Abstract
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.
Highlights
Whereas there has been a multitude of constructions for Calabi–Yau varieties over the years, both compact and non-compact, by far the largest number have been realized with toric geometry as the point d’appui
Compact smooth Calabi–Yau (n − 1)-folds have been constructed [1,2,3,4,5,6,7,8,9,10,11,12] as hypersurfaces corresponding to the anti-canonical divisor within a n-dimensional toric variety coming from a reflexive polytope
A key fact is that X itself is a real cone over a compact Sasaki–Einstein manifold Y of real dimension 2n − 1
Summary
Whereas there has been a multitude of constructions for Calabi–Yau varieties over the years, both compact and non-compact, by far the largest number have been realized with toric geometry as the point d’appui. Perimeter points: lattice points (including vertices) lying on edges toric variety corresponding to n−1; dimC X ( n−1) = n − 1 Fine Stellar Regular triangulation of polytope smooth toric variety (if it exists) obtained from FRS of (Inner) normal fan associated with affine Calabi–Yau (complex) cone over X ( n−1); dimC X = n Sasaki–Einstein manifold, base of X which is real one over Y ; dimR Y = dimR B(C(X ( n−1))) = 2n − 1 components of the Reeb vector, i = 1, .
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
