Deformation Quantization of A-infinity Equivalences
We show that Deformation Quantization of quadratic Poisson structures preserves the $A_\infty$-Morita equivalence of a given pair of Koszul dual $A_\infty$-algebras.
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arXiv subjects
Andrea Ferrario
Publications and source records attributed to Andrea Ferrario.
A note on the Koszul complex in deformation quantization
The aim of this short note is to present a proof of the existence of an $A_\infty$-quasi-isomorphism between the $A_\infty$-$\mathrm S(V^*)$-$\wedge(V)$-bimodule $K$, introduced in \cite{CFFR}, and the Koszul complex $\mathrm K(V)$ of $\mathrm S(V^*)$, viewed as an $A_\infty$-$\mathrm S(V^*)$-$\wedge(V)$-bimodule, for $V$ a finite-dimensional (complex or real) vector space.
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Bimodules and branes in deformation quantization
We prove a version of Kontsevich's formality theorem for two subspaces (branes) of a vector space $X$. The result implies in particular that the Kontsevich deformation quantizations of $\mathrm{S}(X^*)$ and $\wedge(X)$ associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhet's recent paper on Koszul duality in deformation quantization.
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Poisson Sigma Model with branes and hyperelliptic Riemann surfaces
We derive the explicit form of the superpropagators in presence of general boundary conditions (coisotropic branes) for the Poisson Sigma Model. This generalizes the results presented in Cattaneo and Felder's previous works for the Kontsevich angle function used in the deformation quantization program of Poisson manifolds without branes or with at most two branes. The relevant superpropagators for n branes are defined as gauge fixed homotopy operators of a complex of differential forms on n sided polygons P_n with particular "alternating" boundary conditions. In presence of more than three branes we use first order Riemann theta functions with odd singular characteristics on the Jacobian variety of a hyperelliptic Riemann surface (canonical setting). In genus g the superpropagators present g zero modes contributions.
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