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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Citation

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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