Computation of Kontsevich weights of connection and curvature graphs for symplectic Poisson structures (Q2146868)

The paper is based on the MSc dissertation of the second author [Computation of Kontsevich weights of connection and curvature graphs for symplectic Poisson structures. Universität Zürich (2019)]. The goal is to compute the weights \(w_{\Gamma} \in \mathbb{R}\) associated to certain graphs \(\Gamma\), which enter into explicit formulae for three objects related to a ``globalised'' version of Kontsevich's deformation quantisation of Poisson manifolds [\textit{M. Kontsevich}, Lett. Math. Phys. 66, No. 3, 157--216 (2003; Zbl 1058.53065)]; in particular the paper deals with the particular case of nondegenerate Poisson structures, i.e., symplectic manifolds \((M,\omega)\). The ``global'' viewpoint, which allows to consider arbitrary smooth manifolds rather than just open subspaces \(M \subseteq \mathbb{R}^d\), builds on an interpretation of Kontsevich's quantisation as a \emph{Poisson} \(\sigma\)-\emph{model} (first considered by [\textit{A. S. Cattaneo} and \textit{G. Felder}, Commun. Math. Phys. 212, No. 3, 591--611 (2000; Zbl 1038.53088)]), later extended to manifolds with boundary (in the BV-BFV formalism) in work of the first author and collaborators [\textit{A. S. Cattaneo} et al., Commun. Math. Phys. 375, No. 1, 41--103 (2020; Zbl 1443.81053)]. The three objects first appear in Equations (3)--(5); these are respectively Kontsevich's \(\ast\)-product, a certain connection 1-form, and its curvature 2-form. The connection is a one-parameter deformation of a ``classical'' Grothendieck connection \(D_G\), defined on a \emph{Weyl} bundle, which in turn is a formal deformation of (a completion of) the symmetric algebra \(\mathrm{Sym}(T^*M)\) of the cotangent bundle \(T^*M \to M\). Importantly, the equations for horizontal sections of the deformed connection are close to equations derived by Fedosov, in earlier work about the deformation quantisation of symplectic manifolds [\textit{B. V. Fedosov}, J. Differ. Geom. 40, No. 2, 213--238 (1994; Zbl 0812.53034)]. More details about the flat connection \(D_G\), in the context of \emph{formal} geometry (and exponential maps) are provided in Sect.~1.2. The bulk of the paper is \S~2, where the main computations are gathered. The weights \(w_{\Gamma}\) are computed as integrals over (compactifications of) moduli spaces of configurations of points on the upper-half plane and the real line: see Equation (12) and below, and particularly Equation (15) for the integrand differential form \(\omega_{\Gamma}\) -- involving the edges of \(\Gamma\). Two types of vertices in the ``admissible'' graphs lead to three distinct cases, which are dealt with in Sections 2.1, 2.2 and 2.3. The proof of a technical combinatorial passage in Section 2.2 is postponed to the -- unique -- appendix. The final formulae for the curvature involve the hypergeometric function, while a different approach for their computation is given in Section 2.4. In Section 3 the weights are then plugged into the formulae for the \(\ast\)-product, the connection form, and the curvature, respectively in Equations (87), (90) and (93). Then in Section 3.4 the authors consider a modified version of the deformed Grothendieck connection (following aforementioned papers), which relies on Fedosov-type equations, and discuss its existence. Finally in Section 3.5 the general symplectic manifold \((M,\omega)\) is replaced by the cotangent bundle \(N = T^*M\) of a smooth manifold \(M\); in this context two other explicit formulae are provided -- in Equations (106) and (107) -- for lifted versions of the connection 1-form and its curvature, noting the latter undergoes a significant simplification from the general symplectic case.