An introduction to topological quantum field theories.

*(English)*Zbl 0890.57019Topological quantum field theory (TQFT) in 2 dimensions is considered. After an axiomatic approach, valid for all dimensions, it is shown that in 2 dimensions the theory is entirely equivalent to be called a Frobenius algebra. Some standard examples of Frobenius algebras are reviewed. In conclusion, a topological QFT arising from considering projective algebraic varieties (or more generally, compact Kähler manifolds) and related with them its Frobenius algebras named quantum cohomology rings are discussed. The geometry of the integrable \((2+1)\)-sine-Gordon system based on an extension of Darboux’s method of linking the classical Lami system governing triply orthogonal systems of surfaces is considered. A novel reduction of the sine-Gordon system to a system of ordinary differential equations associated with Darboux-type transformations is presented.

Reviewer: A.V.Aminova (Kazan’)

##### MSC:

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |

53Z05 | Applications of differential geometry to physics |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

81T70 | Quantization in field theory; cohomological methods |