Complete Mirror Pairs and Their Naive Stringy Hodge Numbers

<p>The Batyrev-Borisov construction associates a to dual pair of nef-partitions</p><p>\Delta=\Delta₁+\dotsb+\Delta_c and \nabla=\nabla₁+\dotsb+\nabla_c a pair</p><p>of Calabi-Yau complete intersections</p><p>(Y_{\Delta1,\dotsc,\Deltac},Y_{\nabla1,\dotsc,\nablac}) in Gorenstein Fano</p><p>toric varieties (X_\Delta,X_\nabla). These Calabi-Yau varieties are singular</p><p>in general. Batyrev and Nill have developed a generating function $\Est$ for the</p><p>stringy Hodge numbers of Batyrev-Borisov mirror pairs. This function depends</p><p>solely on the combinatorics of the nef-partitions and, under this framework,</p><p>Batyrev-Borisov mirror pairs pass the stringy topological mirror symmetry test</p><p>\hst^p,q(Y_{\Delta1,\dotsc,\Deltac})=\hst^d-p,q(Y_{\nabla1,\dotsc,\nablac}).</p><p>Recently, Aspinwall and Plesser have defined the notion of a complete</p><p>non-reflexive mirror pair $(\scrA,\scrB)$ and used this notion to study</p><p>Calabi-Yau complete intersections in non-Gorenstein toric varieties. Complete</p><p>mirror pairs generalize the notion of a dual pair of almost reflexive Gorenstein</p><p>cones (σ,σ^\bullet) developed by Mavlyutov to propose a</p><p>generalization of the Batyrev-Borisov mirror construction. The only known</p><p>example of either of these two notions is the complete intersection of a quintic</p><p>and a quadric in \PP₂₁₁₁₁₁⁵. We construct $2152$ distinct examples of</p><p>complete mirror pairs and $1077$ distinct examples of dual pairs of almost</p><p>reflexive Gorenstein cones. Additionally, we propose a generalization of Batyrev</p><p>and Nill's stringy $E$-function, called the na\"{i}ve stringy $E$-function</p><p>$\gEst$, that is well-defined for complete mirror pairs.</p>

Author dc:creator

• Fitzpatrick, Brian David

Advisor dc:contributor.advisor

• Aspinwall, Paul S

• Mathematics
• calabi yau
• mirror symmetry
• toric geometry

Handle dc:identifier.uri

https://hdl.handle.net/10161/14531

OAI identifier oai:identifier

oai:dukespace.lib.duke.edu:10161/14531