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Résumé |
A Calabi-Yau ($CY$) algebra is a differential graded (or more generally $A_\infty$) algebra with finite
dimensional cohomology and a certain kind of duality. An example is the de Rham algebra of forms on a
compact closed oriented manifold. Given a $CY$ algebra one can construct a 1+1 dimensional topological
quantum field theory (TQFT).
$CY$ algebras appear on both sides of the mirror symmetry conjecture for compact $CY$ manifolds,
coming from string theory.
We shall explain how to generalize $CY$ algebras to the case where the cohomology of the algebra is not
necessarily
finite dimensional and the duality condition is relaxed. Such algebras are called $pre-CY$ and
examples include the de Rham algebra of forms on a manifold with boundary. We shall explain how to
construct
a 1+1 dimensional TQFT from a $pre-CY$ algebra, via acyclic directed ribbon graphs.
$Pre-CY$ algebras appear on both sides of the mirror symmetry conjecture for open $CY$ manifolds and
for Fano manifolds. This is joint work with Maxim Kontsevich.
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