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Topological conformal field theories and Calabi-Yau categories. (English) Zbl 1171.14038

Summary: This is the first of two papers [see also Geom. Topol. 11, 1539–1579 (2007; Zbl 1139.32006)] which construct a purely algebraic counterpart to the theory of Gromov-Witten invariants (at all genera). These Gromov-Witten type invariants depend on a Calabi-Yau \(A_{\infty}\) category, which plays the role of the target in ordinary Gromov-Witten theory. When we use an appropriate \(A_{\infty}\) version of the derived category of coherent sheaves on a Calabi-Yau variety, this constructs the \(B\) model at all genera. When the Fukaya category of a compact symplectic manifold \(X\) is used, it is shown, under certain assumptions, that the usual Gromov-Witten invariants are recovered. The assumptions are that open-closed Gromov-Witten theory can be constructed for \(X\), and that the natural map from the Hochschild homology of the Fukaya category of \(X\) to the ordinary homology of \(X\) is an isomorphism.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T45 Topological field theories in quantum mechanics

Citations:

Zbl 1139.32006
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References:

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