The Calabi-Yau Landscape: Beyond the Lampposts Mehmet Demirtas Cornell University String Pheno Series, 2020 Based on works with (various subsets of): Manki Kim, Cody Long, Liam McAllister, Jakob Moritz, Mike Stillman, Andres Rios Tascon
What is possible in quantum gravity? • de-Sitter solutions? • Quintessence? • Super-Planckian field ranges? • Global symmetries?
What is possible in quantum gravity? • de-Sitter solutions? • Quintessence? • Super-Planckian field ranges? • Global symmetries? What is generic in quantum gravity? • Ultralight axions? • Exponential hierarchies? • Light dark sectors? • Light moduli?
What is possible in quantum gravity? • de-Sitter solutions? • Quintessence? • Super-Planckian field ranges? • Global symmetries? A primary method: Study solutions of string theory. What is generic in quantum gravity? • Ultralight axions? • Exponential hierarchies? • Light dark sectors? • Light moduli?
What is possible in quantum gravity? • de-Sitter solutions? • Quintessence? • Super-Planckian field ranges? • Global symmetries? A primary method: Study solutions of string theory. What is generic in quantum gravity? • Ultralight axions? • Exponential hierarchies? • Light dark sectors? • Light moduli? Can answer for: Weakly coupled compactifications of superstring theories.
To get started: Compactifications on simple Calabi-Yau (CY) manifolds with small Hodge numbers.
To get started: Compactifications on simple Calabi-Yau (CY) manifolds with small Hodge numbers. Picture taken from Aliexpress.com. (You can buy this lamppost!)
To get started: Compactifications on simple Calabi-Yau (CY) manifolds with small Hodge numbers. However: this is an exponentially small fraction of the String Landscape. • Number of (known) topologically inequivalent CY manifolds increases exponentially with . [MD, McAllister, Rios Tascon, hep-th/2008.01730] • Number of flux vacua in type IIB (F- Theory) compactifications increases exponentially with ( ). [Denef, Douglas, hep-th/0404116] [Denef, Douglas, hep-th/0411183] [Taylor, Wang, hep-th/1511.03209] Picture taken from Aliexpress.com. (You can buy this lamppost!)
We can now construct CY threefolds with largest known Hodge numbers and compute relevant topological data. Kreuzer-Skarke
Outline I. CY 3 ’s from Triangulations II. Holomorphic Cycles Application: Ultralight Axions III. 3-cycles Application: Towards KKLT
A Quick Review • This talk: CY threefolds.
A Quick Review • This talk: CY threefolds. • Largest known set of CY threefolds: hypersurfaces in toric varieties. [Batyrev, alg-geom/9310003] [Kreuzer, Skarke, hep-th/0002240]
A Quick Review • This talk: CY threefolds. • Largest known set of CY threefolds: hypersurfaces in toric varieties. [Batyrev, alg-geom/9310003] [Kreuzer, Skarke, hep-th/0002240] The construction: [Batyrev, alg-geom/9310003] 1. Take a 4D reflexive lattice polytope Reflexive: the only interior point of the polytope (and its dual) is the origin.
A Quick Review • This talk: CY threefolds. • Largest known set of CY threefolds: hypersurfaces in toric varieties. [Batyrev, alg-geom/9310003] [Kreuzer, Skarke, hep-th/0002240] The construction: [Batyrev, alg-geom/9310003] 1. Take a 4D reflexive lattice polytope Reflexive: the only interior point of the polytope (and its dual) is the origin. 2. Obtain a (fine, regular, star) triangulation
A Quick Review • This talk: CY threefolds. • Largest known set of CY threefolds: hypersurfaces in toric varieties. [Batyrev, alg-geom/9310003] [Kreuzer, Skarke, hep-th/0002240] The construction: [Batyrev, alg-geom/9310003] 1. Take a 4D reflexive lattice polytope Reflexive: the only interior point of the polytope (and its dual) is the origin. 2. Obtain a (fine, regular, star) triangulation This triangulation defines a fan, which describes a toric variety V that has a CY hypersurface X.
The number of reflexive lattice polytopes: In 2D: 16 ●
The number of reflexive lattice polytopes: In 2D: 16 ●
The number of reflexive lattice polytopes: In 2D: 16 ● In 3D: 4,319 ●
The number of reflexive lattice polytopes: In 2D: 16 ● In 3D: 4,319 ● In 4D: 473,800,776 ● [Kreuzer, Skarke, hep-th/0002240]
Reflexive polytopes in 4 dimensions
Reflexive polytopes in 4 dimensions
Reflexive polytopes in 4 dimensions
The number of reflexive lattice polytopes: In 2D: 16 ● In 3D: 4,319 ● In 4D: 473,800,776 ● [Kreuzer, Skarke, hep-th/0002240] Number of triangulations Number of lattice points on the polytope ● ● ●
The number of reflexive lattice polytopes: In 2D: 16 ● In 3D: 4,319 ● In 4D: 473,800,776 ● [Kreuzer, Skarke, hep-th/0002240] Number of triangulations Number of lattice points on the polytope ● ● ● How many CY 3 hypersurfaces are there?
