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An introduction to topological quantum field theories. (English) Zbl 0890.57019
Topological 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.

##### 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