General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications Generalized Complete Intersection Calabi-Yau (gCICY) Xin Gao Work with: Lara B. Anderson, Fabio Apruzzi, James Gray, Seung-Joo Lee, arXiv: 1507.03235, 1510.xxxxx 24.Oct, 2015 @ Duke University, Southeastern mathematical string theory meeting
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications Outline General Motivations 1 Construction of gCICY 2 Construction of Sections 3 Redundancies 4 Classifications 5 Physical Applications 6
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications Outline General Motivations 1 Construction of gCICY 2 Construction of Sections 3 Redundancies 4 Classifications 5 Physical Applications 6
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications Why Calabi-Yau From string to the real wold: 10D → 4D What we want: N = 1 Supersymmetry with chiral spectrum Best under control: N = 1 Flux Compactification Het string on CY 3 Type IIA/B on CY 3 with orientifold (include Type I ∼ = Type IIB orientifold with O 9 -plane) (Aux 12D) F-theory on CY 4 (11D) M-theory on CY 3 × S 1 / Z 2 or on M 7 with G 2 holonomy . . .
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications Why Calabi-Yau From string to the real wold: 10D → 4D What we want: N = 1 Supersymmetry with chiral spectrum Best under control: N = 1 Flux Compactification Het string on CY 3 Type IIA/B on CY 3 with orientifold (include Type I ∼ = Type IIB orientifold with O 9 -plane) (Aux 12D) F-theory on CY 4 (11D) M-theory on CY 3 × S 1 / Z 2 or on M 7 with G 2 holonomy . . . ⇒ Calabi-Yau threefold CY 3 or fourfold CY 4 .
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications What is Calabi-Yau Calabi-Yau n-folds is a complex n-dimentional compacted K¨ ahler Manifold satisfied: c 1 ( M ) = 0 ∈ H 2 ( M, Z ) . K M = ∧ n T ∗ (1 , 0)( M ) is trivial since c 1 ( K M ) = − c 1 ( M ) . Unique nowhere vanishing holomophic n-form, Ω n ∈ Ω n, 0 ( M ) , d Ω n = 0 The Ricci tensor vanish, i.e. R mn = 0 The holonomy group of M is SU ( n )
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications What is Calabi-Yau Calabi-Yau n-folds is a complex n-dimentional compacted K¨ ahler Manifold satisfied: c 1 ( M ) = 0 ∈ H 2 ( M, Z ) . K M = ∧ n T ∗ (1 , 0)( M ) is trivial since c 1 ( K M ) = − c 1 ( M ) . Unique nowhere vanishing holomophic n-form, Ω n ∈ Ω n, 0 ( M ) , d Ω n = 0 The Ricci tensor vanish, i.e. R mn = 0 The holonomy group of M is SU ( n ) Calabi-Yau 3,4-folds
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications How to construct Calabi-Yau calculable Toric Calabi-Yau Borisov, Batyrev, Cox, Kreuzer, Skarke . . . ... → 473,800,776 reflexive polyhedra in 4D Hypersurface ֒ Kreuzer,Skarke,Altman,Gray,He,Jejjala,Nelson,. . . Hypersurface ֒ → weighted project space Kreuzer,Skarke,. . . Complete Intersection Calabi-Yau (CICY) Complete intersection hypersurfaces ֒ → Product of projective spaces 7,890 configuration matrices for CY3 Hubsch,Candelas,Dale,Lutaken,Schimmrigk,Green,. . . 921,49 configuration matrices for CY4 Gray,Haupt,Lukas,. . .
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications How to construct Calabi-Yau calculable Toric Calabi-Yau Borisov, Batyrev, Cox, Kreuzer, Skarke . . . ... → 473,800,776 reflexive polyhedra in 4D Hypersurface ֒ Kreuzer,Skarke,Altman,Gray,He,Jejjala,Nelson,. . . Hypersurface ֒ → weighted project space Kreuzer,Skarke,. . . Complete Intersection Calabi-Yau (CICY) Complete intersection hypersurfaces ֒ → Product of projective spaces 7,890 configuration matrices for CY3 Hubsch,Candelas,Dale,Lutaken,Schimmrigk,Green,. . . 921,49 configuration matrices for CY4 Gray,Haupt,Lukas,. . . Generalized Complete Intersection Calabi-Yau Manifolds (gCICY)
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications CICY 3-folds � P 2 � 1 1 1 X = P 4 3 1 1 X ≡ X 1 ∩ X 2 ∩ X 3 ֒ = P 2 × P 4 → A ∼ X a : p a ( x 1 , x 2 ) = 0 , a = 1 , 2 , 3 . || p 1 || = (1 , 3) , || p 2 , 3 || = (1 , 1) . x 1 = ( x 0 1 : x 2 1 : x 3 x 2 = ( x 0 2 : · · · : x 5 1 ) , 2 ) . p 1 ∈ H 0 ( A , O (1 , 3)) , p 2 , 3 ∈ H 0 ( A , O (1 , 1)) .
