2 < −1.I Splitting off E1 and E2 in combination, can get h0 = 17, 18, ..., 20.
I Cannot get h0 = 16 this way!
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 12 / 18
Application to toy example
A toy F-theory model
Compact fourfold Y4 → B3 with B3 a hypersurface in toric space. F-theory on Y4 hasSU(5)× U(1) gauge symmetry, with SU(5) supported on S ∼= dP3 ⊂ B3 [Bies, 17].
In a “U(1)-flux” gauge background [Grimm/Weigand, 10], we have chiral spectrum:χ(101) = 3 , χ(5−2) = −18 , χ(53) = 15.
Focus on C53 ≡ C , with g = 24, polynomial has 44 coefficients; deg(L53) = 38.
Finding splitting C → C̃ ∪ P1 easy in this case (dP3 has 6 rigid divisors).I Only two of them (E1,2) lead to jump when split off. They have L · E1,2 < −1.I Splitting off E1 and E2 in combination, can get h0 = 17, 18, ..., 20.I Cannot get h0 = 16 this way!
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 12 / 18
“Moduli” space of jumps
Origin of jumps
Ca = C̃ ∪ C ′: count s ∈ H0(Ca, L|Ca) via “gluing” conditions along C̃ ∩ C ′ 6= ∅.In particular, sufficient condition for jump when C ′ ∼= P1 andL · P1 < −1, DL · [Ca] > 2g − 2.
I In such cases: curve Ca is non-generic.I Occurs often when (many) coefficients of P set to 0.
Cases not predicted by decision tree: L = L|Ca non-generic.E.g.: L = p1 − p2, becomes trivial when p1 = p2 in Pic(Ca).
I No change of topological data explains failure of trained algorithm.I Requires precise “alignment” of L and Ca, unlikely to achieve with ak ∈ {0, 1}.I Non-genericity quantified by Brill–Noether theory.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 13 / 18
“Moduli” space of jumps
Origin of jumps
Ca = C̃ ∪ C ′: count s ∈ H0(Ca, L|Ca) via “gluing” conditions along C̃ ∩ C ′ 6= ∅.In particular, sufficient condition for jump when C ′ ∼= P1 andL · P1 < −1, DL · [Ca] > 2g − 2.
I In such cases: curve Ca is non-generic.I Occurs often when (many) coefficients of P set to 0.
Cases not predicted by decision tree: L = L|Ca non-generic.E.g.: L = p1 − p2, becomes trivial when p1 = p2 in Pic(Ca).
I No change of topological data explains failure of trained algorithm.I Requires precise “alignment” of L and Ca, unlikely to achieve with ak ∈ {0, 1}.I Non-genericity quantified by Brill–Noether theory.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 13 / 18
“Moduli” space of jumps
Origin of jumps
Ca = C̃ ∪ C ′: count s ∈ H0(Ca, L|Ca) via “gluing” conditions along C̃ ∩ C ′ 6= ∅.In particular, sufficient condition for jump when C ′ ∼= P1 andL · P1 < −1, DL · [Ca] > 2g − 2.
I In such cases: curve Ca is non-generic.I Occurs often when (many) coefficients of P set to 0.
Cases not predicted by decision tree: L = L|Ca non-generic.E.g.: L = p1 − p2, becomes trivial when p1 = p2 in Pic(Ca).
I No change of topological data explains failure of trained algorithm.I Requires precise “alignment” of L and Ca, unlikely to achieve with ak ∈ {0, 1}.
I Non-genericity quantified by Brill–Noether theory.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 13 / 18
“Moduli” space of jumps
Origin of jumps
Ca = C̃ ∪ C ′: count s ∈ H0(Ca, L|Ca) via “gluing” conditions along C̃ ∩ C ′ 6= ∅.In particular, sufficient condition for jump when C ′ ∼= P1 andL · P1 < −1, DL · [Ca] > 2g − 2.
I In such cases: curve Ca is non-generic.I Occurs often when (many) coefficients of P set to 0.
Cases not predicted by decision tree: L = L|Ca non-generic.E.g.: L = p1 − p2, becomes trivial when p1 = p2 in Pic(Ca).
I No change of topological data explains failure of trained algorithm.I Requires precise “alignment” of L and Ca, unlikely to achieve with ak ∈ {0, 1}.I Non-genericity quantified by Brill–Noether theory.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 13 / 18
“Moduli” space of jumps
Brill–Noether theory
Assume C is a smooth Riemann surface of genus g . The Abel–Jacobi map ϕ gives a map
Picd(C )ϕd−→ J(C ) ∼= Cg/Λ .
For n ∈ Z≥0, define Gnd = {ϕd(L) | h0(C ,L) = n} ⊂ J(C ). Then Brill–Noether theory says:
dim(Gnd ) ≥ ρ(n, d) := g − n(n − (d − g + 1)) ≡ g − h0 h1 .
If ρ(n, d) > 0, then there are degree d line bundles with h0 = n on C .
For the F-theory toy-model (g = 24, d = 38, generically h0 = 15): ρ(16, 38) = 24− 16 · 1 = 8.So jump by 1 expected (and found) on smooth curve.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 14 / 18
“Moduli” space of jumps
Brill–Noether theory
Assume C is a smooth Riemann surface of genus g . The Abel–Jacobi map ϕ gives a map
Picd(C )ϕd−→ J(C ) ∼= Cg/Λ .
