Black hole condensation and the web of Calabi-Yau manifolds

Complete classification of reflexive polyhedra in four dimensions

“…as subgroups of GL(n,Z) and not as rotations). The corresponding self mirror Calabi-Yau manifold has Hodge numbers (20,20). The polytope with the largest order, namely 128, of Mfi nes t/Mcoarsest is determined by the weight system (1,1,1,1,4)/8.…”

“…In a similar way the model with Hodge numbers (2,29) comes from modding the bicubic hypersurface in P 2 x F 2 by a fixed point free Z3. The hypersurface with Hodge numbers (7,19) corresponds to a dual pair of polytopes where A* has 9 vertices and 12 lattice points and A has 10 vertices and 24 lattice points; for the model with Hodge numbers (5,20) …”

“…While this question was originally asked by a mathematician [17], it was later found that physical processes [18,19] may interpolate between the corresponding different branches of the moduli space. Here, again, an important role is played by reflexive polyhedra: They were used to demonstrate the connectedness of the space of Calabi-Yau hypersurfaces in weighted projective spaces [20,21]. In terms of polyhedra, two Calabi-Yau manifolds are connected if one of the corresponding polytopes is a subpolyhedron of the other or, more generally, if between the two polyhedra there is a chain of polyhedra that are mutually connected in this way.…”

Complex/Symplectic Mirrors

“…2 Implicit in the use of the word "conifold" is the assumption that several cycles do not collapse together in a single point of the manifold M. More general cases are also interesting to consider, see for example [26] for the complex case and [4] for the symplectic case.…”

“…For more details on this physically well-studied case, the reader might want to consult [24,25,26,14].…”