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On the other hand the Hochschild homology of $X$ also has the Mukai pairing (see [1]). If $X$ is Calabi-Yau, this pairing arises from the action of the class of a genus 0 Riemann-surface with two incoming closed boundaries and no outgoing boundary in $\\text{H}_{0}({\\mathcal M}_0(2,0))$ on the algebra of closed states of a version of the B-Model on $X$. 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Ramadoss","submitted_at":"2008-05-12T23:20:01Z","abstract_excerpt":"Let $X$ be a smooth proper scheme over a field of characteristic 0. Following D. Shklyarov [10], we construct a (non-degenerate) pairing on the Hochschild homology of $\\per{X}$, and hence, on the Hochschild homology of $X$. On the other hand the Hochschild homology of $X$ also has the Mukai pairing (see [1]). If $X$ is Calabi-Yau, this pairing arises from the action of the class of a genus 0 Riemann-surface with two incoming closed boundaries and no outgoing boundary in $\\text{H}_{0}({\\mathcal M}_0(2,0))$ on the algebra of closed states of a version of the B-Model on $X$. 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