Machine Learning Landscape - ICTP Trieste Machine Learning Line Bundle Cohomology Daniel Kläwer DK, Lorenz Schlechter — arXiv:1809.02547
Outline Machine Learning Landscape vs. Machine Learning Swampland Landscape: Line Bundle Cohomology Traditional ML Approach Approach 2
Size of the Landscape • The String Landscape is vast. • Classic estimate for # of type IIB flux vacua: # > 10 500 • Exist even much larger numbers in the literature: # > 10 272,000 Taylor, Wang 2015 • Proof of finiteness does not exist! But it is known that the # Grassi 1991; Gross 1993 of elliptically fibered CY threefolds is finite • Recent work suggests that the # of Calabi-Yau threefolds Anderson, Gao, Gray, Lee 2017 (needed for compactification of type II/het.) could in fact be Huang, Taylor 2018 finite, because “almost all” are elliptically fibered 3
Landscape vs. Swampland Vafa 2005 • String Landscape: Set of effective field theories that can be UV completed to a string theory vacuum • String Swampland: The complement of seemingly consistent effective field theories which do not arise from string theory • Swampland Conjectures: Conjectured properties of theories that SM are able to discriminate between the landscape and swampland 4
String Phenomenology, July 2018 A Web of Conjectures… non-SUSY AdS global symmetries (Refined) Weak Gravity Completeness Swampland Conjecture Conjecture Distance Conjecture Lattice WGC ? ? M pl | ∇ V | ≥ cV de Sitter ??? 5
Machine Learning Landscape 2018 A Web of Conjectures… Emergence? non-SUSY AdS global symmetries (Refined) Weak Gravity Completeness Swampland Conjecture Conjecture Distance Conjecture Lattice WGC Ooguri, Palti, Shiu, Vafa M pl | ∇ V | ≥ cV arXiv:1810.05506 See also: Andriot, Roupec 2018 pl min ( ∇ i ∇ j V ) ≤ − c ′ � V M 2 + many others Spin-2 Conjecture ??? DK, Lüst, Palti arXiv:1811.07908 de Sitter 6
Problems of the Landscape Type • Often study a highly nonlinear map of the type: input data properties of vacua ℤ I ℤ O Toric data SUSY? Fluxes Gauge group Vector Bundles # representations # branes Vacuum energy … … ‣ Classify inequivalent input data • Possible questions: ‣ SM problem: find pre-images of given U ( x ) ⊂ ℤ O ‣ Find approximate parameter distributions that emulate the distribution in string vacua … 7
Problems of the Swampland Type • The known swampland conjectures are mostly of the following form: ‣ A bound on a certain quantity in the Landscape Q ( ℤ I ) < 𝒫 (1) ‣ Exclusion of a property (¬ SUSY ) ∪ ( V < 0) ∪ stable = False ‣ Can we violate it parametrically? • Possible Questions: ‣ Can we bend it? What is the number? 𝒫 (1) ‣ How likely is it that the inequality is saturated? ≈ • Very important to give generic predictions from string theory! 8
CYs and Toric Varieties • Anti-canonical hypersurfaces in toric varieties many Calabi-Yau manifolds 5 ℂ P I − 1 [ x 1 : … : x I ] ∑ • Prototype: in i =1 x 5 ℂ P 4 i = 0 • More general: toric variety described in terms of homogeneous coordinates I and scaling relations R • Example: K3 hypersurface in weighted projective space ℙ 3 1112 [ x 1 : x 2 : x 3 : x 4 : x 5 ] ∼ [ α x 1 : α x 2 : α x 3 : x 4 : α 2 x 5 ] [ x 1 : x 2 : x 3 : x 4 : x 5 ] ∼ [ x 1 : x 2 : x 3 : β x 4 : β x 5 ] • Line bundles = divisors: ℒ = 𝒫 X ( D ) = 𝒫 X ( m , n ) D = mD 1 + nD 2 D 1 = { x 1 = 0} ∼ { x 2 = 0} ∼ { x 3 = 0} D 2 = { x 4 = 0} 9
CYs and Toric Varieties • Anti-canonical hypersurfaces in toric varieties many Calabi-Yau manifolds 5 ℂ P I − 1 [ x 1 : … : x I ] ∑ • Prototype: in i =1 x 5 ℂ P 4 i = 0 • More general: toric variety described in terms of homogeneous coordinates I and scaling relations R • Example: K3 hypersurface in weighted projective space ℙ 3 1112 [ x 1 : x 2 : x 3 : x 4 : x 5 ] ∼ [ α x 1 : α x 2 : α x 3 : x 4 : α 2 x 5 ] [ x 1 : x 2 : x 3 : x 4 : x 5 ] ∼ [ x 1 : x 2 : x 3 : β x 4 : β x 5 ] • Line bundles = divisors: ℒ = 𝒫 X ( D ) = 𝒫 X ( m , n ) D = mD 1 + nD 2 D 1 = { x 1 = 0} ∼ { x 2 = 0} ∼ { x 3 = 0} D 2 = { x 4 = 0} 10
Line Bundle Cohomology - Why? • CY compactifications of the heterotic string or F-theory: interested in sheaf cohomology groups , where X is the CY itself or a sub-manifold of it. H i ( X , ℒ ) • Dimensions count massless modes in the 4d theory h i ( X , ℒ ) • Heterotic String/F-theory: e.g. chiral fermions • For us, they simply define a non-linear map of the type discussed before h i ( X , 𝒫 X ( m 1 , …, m I ) ) : ℤ I → ℤ dim ( X ) , i = 1,…, dim ( X ) Line bundle integers Cohomology dimensions 11
