Introduces G-Hermitian 2-vector spaces via fixed points of an O(2)-action on 2Vect and criteria for positive pairings to generalize the Hermitian-to-Hilbert passage, with an outline for inductive higher-dimensional versions.
Topological conformal field theories and Calabi-Yau categories
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abstract
This is the first of two papers 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-infinity category, which plays the role of the target in ordinary Gromov-Witten theory. When we use an appropriate A-infinity 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 a good theory of open-closed Gromov-Witten invariants exists 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.
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2026 1verdicts
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The many faces of higher Hilbert spaces
Introduces G-Hermitian 2-vector spaces via fixed points of an O(2)-action on 2Vect and criteria for positive pairings to generalize the Hermitian-to-Hilbert passage, with an outline for inductive higher-dimensional versions.