Complete Mirror Pairs and Their Naive Stringy Hodge Numbers
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2017
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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 pair
of Calabi-Yau complete intersections
$(Y_{\Delta_1,\dotsc,\Delta_c},Y_{\nabla_1,\dotsc,\nabla_c})$ in Gorenstein Fano
toric varieties $(X_\Delta,X_\nabla)$. These Calabi-Yau varieties are singular
in general. Batyrev and Nill have developed a generating function $\Est$ for the
stringy Hodge numbers of Batyrev-Borisov mirror pairs. This function depends
solely 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 complete
non-reflexive mirror pair $(\scrA,\scrB)$ and used this notion to study
Calabi-Yau complete intersections in non-Gorenstein toric varieties. Complete
mirror pairs generalize the notion of a dual pair of almost reflexive Gorenstein
cones $(\sigma,\sigma^\bullet)$ developed by Mavlyutov to propose a
generalization of the Batyrev-Borisov mirror construction. The only known
example of either of these two notions is the complete intersection of a quintic
and a quadric in $\PP_{211111}^5$. We construct $2152$ distinct examples of
complete mirror pairs and $1077$ distinct examples of dual pairs of almost
reflexive Gorenstein cones. Additionally, we propose a generalization of Batyrev
and Nill's stringy $E$-function, called the na\"{i}ve stringy $E$-function
$\gEst$, that is well-defined for complete mirror pairs.
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Fitzpatrick, Brian David (2017). Complete Mirror Pairs and Their Naive Stringy Hodge Numbers. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/14531.
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