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications CICY 3-folds � P 2 � 1 1 1 X = P 4 3 1 1 X ≡ X 1 ∩ X 2 ∩ X 3 ֒ = P 2 × P 4 → A ∼ X a : p a ( x 1 , x 2 ) = 0 , a = 1 , 2 , 3 . || p 1 || = (1 , 3) , || p 2 , 3 || = (1 , 1) . x 1 = ( x 0 1 : x 2 1 : x 3 x 2 = ( x 0 2 : · · · : x 5 1 ) , 2 ) . p 1 ∈ H 0 ( A , O (1 , 3)) , p 2 , 3 ∈ H 0 ( A , O (1 , 1)) . 3 dim. c 1 = 0 . h 1 , 1 = 3 , h 2 , 1 = 63 . Smooth by Bertini’s theorem
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications CICY 3-folds � P 2 � 1 1 1 X = P 4 3 1 1 X ≡ X 1 ∩ X 2 ∩ X 3 ֒ = P 2 × P 4 → A ∼ X a : p a ( x 1 , x 2 ) = 0 , a = 1 , 2 , 3 . || p 1 || = (1 , 3) , || p 2 , 3 || = (1 , 1) . x 1 = ( x 0 1 : x 2 1 : x 3 x 2 = ( x 0 2 : · · · : x 5 1 ) , 2 ) . p 1 ∈ H 0 ( A , O (1 , 3)) , p 2 , 3 ∈ H 0 ( A , O (1 , 1)) . 3 dim. c 1 = 0 . h 1 , 1 = 3 , h 2 , 1 = 63 . Smooth by Bertini’s theorem Generalized To Drop: Positive semi-definite entries.
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications gCICY P 1 1 1 − 1 1 P 1 X = 1 1 1 − 1 P 5 3 1 1 1
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications gCICY P 1 1 1 − 1 1 P 1 X = 1 1 1 − 1 P 5 3 1 1 1 Still complete intersection? h 0 ( P 1 × P 1 × P 5 , O (1 , − 1 , 1)) = 0 . → A is not algebraic complete intersection. X ֒ Calabi-Yau? Smooth?
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications gCICY P 1 P 1 1 1 − 1 1 1 1 P 1 P 1 X = 1 1 1 − 1 M = 1 1 P 5 P 5 3 1 1 1 3 1 � 2 � 1 X ֒ − → M ֒ − → A
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications gCICY P 1 P 1 1 1 − 1 1 1 1 P 1 P 1 X = 1 1 1 − 1 M = 1 1 P 5 P 5 3 1 1 1 3 1 � 2 � 1 X ֒ − → M ֒ − → A � : h 0 ( M , O M (1 , − 1 , 1)) = h 0 ( M , O M ( − 1 , 1 , 1)) = 1 2 ⇒ Polynomial description in M “ ≡ ” Rational description by x ∈ A � , 2 1 � are algebraic complete intersection.
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications gCICY P 1 P 1 1 1 − 1 1 1 1 P 1 P 1 X = 1 1 1 − 1 M = 1 1 P 5 P 5 3 1 1 1 3 1 � 2 � 1 X ֒ − → M ֒ − → A � : h 0 ( M , O M (1 , − 1 , 1)) = h 0 ( M , O M ( − 1 , 1 , 1)) = 1 2 ⇒ Polynomial description in M “ ≡ ” Rational description by x ∈ A � , 2 1 � are algebraic complete intersection. Rational description ⇒ “non-polynomail ” deformations Candelas, De La Ossa, Font, Katz, Morrison, Green, Hubsch, Mavlyutov,. . .
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications gCICY P 1 P 1 1 1 − 1 1 1 1 P 1 P 1 X = 1 1 1 − 1 M = 1 1 P 5 P 5 3 1 1 1 3 1 � 2 � 1 X ֒ − → M ֒ − → A � : h 0 ( M , O M (1 , − 1 , 1)) = h 0 ( M , O M ( − 1 , 1 , 1)) = 1 2 ⇒ Polynomial description in M “ ≡ ” Rational description by x ∈ A � , 2 1 � are algebraic complete intersection. Rational description ⇒ “non-polynomail ” deformations Candelas, De La Ossa, Font, Katz, Morrison, Green, Hubsch, Mavlyutov,. . . The effective cone of M is larger than the one in A
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications Outline General Motivations 1 Construction of gCICY 2 Construction of Sections 3 Redundancies 4 Classifications 5 Physical Applications 6
General Motivations Construction of gCICY Construction of Sections Redundancies Classifications Physical Applications Definition P n 1 a 1 a 1 · · · 1 K P n 2 a 2 a 2 · · · 1 K M = [ n || { a α } ] = . . . ... . . . . . . P nm a m a m · · · 1 K m � dim C M = n r − K , r =1 Standard Complete Intersection M : { p α ( x r ) = 0 } , α = 1 , 2 , . . . , K ; r = 1 , . . . , m.
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