Communications in Mathematical Physics

, Volume 361, Issue 1, pp 155–204 | Cite as

Calabi–Yau Volumes and Reflexive Polytopes

  • Yang-Hui He
  • Rak-Kyeong Seong
  • Shing-Tung Yau
Open Access


We study various geometrical quantities for Calabi–Yau varieties realized as cones over Gorenstein Fano varieties, obtained as toric varieties from reflexive polytopes in various dimensions. Focus is made on reflexive polytopes up to dimension 4 and the minimized volumes of the Sasaki–Einstein base of the corresponding Calabi–Yau cone are calculated. By doing so, we conjecture new bounds for the Sasaki–Einstein volume with respect to various topological quantities of the corresponding toric varieties. We give interpretations about these volume bounds in the context of associated field theories via the AdS/CFT correspondence.


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Authors and Affiliations

  1. 1.Merton College, University of OxfordOxfordUK
  2. 2.Department of MathematicsCity, University of LondonLondonUK
  3. 3.School of PhysicsNanKai UniversityTianjinPeople’s Republic of China
  4. 4.Angstrom Laboratory, Department of Physics and AstronomyUppsala UniversityUppsalaSweden
  5. 5.Jefferson Physical Laboratory, Center of Mathematical Sciences and Applications, Department of MathematicsHarvard UniversityCambridgeUSA

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