Line Bundle Cohomology - Why? • CY compactifications of the heterotic string or F-theory: interested in sheaf cohomology groups , where X is the CY itself or a sub-manifold of it. H i ( X , ℒ ) • Dimensions count massless modes in the 4d theory h i ( X , ℒ ) • Heterotic String/F-theory: e.g. chiral fermions • For us, they simply define a non-linear map of the type discussed before h i ( X , 𝒫 X ( m 1 , …, m I ) ) : ℤ I → ℤ dim ( X ) , i = 1,…, dim ( X ) Line bundle integers Cohomology dimensions 12
The Traditional Approach • Several algorithms are known for the computation of line bundle cohomology • In various cases, exact formulae for toric varieties are known • Not so much for hypersurfaces • Analytic results for CICYs in products of projective spaces Lukas, Constantin 2018 h 0 ( ℂ P n , 𝒫 ℂ P n ( k ) ) = ( k + n n ) • Bott formula: Polynomial of degree n! • We use the cohomCalg algorithm Blumenhagen, Jurke, Rahn, Roschy 2010 • Computes for us cohomology of line bundles on the toric ambient space X 13
From Ambient Space to Hypersurface • Line bundles on the ambient space pull back to the hypersurface 𝒫 X ( D ) 𝒫 H ( D ) • Relation is given by the Koszul sequence: res m 0 → 𝒫 X ( D − H ) → 𝒫 X ( D ) → 𝒫 H ( D ) → 0 • Induces long exact sequence of cohomology groups res m δ ⋯ → H i ( 𝒫 X ( D − H )) → H i ( 𝒫 X ( D )) → H i ( 𝒫 H ( D )) → H i +1 ( 𝒫 X ( D − H )) → ⋯ • Strategy: 1. Cut long sequence into short ones if we hit a zero 2. Use the fact that the alternating sum of dimensions is zero 3. Try to solve this linear system for the in terms of h i ( 𝒫 H ( D )) the and h i ( 𝒫 X ( D )) h i ( 𝒫 X ( D − H )) 4. If this procedure does not give a unique result, need to introduce further cuts and need info about maps! 14
The Data Toric variety: dP 1 15
The Data K3 hypersurface in ℙ 3 1112 16
NN Approaches for Learning Cohomolgy 1. Classification • Output is probability Bull, He, Jejjala, Mishra 2018 for class membership h 0 ( X , ℒ ) = 1 m 1 • Bad: Have to bin/cut h 0 ( X , ℒ ) = 2 m 2 off + output scales NN h 0 ( X , ℒ ) = 3 with ⋯ m i ⋯ m I Cannot extrapolate h 0 ( X , ℒ ) > h 0 to larger than m i max training set 2. Regression Ruehle 2017 • Output should reproduce directly the h i h 0 ( X , ℒ ) m 1 • Round to nearest integer h 1 ( X , ℒ ) m 2 • Extrapolation limited by NN h 2 ( X , ℒ ) ⋯ used float point precision ⋯ m I • Can make use of h d ( X , ℒ ) correlations between h i 17
Regression by Neural Networks 3 fully connected leaky ReLU layers with 100 neurons 81^2=6561 data points 60% used for training Normalized data Batch size: 300 Training time: 2min on Desktop CPU K3 hypersurface in ℙ 3 1112 ≈ 50% accuracy but: easy to learn the zeros… 18
Regression by Neural Networks Problems: • Regression works for simple examples (almost 100% accuracy for a dP1 toric ambient space), but fails for more complicated ones (hypersurfaces). • Later: partly due to high frequency oscillation in data • Extrapolation fails also due to finite float precision m max Can we do better? General lesson: Yes , if we understand our data… 19
Hirzebruch-Riemann-Roch • The HRR theorem allows us to easily compute an analytic expression for the holomorphic Euler characteristic of a line bundle χ ( X , ℒ ) = ∫ X ch ( ℒ ) td ( X ) • The Euler characteristic is just the alternating sum of ranks dim ℂ ( X ) ∑ ( − 1) i h i ( X , ℒ ) χ ( X , ℒ ) = i =0 • It is a polynomial of degree in the line bundle integers dim ℂ ( X ) • If the alternating sum of cohomologies is that simple, it is reasonable to expect that locally they are also individually polynomial, with pairwise discontinuities or kinks at loci of codimension 1 ≥ 20
A Closer Look at the Data… K3 hypersurface in ℙ 3 1112 21
The Classification Problem • Visually: Data is locally described by polynomial • Aim: identify the phase boundaries in the space of line bundle integers • Look for locations where the map h i ( X , 𝒫 X ( m 1 , …, m I ) ) : ℤ I → ℤ dim ( X ) , i = 1,…, dim ( X ) is not differentiable • Get a cone structure in input space not quite actually… (see later) • Use unsupervised learning for this! • Once done, just fit a polynomial within each phase! 22
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