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We construct a family of oriented extended topological field theories using the AKSZ construction in derived algebraic geometry, which can be viewed as an algebraic and topological version of the classical AKSZ field theories that occur in…

A construction of gauge-invariant observables is suggested for a class of topological field theories, the AKSZ sigma-models. The observables are associated to extensions of the target Q-manifold of the sigma model to a Q-bundle over it with…

We discuss unifying features of topological field theories in 2, 3 and 4 dimensions. This includes relations among enumerative geometry (2d topological field theory) link invariants (3d Chern-Simons theory) and Donaldson invariants (4d…

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…

We propose a new topological field theory on generalized complex geometry in two dimension using AKSZ formulation. Zucchini's model is $A$ model in the case that the generalized complex structuredepends on only a symplectic structure. Our…

This is an introductory review of topological field theories (TFTs) called AKSZ sigma models. The AKSZ construction is a mathematical formulation for the construction and analyses of a large class of TFTs, inspired by the Batalin-Vilkovisky…

We prove that Donaldson-Thomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory…

Using the AKSZ prescription we construct 2D and 3D topological field theories associated to generalized complex manifolds. These models can be thought of as 2D and 3D generalizations of A- and B-models. Within the BV framework we show that…

We extend the AKSZ formulation of the Poisson sigma model to more general target spaces, and we develop the general theory of graded geometry for poly-symplectic and poly-Poisson structures. In particular we prove a Schwarz-type theorem and…

We define a new type of Hall algebras associated e.g. with quivers with polynomial potentials. The main difference with the conventional definition is that we use cohomology of the stack of representations instead of constructible sheaves…

We present a construction of Donaldson-Thomas invariants for three-dimensional projective Calabi-Yau Deligne-Mumford stacks. We also study the structure of these invariants for etale gerbes over such stacks.

Topological quantum field theories can be used to probe topological properties of low dimensional manifolds. A class of these theories known as Schwarz type theories, comprise of Chern-Simons theories and BF theories. In three dimensions…

We reformulate and motivate AKSZ-type topological field theories in pedestrian terms, explaining how they arise as the most general Schwartz-type topological actions subject to a simple constraint, and how they generalize Chern$-$Simons…

In this paper we discuss topological BF theories in 3 and 4 dimensions. Observables are associated to ordinary knots and links (in 3 dimensions) and to 2-knots (in 4 dimensions). The vacuum expectation values of such observables give a wide…

We show that equivariant Donaldson polynomials of compact toric surfaces can be calculated as residues of suitable combinations of Virasoro conformal blocks, by building on AGT correspondence between N = 2 supersymmetric gauge theories and…

I review some recent results on four-manifold invariants which have been obtained in the context of topological quantum field theory. I focus on three different aspects: (a) the computation of correlation functions, which give explicit…

This review gives an introduction to cohomological Donaldson-Thomas theory: the study of a cohomology theory on moduli spaces of sheaves on Calabi-Yau threefolds, and of complexes in 3-Calabi-Yau categories, categorifying their numerical DT…

A brief introduction to Topological Quantum Field Theory as well as a description of recent progress made in the field is presented. I concentrate mainly on the connection between Chern-Simons gauge theory and Vassiliev invariants, and…

We generalize the AKSZ construction of topological field theories to allow the target manifolds to have possibly-degenerate (homotopy) Poisson structures. Classical AKSZ theories, which exist for all oriented spacetimes, are described in…

A class of two dimensional field theories, based on (generically degenerate) Poisson structures and generalizing gravity-Yang-Mills systems, is presented. Locally, the solutions of the classical equations of motion are given. A general…