For n ∈ Z≥0, define Gnd = {ϕd(L) | h0(C ,L) = n} ⊂ J(C ). Then Brill–Noether theory says:
dim(Gnd ) ≥ ρ(n, d) := g − n(n − (d − g + 1)) ≡ g − h0 h1 .
If ρ(n, d) > 0, then there are degree d line bundles with h0 = n on C .
For the F-theory toy-model (g = 24, d = 38, generically h0 = 15): ρ(16, 38) = 24− 16 · 1 = 8.So jump by 1 expected (and found) on smooth curve.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 14 / 18
“Moduli” space of jumps
Brill–Noether theory
Assume C is a smooth Riemann surface of genus g . The Abel–Jacobi map ϕ gives a map
Picd(C )ϕd−→ J(C ) ∼= Cg/Λ .
For n ∈ Z≥0, define Gnd = {ϕd(L) | h0(C ,L) = n} ⊂ J(C ). Then Brill–Noether theory says:
dim(Gnd ) ≥ ρ(n, d) := g − n(n − (d − g + 1)) ≡ g − h0 h1 .
If ρ(n, d) > 0, then there are degree d line bundles with h0 = n on C .
For the F-theory toy-model (g = 24, d = 38, generically h0 = 15): ρ(16, 38) = 24− 16 · 1 = 8.So jump by 1 expected (and found) on smooth curve.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 14 / 18
“Moduli” space of jumps
“Moduli” space of jumps
Number of vector-like pairs induces a stratification of complex structure moduli space.I Brill–Noether describes strata with smooth curves, provides upper bound on h0 [Watari, 16].I Can be violated by curve splitting.
In global models: changing complex structure parameters affect genericity of line bundle(measured by Brill–Noether) and curve (determined by “split-type”) democratically=⇒ strata of both types in moduli space.
For a given pair ([C ], L), can be summarized in a Hasse-diagram.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 15 / 18
“Moduli” space of jumps
“Moduli” space of jumps
Number of vector-like pairs induces a stratification of complex structure moduli space.I Brill–Noether describes strata with smooth curves, provides upper bound on h0 [Watari, 16].I Can be violated by curve splitting.
In global models: changing complex structure parameters affect genericity of line bundle(measured by Brill–Noether) and curve (determined by “split-type”) democratically=⇒ strata of both types in moduli space.
For a given pair ([C ], L), can be summarized in a Hasse-diagram.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 15 / 18
“Moduli” space of jumps
“Moduli” space of jumps
Number of vector-like pairs induces a stratification of complex structure moduli space.I Brill–Noether describes strata with smooth curves, provides upper bound on h0 [Watari, 16].I Can be violated by curve splitting.
In global models: changing complex structure parameters affect genericity of line bundle(measured by Brill–Noether) and curve (determined by “split-type”) democratically=⇒ strata of both types in moduli space.
For a given pair ([C ], L), can be summarized in a Hasse-diagram.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 15 / 18
“Moduli” space of jumps
Example (g = 5, d = 4, χ = 0):
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 16 / 18
Summary & Outlook
Summary
Explicitly computing vector-like spectrum in global F-theory models is hard.
Using machine learning techniques, can gain intuition about computationally challengingcases.
I Qualitatively different than previous machine learning studies of line bundle cohomologies,because both line bundle and curve topology change simultaneously.
Both changes source jumps in cohomologies, captured by Hasse-type diagrams.I Reflect fact that vector-like spectra induce stratification on complex structure moduli.
See also talk by Martin Bies in “Summer Series on String Pheno”, June 16.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 17 / 18
Summary & Outlook
Summary
Explicitly computing vector-like spectrum in global F-theory models is hard.
Using machine learning techniques, can gain intuition about computationally challengingcases.
I Qualitatively different than previous machine learning studies of line bundle cohomologies,because both line bundle and curve topology change simultaneously.
Both changes source jumps in cohomologies, captured by Hasse-type diagrams.I Reflect fact that vector-like spectra induce stratification on complex structure moduli.
See also talk by Martin Bies in “Summer Series on String Pheno”, June 16.
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 17 / 18
Summary & Outlook
Open problems of ...
...technical nature: extend to non-pull-back & “fractional” pull-back bundles, combinestratification diagrams for several curves in one global model, ...
...conceptual nature: compute vector-like spectrum for pseudo-real representations,incorporate gauge backgrounds with non-vertical G4 (flux moduli dependence!),(geometric) symmetries protecting vector-like pairs, ...
...practical nature: apply to model building, (S)CFTs, swampland program, ...
Thank you!
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 18 / 18
Summary & Outlook
Open problems of ...
...technical nature: extend to non-pull-back & “fractional” pull-back bundles, combinestratification diagrams for several curves in one global model, ...
...conceptual nature: compute vector-like spectrum for pseudo-real representations,incorporate gauge backgrounds with non-vertical G4 (flux moduli dependence!),(geometric) symmetries protecting vector-like pairs, ...
...practical nature: apply to model building, (S)CFTs, swampland program, ...
Thank you!
Ling Lin (CERN) Decision trees, Brill–Noether theory, and vector-like spectra in F-theory June 11, 2020 18 / 18