Abstract
This thesis is divided into three parts.
1: In Part 1 we give a construction for globalization within the mathematical construction of the Batalin-Vilkovisky and Batalin-Fradkin-Vilkovisky formalisms (BV-BFV) for certain type of theories. This is done by using methods of formal geometry as in [17, 64]. In particular, the focus lies on AKSZ theories of a special type which we call "split". We extend the perturbative quantization on manifolds with boundary to a global type, where we replace the modified Qunatum Master Equation, describing a gauge condition for quantization, to a differential version which we call the "modified differential Quantum Master Equation" and show that it is consistent with the cohomological methods described in [38]. Moreover, we explain how the state and the boundary operator in this setting behave for a change of depending choices.
2: In Part 2 we consider thte Poisson Sigma Model which is an example of a split AKSZ type Sigma Model. The Poisson Sigma Model is important in relation to deformation quantization and Kontsevich's star product construction and it is an example of a non-trivial theory where the Batalin-Vilkovsky formalism is in fact needed for quantization (the BRST formalism [13, 12, 11, 103], which is another way of dealing with gauge theories, is only alid for linear Poisson structures). We apply the constructions developed in the first part to the Poisson Sigma Model with the difference that we consider mixed boundary conditions (corners). This is in fact important for the quantization of the relational symplectic groupoid which is ultimately linked to the Poisson Sigma Model. However, considering these type of mixed conditions, the modified differential Quantum master Equation does not hold. In fact, it is spoilt by "curvature terms" arising from Kontsevich's L∞-morphism. To fix this issue, one has to extend the constructions of the first part by adding some counter terms to the action and use a Fedosov-type globalization construction for Poisson manifolds. We call the extended construction the "twisted theory". We show that the modified differential Quantum Master Equation holds for this construction and that everything is again consistent with the cohomolgical methods as before.
3: In Part 3 we construct a global version of a trace formula which was given by Cattaneo and Felder in [28] using the formal global extension of the Poisson Sigma Model on the disk. The globalization construction involves similar techniques as in part 2, such as formal geometry, where we use a Fedosov-type equation to show the trace property. Moreover, we describe its connection to the Nest-Tsygan theorem (algebraic index theorem) [87] and the Tamarkin-Tsygan theorem [100]. we also show how this trace is reduced to a trace-construction for symplectic manifolds presented by Grady, Li and Li in [69] for the case of cotangent bundle.
Moshayedi, Nima