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arXiv:0707.1512v3[math.GT]19Aug2009 MIRROR DUALITY IN A JOYCE MANIFOLD
 

Summary: arXiv:0707.1512v3[math.GT]19Aug2009
MIRROR DUALITY IN A JOYCE MANIFOLD
SELMAN AKBULUT, BARIS EFE, AND SEMA SALUR
Abstract. Previously the two of the authors defined a notion of dual Calabi-
Yau manifolds in a G2 manifold, and described a process to obtain them. Here
we apply this process to a compact G2 manifold, constructed by Joyce, and as a
result we obtain a pair of Borcea-Voisin Calabi-Yau manifolds, which are known
to be mirror duals of each other.
1. Introduction
Recall that G2 is the simple Lie group which can be identified with the subgroup
G2 = {A GL(7, R) | A
0 = 0 }
where 0 = e123 + e145 + e167 + e246 - e257 - e347 - e356 with eijk = dxi dxj dxk
(for more information on G2 manifolds the reader can consult [Br1], [Br2], [HL],
[J], [AS2]). We say a 7-manifold M7 has a G2 structure if there is a 3-form
3(M) such that at each p M the pair (Tp(M), (p)) is (pointwise) isomorphic to
(T0(R7), 0) (this condition is equivalent to reducing the tangent frame bundle to a
G2-bundle). A manifold with G2 structure (M, ) is called a G2 manifold (integrable
G2 structure) if at each point p M there is a chart (U, p) (R7, 0) on which
equals to 0 up to second order term, i.e. on the image of the open set U we can write

  

Source: Akbulut, Selman - Department of Mathematics, Michigan State University

 

Collections: Mathematics