The number of reflexive lattice polytopes: In 2D: 16 ● In 3D: 4,319 ● In 4D: 473,800,776 ● [Kreuzer, Skarke, hep-th/0002240] Number of triangulations Number of lattice points on the polytope ● ● ● How many CY 3 hypersurfaces are there? • Not known. • We recently proved an upper bound of . [MD, McAllister, Rios Tascon, hep-th/2008.01730]
Holomorphic Cycles Notation: • : 4D Ambient Variety, X: Calabi-Yau threefold hypersurface
Holomorphic Cycles Notation: • : 4D Ambient Variety, X: Calabi-Yau threefold hypersurface • Mori cone: is the cone of effective curves.
Holomorphic Cycles Notation: • : 4D Ambient Variety, X: Calabi-Yau threefold hypersurface • Mori cone: is the cone of effective curves. • Kähler cone: is the set of cohomology classes of Kähler forms.
Holomorphic Cycles Notation: • : 4D Ambient Variety, X: Calabi-Yau threefold hypersurface • Mori cone: is the cone of effective curves. • Kähler cone: is the set of cohomology classes of Kähler forms. • are dual cones:
Holomorphic Cycles Notation: • : 4D Ambient Variety, X: Calabi-Yau threefold hypersurface • Mori cone: is the cone of effective curves. • Kähler cone: is the set of cohomology classes of Kähler forms. • are dual cones: • No general algorithm for computing in hypersurfaces. • Can compute on a case-by-case basis. • Can compute .
Holomorphic Cycles • Volumes of 2-cycles , 4-cycles , and itself are determined by the Kähler form and the intersection numbers: where span .
Holomorphic Cycles • Volumes of 2-cycles , 4-cycles , and itself are determined by the Kähler form and the intersection numbers: where span . • Stretched Kähler cone:
Kähler cone Mori cone generator Kähler cone generator
Stretched Kähler cone Mori cone generator Kähler cone generator
Holomorphic Cycles • Volumes of 2-cycles , 4-cycles , and itself are determined by the Kähler form and the intersection numbers: where span . • Stretched Kähler cone: estimate for the convergence of the worldsheet instanton expansion and the control of the expansion. [Candelas, De La Ossa, Green, Parkes, ‘90]
Recent Advances • Recap: Need to triangulate a reflexive polytope and compute intersection numbers . • Can be done via open source math software, like Sage.
Recent Advances • Recap: Need to triangulate a reflexive polytope and compute intersection numbers . • Can be done via open source math software, like Sage. • Many, systematic studies. [Braun, Walliser, hep-th/1106.4529] [Braun, Lukas, Sun, hep-th/1704.07812] [Blumenhagen, Gao, Rahn, Shukla, hep-th/1205.2485] [Altman, He, Jejjala, Nelson, hep-th/1706.09070] [Gao, Shukla, hep-th/1307.1139] [Long, McAllister, Stout, hep-th/1603.01259] [Cicoli, Ciupke, Mayrhofer, Shukla, hep-th/1801.05434] [Altman, Gray, He, Jejjala, Nelson, hep-th/1411.1418] [Carifio, Cunningham, Halverson, Krioukov, Long, Nelson, hep-th/1711.06685] [Cicoli, Muia, Shukla, hep-th/1611.04612] … many more!
Recent Advances • Recap: Need to triangulate a reflexive polytope and compute intersection numbers . • Can be done via open source math software, like Sage. • Many, systematic studies. [Braun, Walliser, hep-th/1106.4529] [Braun, Lukas, Sun, hep-th/1704.07812] [Blumenhagen, Gao, Rahn, Shukla, hep-th/1205.2485] [Altman, He, Jejjala, Nelson, hep-th/1706.09070] [Gao, Shukla, hep-th/1307.1139] [Long, McAllister, Stout, hep-th/1603.01259] [Cicoli, Ciupke, Mayrhofer, Shukla, hep-th/1801.05434] [Altman, Gray, He, Jejjala, Nelson, hep-th/1411.1418] [Carifio, Cunningham, Halverson, Krioukov, Long, Nelson, hep-th/1711.06685] [Cicoli, Muia, Shukla, hep-th/1611.04612] … many more! • Few, limited studies. [Long, McAllister, McGuirk, hep-th/1407.0709] [Long, McAllister, Stout, hep-th/1603.01259] [Halverson, Long, hep-th/2001.00555]
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