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 Web of threefold bases in F-theory Yi-Nan Wang 1 / 33

Exploring a huge oriented graph Web of threefold bases in F-theory Yi-Nan Wang 2 / 33

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 − 4 K B and − 6 K 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. Web of threefold bases in F-theory Yi-Nan Wang 3 / 33

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 . � σ 2 is either another cone in Σ or • For any two cones σ 1 , σ 2 ∈ Σ, σ 1 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 3 =0 (0,0,1) (0,1,0) z 2 =0 (1,0,0) z 1 =0 z 4 =0 (-1,-1,-1) Web of threefold bases in F-theory Yi-Nan Wang 4 / 33

Toric threefolds z 3 =0 (0,0,1) (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 ) + 3. • 2D 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. Web of threefold bases in F-theory Yi-Nan Wang 5 / 33

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 i D i ⇔ points p in the dual lattice Z 3 : { p ∈ Z 3 , ∀ v i , � p , v i � ≥ − a i } . (2) z � p , v i � + a i � m p = (3) i i Web of threefold bases in F-theory Yi-Nan Wang 6 / 33

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 i D i ⇔ points p in the dual lattice Z 3 : { p ∈ Z 3 , ∀ v i , � p , v i � ≥ − a i } . (2) z � p , v i � + a i � m p = (3) i i 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) Web of threefold bases in F-theory Yi-Nan Wang 6 / 33

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 ˜ v = v i + v j + v k . (2) Blow up a curve v i v j : add another ray ˜ v = v i + v j . Web of threefold bases in F-theory Yi-Nan Wang 7 / 33

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). Web of threefold bases in F-theory Yi-Nan Wang 8 / 33

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 . Web of threefold bases in F-theory Yi-Nan Wang 9 / 33

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 (7) 2 . • 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. Web of threefold bases in F-theory Yi-Nan Wang 10 / 33

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 (8) 2 . -3 -3 SO(8) SO(8) z 2 =0 z 1 =0 Web of threefold bases in F-theory Yi-Nan Wang 11 / 33

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 (8) 2 . -3 -3 SO(8) SO(8) z 2 =0 z 1 =0 Blow up the point z 1 = z 2 =0 -4 -4 z 2 =0 SO(8) SO(8) -1 z 1 =0 Web of threefold bases in F-theory Yi-Nan Wang 11 / 33

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) z 2 =0 z 1 =0 Blow up the point z 1 = z 2 =0 -4 -4 z 2 =0 SO(8) SO(8) -1 z 1 =0 • Similar statement holds for 4D as well? Classify 4D N = 1 SCFT using their Higgs branch? Web of threefold bases in F-theory Yi-Nan Wang 12 / 33

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. Web of threefold bases in F-theory Yi-Nan Wang 13 / 33

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. Web of threefold bases in F-theory Yi-Nan Wang 13 / 33

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 ) � l ′ (Θ) − 4 = |F| + |G| − (10) Θ ∈ ∆ , dim Θ=2 � l ′ (Θ i ) · l ′ (Θ ∗ + i ) . Θ i ∈ ∆ , Θ ∗ i ∈ ∆ ∗ , dim(Θ i )=dim(Θ ∗ i )=1 l ′ (Θ) is the number of interior points on a facet Θ. Web of threefold bases in F-theory Yi-Nan Wang 14 / 33

Approach 1 Random walk on the toric threefold landscape (1510.04978 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. Web of threefold bases in F-theory Yi-Nan Wang 15 / 33

Approach 1 Random walk on the toric threefold landscape (1510.04978 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) 4 × 10 − 6 13 . 6 2 . 0 9 . 7 SO(8) F 4 E 6 E 7 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 ∼ 10 48 . max( h 1 , 1 ( B )) ∼ 120. Web of threefold bases in F-theory Yi-Nan Wang 15 / 33