Complete Mirror Pairs and Their Naive Stringy Hodge Numbers

Abstract

The Batyrev-Borisov construction associates a to dual pair of nef-partitions$\Delta=\Delta_1+\dotsb+\Delta_c$ and $\nabla=\nabla_1+\dotsb+\nabla_c$ a pairof Calabi-Yau complete intersections$(Y_{\Delta_1,\dotsc,\Delta_c},Y_{\nabla_1,\dotsc,\nabla_c})$ in Gorenstein Fanotoric varieties $(X_\Delta,X_\nabla)$. These Calabi-Yau varieties are singularin general. Batyrev and Nill have developed a generating function $\Est$ for thestringy Hodge numbers of Batyrev-Borisov mirror pairs. This function dependssolely on the combinatorics of the nef-partitions and, under this framework,Batyrev-Borisov mirror pairs pass the stringy topological mirror symmetry test$\hst^{p,q}(Y_{\Delta_1,\dotsc,\Delta_c})=\hst^{d-p,q}(Y_{\nabla_1,\dotsc,\nabla_c})$.Recently, Aspinwall and Plesser have defined the notion of a completenon-reflexive mirror pair $(\scrA,\scrB)$ and used this notion to studyCalabi-Yau complete intersections in non-Gorenstein toric varieties. Completemirror pairs generalize the notion of a dual pair of almost reflexive Gorensteincones $(\sigma,\sigma^\bullet)$ developed by Mavlyutov to propose ageneralization of the Batyrev-Borisov mirror construction. The only knownexample of either of these two notions is the complete intersection of a quinticand a quadric in $\PP_{211111}^5$. We construct $2152$ distinct examples ofcomplete mirror pairs and $1077$ distinct examples of dual pairs of almostreflexive Gorenstein cones. Additionally, we propose a generalization of Batyrevand Nill's stringy $E$-function, called the na\"{i}ve stringy $E$-function$\gEst$, that is well-defined for complete mirror pairs.