Schwarz Type Topological Quantum Field Theories

TL;DR

This paper reviews Schwarz type topological quantum field theories, including Chern-Simons and BF theories, highlighting their role in understanding low-dimensional topology, knot invariants, and four-dimensional Donaldson invariants.

Contribution

It provides a comprehensive overview of Schwarz type theories and their applications in topology and quantum gravity across multiple dimensions.

Findings

Chern-Simons theories capture knot and link invariants in 3D.

BF theories generalize to higher dimensions, describing complex topological invariants.

These theories connect quantum field approaches with topological properties of manifolds.

Abstract

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 both capture the properties of knots and links leading to invariants characterising them. These can also be used to construct three-manifold invariants. Three dimensional gravity is described by these field theories. BF theories exist also in higher dimensions. In four dimensions, these describe two-dimensional generalization of knots as well as Donaldson invariants.