We study the behavior of Donaldsons invariants of 4-manifolds based on the moduli space of anti self-dual connections (instantons) in the perturbative field theory setting where the underlying source manifold has boundary. It is well-known that these
invariants take values in the instanton Floer homology groups of the boundary 3-manifold. Gluing formulae for these constructions lead to a functorial topological field theory description according to a system of axioms developed by Atiyah, which can be also regarded in the setting of perturbative quantum field theory, as it was shown by Witten, using a version of supersymmetric Yang-Mills theory, known today as Donaldson-Witten theory. One can actually formulate an AKSZ model which recovers this theory for a certain gauge-fixing. We consider these constructions in a perturbative quantum gauge formalism for manifolds with boundary that is compatible with cutting and gluing, called the BV-BFV formalism, which was recently developed by Cattaneo, Mnev and Reshetikhin. We prove that this theory satisfies a modified Quantum Master Equation and extend the result to a global picture when perturbing around constant background fields. Additionally, we relate these constructions to Nekrasovs partition function by treating an equivariant version of Donaldson-Witten theory in the BV formalism. Moreover, we discuss the extension, as well as the relation, to higher gauge theory and enumerative geometry methods, such as Gromov-Witten and Donaldson-Thomas theory and recall their correspondence conjecture for general Calabi-Yau 3-folds. In particular, we discuss the corresponding (relative) partition functions, defined as the generating function for the given invariants, and gluing phenomena.
We describe a globalization construction for the Rozansky-Witten model in the BV-BFV formalism for a source manifold with and without boundary in the classical and quantum case. After having introduced the necessary background, we define an AKSZ sigm
a model, which, upon globalization through notions of formal geometry extended appropriately to our case, is shown to reduce to the Rozansky-Witten model. The relations with other relevant constructions in the literature are discussed. Moreover, we split the model as a $BF$-like theory and we construct a perturbative quantization of the model in the quantum BV-BFV framework. In this context, we are able to prove the modified differential Quantum Master Equation and the flatness of the quantum Grothendieck BFV operator. Additionally, we provide a construction of the BFV boundary operator in some cases.
These are lecture notes for the course Poisson geometry and deformation quantization given by the author during the fall semester 2020 at the University of Zurich. The first chapter is an introduction to differential geometry, where we cover manifold
s, tensor fields, integration on manifolds, Stokes theorem, de Rhams theorem and Frobenius theorem. The second chapter covers the most important notions of symplectic geometry such as Lagrangian submanifolds, Weinsteins tubular neighborhood theorem, Hamiltonian mechanics, moment maps and symplectic reduction. The third chapter gives an introduction to Poisson geometry where we also cover Courant structures, Dirac structures, the local splitting theorem, symplectic foliations and Poisson maps. The fourth chapter is about deformation quantization where we cover the Moyal product, $L_infty$-algebras, Kontsevichs formality theorem, Kontsevichs star product construction through graphs, the globalization approach to Kontsevichs star product and the operadic approach to formality. The fifth chapter is about the quantum field theoretic approach to Kontsevichs deformation quantization where we cover functional integral methods, the Moyal product as a path integral quantization, the Faddeev-Popov and BRST method for gauge theories, infinite-dimensional extensions, the Poisson sigma model, the construction of Kontsevichs star product through a perturbative expansion of the functional integral quantization for the Poisson sigma model for affine Poisson structures and the general construction.
These are lecture notes written at the University of Zurich during spring 2014 and spring 2015. The first part of the notes gives an introduction to probability theory. It explains the notion of random events and random variables, probability measure
s, expectation, distributions, characteristic function, independence of random variables, types of convergence and limit theorems. The first part is separated into two different chapters. The first chapter is about combinatorial aspects of probability theory and the second chapter is the actual introduction to probability theory, which contains the modern probability language. The second part covers conditional expectations, martingales and Markov chains, which are easily accessible after reading the first part. The chapters are exactly covered in this order and go into some more details of the respective topic.
These notes give an introduction to the quantization procedure called geometric quantization. It gives a definition of the mathematical background for its understanding and introductions to classical and quantum mechanics, to differentiable manifolds
, symplectic manifolds and the geometry of line bundles and connections. Moreover, these notes are endowed with several exercises and examples.
Using the quantum construction of the BV-BFV method for perturbative gauge theories, we show that the obstruction for quantizing a codimension 1 theory is given by the second cohomology group with respect to the boundary BRST charge. Moreover, we giv
e an idea for the algebraic construction of codimension $k$ quantizations in terms of $mathbb{E}_k$-algebras and higher shifted Poisson structures by formulating a higher version of the quantum master equation.
We introduce the notions of relational groupoids and relational convolution algebras. We provide various examples arising from the group algebra of a group $G$ and a given normal subgroup $H$. We also give conditions for the existence of a Haar syste
m of measures on a relational groupoid compatible with the convolution, and we prove a reduction theorem that recovers the usual convolution of a Lie groupoid.
We consider a construction of observables by using methods of supersymmetric field theories. In particular, we give an extension of AKSZ-type observables using the Batalin-Vilkovisky structure of AKSZ theories to a formal global version with methods
of formal geometry. We will consider the case where the AKSZ theory is split which will give an explicit construction for formal vector fields on base and fiber within the formal global action. Moreover, we consider the example of formal global generalized Wilson surface observables whose expectation values are invariants of higher-dimensional knots by using $BF$ field theory. These constructions give rise to interesting global gauge conditions such as the differential Quantum Master Equation and further extensions.
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 grap
hs 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.
A globalized version of a trace formula for the Poisson Sigma Model on the disk is presented by using its formal global picture in the setting of the Batalin-Vilkovisky formalism. This global construction includes the concept of zero modes. Moreover,
for the symplectic case of the Poisson Sigma Model with cotangent target, the globalized trace reduces to a symplectic construction which was presented by Grady, Li and Li for 1-dimensional Chern-Simons theory (topological quantum mechanics). In addition, the connection between this formula and the Nest-Tsygan theorem and the Tamarkin-Tsygan theorem is explained.
