Formal connections in deformation quantization

By Paolo Masulli
PhD Dissertations
October 2014

The field of this thesis is deformation quantization, and we consider mainly symplectic manifolds equipped with a star product.

After reviewing basics in complex geometry, we introduce quantization, focusing on geometric quantization and deformation quantization. The latter is defined as a star product on a Poisson manifold that is in general non-commutative and corresponds to the composition of the quantized observables.

While in general it is difficult to express a star product globally on a curved manifold in an explicit way, we consider a case where this is possible, namely that of a Kähler manifold. Gammelgaard gave an explicit formula for a class of star products in this setting. We review his construction, which is combinatorial and based on a certain family of graphs and extend it, to provide the graph formalism with the notions of composition and differentiation.

We shall focus our attention on symplectic manifolds equipped with a family of star products, indexed by a parameter space. In this situation we can define a connection in the trivial bundle over the parameter space with fibres the formal smooth functions on the manifold, which relates the star products in the family and is called a formal connection. We study the question of classifying such formal connections. To each star product we can associate a certain cohomology class called the characteristic class. It turns out that a formal connection exists if and only if all the star products in the family have the same characteristic class, and that formal connections form an affine space over the derivations of the star products.

Moreover, if the parameter space for the family of star products is contractible, we obtain that any two flat formal connections are gauge equivalent via a selfequivalence of the family of star products. Afterwards we study the problem of trivializing a formal connection, that is to define a differential operator on the manifold which makes any section of the bundle parallel with respect to the connection. To approach the problem we use the graph formalism described above to encode it in graph terms. This allows us to express the equations determining a trivialization of the formal connection completely in graph terms, and solving them amounts to finding a linear combination of graphs whose derivative is equal to a given expression. We shall also look at another approach to the problem that is more calculative. Moreover we use the graph formalism to give a set of recursive equations determining the formal connection for a given family of star products.

Thesis advisor: Jørgen Ellegard Andersen
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