 
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 BorceaVoisin CalabiYau 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 7manifold M7 has a G2 structure if there is a 3form
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
G2bundle). 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
