Geometry of 2D topological field theories. (English) Zbl 0841.58065

Francaviglia, M. (ed.) et al., Integrable systems and quantum groups. Lectures given at the 1st session of the Centro Internazionale Matematico Estivo (CIME) held in Montecatini Terme, Italy, June 14-22, 1993. Berlin: Springer-Verlag. Lect. Notes Math. 1620, 120-348 (1996).
The author gave lectures on the geometry of 2D topological field theories and here are the lecture notes. Two hundred and twenty nine pages of rather esoteric material was covered in just six lectures. At the centre of the author’s theme is the task of finding a quasi-homogeneous function of \(n\) variables where the third derivatives are structure constants of an associative algebra \(A\). This leads to a rather complicated overdetermined system of third order partial differential equations to which the author gives the name WDVV-equation: WDVV being the abbreviation for Witten-Dijkgraaf – E. Verlinde – H. Verlinde. The algebra mentioned above is given the name Frobenius algebra and is a commutative associative \(\mathbb{C}\)-algebra with unit \(e\) and is equipped with a \(\mathbb{C}\)-bilinear symmetric non-degenerate inner-product that satisfies \(\langle ab,c \rangle = \langle a,bc \rangle\). This leads to the idea of Frobenius manifold which is a manifold that carries the Frobenius algebra at each point on the tangent bundle and has certain other related properties. Having stated the problem in the first lecture, in the second lecture the author introduces the subject matter of topological conformal field theories and their moduli. The remaining four lectures are based on six of the author’s earlier papers. These lecture notes contain ten appendices which contain the really new material of this work. In a short review of this kind it is not possible to describe everything in detail but the titles of the appendices give a good indication of the contents and their flavour: A. Polynomial solutions of WDVV. Towards classification of Frobenius manifolds with good analytic properties. B. Symmetries of WDVV. Twisted Frobenius manifolds. C. WDVV and Chazy equation. Affine connections on curves with projective structure. D. Geometry of flat pencils of metrics. E. WDVV and PainlevĂ© – VI. F. Analytic continuation of solutions of WDVV and braid group. G. Monodromy group of Frobenius manifold. H. Generalized hypergeometric equation associated with a Frobenius manifold and its monodromy. I. Determination of a superpotential of a Frobenius manifold. J. Extended complex crystallographic groups and twisted Frobenius manifolds. An overdetermined system of third-order partial differential equations in \(n\) variables arising in the study of a two-dimensional quantum field theory is hardly something that an ordinary mortal would find exciting, but those who have the patience to work through these notes will be rewarded with a rich variety of algebra, analysis, geometry and topology nicely interwoven and interacting to produce a most fascinating structure. Methods developed here are likely to be useful in many related and even apparently unrelated problems.
For the entire collection see [Zbl 0829.00029].


58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
81T10 Model quantum field theories
58J90 Applications of PDEs on manifolds
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
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