Decision trees, Brill Noether theory, and vector-like ... · Computingvector-likespectruminF-theory Chiralmatterin4dF-theory F-theory˘=typeIIBonB 3 with(p;q)-7-branesatﬁniteg s,bygeometrizingbackreactionong

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?


Motivation







Motivation







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


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]






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,



























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].



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].



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”?


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.





















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.



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.



The data set










The data set










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


samples = 2385value = [1541, 844]

class = jump

False


samples = 1440value = [160, 1280]

class = no jump






samples = 77value = [13, 64]class = no jump




samples = 23value = [2, 21]

class = no jump

gini = 0.198samples = 18value = [2, 16]

class = no jump


class = no jump




samples = 1305value = [1041, 264]

class = jump












class = no jump


value = [21, 21]class = jump




class = no jump




samples = 885value = [633, 252]

class = jump



class = jump


samples = 674value = [462, 212]

class = jump



class = jump




samples = 308value = [186, 122]

class = jump





class = jump










class = no jump


class = no jump

gini = 0.0samples = 52value = [52, 0]class = jump



class = jump



class = jump



class = jump

split type <= 19.5gini = 0.444samples = 9value = [3, 6]

class = no jump



class = no jump


class = no jump




class = jump


class = no jump



class = jump






class = jump



class = jump



class = jump


class = no jump



samples = 101value = [96, 5]class = jump


class = no jump
















class = jump




class = jump






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



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.


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!







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!







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!


“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.



Origin of jumps







Origin of jumps




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.



Origin 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.






















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.















Example (g = 5, d = 4, χ = 0):


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.


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.


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!


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!


Documents

Decision trees, Brill Noether theory, and vector-like ... · Computingvector-likespectruminF-theory Chiralmatterin4dF-theory F-theory˘=typeIIBonB 3 with(p;q)-7-branesatﬁniteg s,bygeometrizingbackreactionong