Computation of Kontsevich Weights of Connection and Curvature Graphs for Symplectic Poisson Structures

TL;DR

This paper explicitly computes Kontsevich graph weights related to connection and curvature in symplectic Poisson structures, expressing them via hypergeometric functions and simplifying for cotangent bundles.

Contribution

It provides explicit formulas for Kontsevich graph weights in symplectic cases, including connections to hypergeometric functions and simplifications for cotangent bundles.

Findings

Weights expressed in terms of hypergeometric functions

Simplified formulas for curvature graphs

Explicit computations for cotangent bundles

Abstract

We give a detailed explicit computation of weights of Kontsevich graphs which arise from connection and curvature terms within the globalization picture for the special case of symplectic manifolds. We will show how the weights for the curvature graphs can be explicitly expressed in terms of the hypergeometric function as well as by a much simpler formula combining it with the explicit expression for the weights of its underlined connection graphs. Moreover, we consider the case of a cotangent bundle, which will simplify the curvature expression significantly.