We propose an extension of quantum Teichmüller theory to a TQFT which gives the exact solution of quantum Chern-Simons theory with gauge group SL(2,C). The principal ingredient in our construction is Faddeev's quantum dilogarithm.

Supersymmetric gauge theories have played a central role in applications of quantum field theory to mathematics. Topologically twisted supersymmetric gauge theories often admit a rigorous mathematical description: for example, the Donaldson invariants of a 4-manifold can be interpreted as the correlation functions of a topologically twisted N=2 gauge theory. The aim of these lectures is to describe a mathematical formulation of partially-twisted supersymmetric gauge theories. These partially twisted theories are intermediate in complexity between the physical theory and the topologically twisted theories. If time permits, I will explain how the operators of such a theory form a 2 complex dimensional analog of a vertex algebra; and, following Kapustin and Witten, I will analyze the relationship between dimensional reductions of the N=4 theory and the A- and B-models on the Hitchin moduli spaces.

The close relation between locality and algebraic structures in quantum physics is demonstrated in classical field theory and in perturbatively renormalized quantum field theory. In particular theories on generic globally hyperbolic spacetimes are constructeded, and the structure of the renormalization group is analysed. The general methods are applied to scalar field theory, Yang Mills theory and gravity.

Factorization homologies (also known as chiral or higher Hochschild homology) are homology theories similar to the usual homology but where the excision axiom is replaced by a locality axiom similar to those of Topological Field Theories. They are interpolating between manifolds and algebras. We will explain their axiomatic characterization, their link with mapping spaces and explain their applications to the study of $E_n$-algebras and higher Deligne conjecture. The first lecture will focus on the case of commutative algebras (which has its own features and is essentially topological) and the second lecture we will study the case of $E_n$-algebras (which is more geometric).

We review the geometry of the moduli spaces of vacua of the theories with eight supercharges in four and three dimensions, their quantization and applications to topological field theories.

The lectures will be focused on semiclassical quantization of classical topological field theories. Known
examples of such theories have gauge symmetry. Among basic examples are the BF thpory, the Chern-
Simons and more general theories of AKSZ type (Aleksandorv, Konstevich, Schwarz and Zaboronsky).
First we will discuss BV (Batalin-Vilkovisky) extensions of these theories, then the formal semiclassical
quantization of such theories in the BV setting. At the end we will discuss the problem of analytical
continuation of the formal semiclassical quantization and the comparison with combinatorial formulae for
invariants of 3-manifolds.

Lecture notes in pdf: lecture 1 (Alberto Cattaneo), lecture 2 (Alberto Cattaneo), lecture 3 (Pavel Mnev)