These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories. Introduction. Lecture 1. WDVV equations and Frobenius manifolds. {Appendix A.} Polynomial solutions of WDVV. {Appendix B.} Symmetriies of WDVV. Twisted Frobenius manifolds. {Appendix C.} WDVV and Chazy equation. Affine connections on curves with projective structure. Lecture 2. Topological conformal field theories and their moduli. Lecture 3. Spaces of isomonodromy deformations as Frobenius manifolds. {Appendix D.} Geometry of flat pencils of metrics. {Appendix E.} WDVV and Painlev\'e-VI. {Appendix F.} Branching of solutions of the equations of isomonodromic deformations and braid group. {Appendix G.} Monodromy group of a Frobenius manifold. {Appendix H.} Generalized hypergeometric equation associated to a Frobenius manifold and its monodromy. {Appendix I.} Determination of a superpotential of a Frobenius manifold. Lecture 4. Frobenius structure on the space of orbits of a Coxeter group. {Appendix J.} Extended complex crystallographic groups and twisted Frobenius manifolds. Lecture 5. Differential geometry of Hurwitz spaces. Lecture 6. Frobenius manifolds and integrable hierarchies. Coupling to topological gravity.

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Geometry of 2D topological field theories

The aim of this note is to give a summary of the study of primitive forms and period mappings associated with a universal unfolding F of an isolated hypersurface singularity, which was studied in [35]. There is already a summary [36] on the subject. Compared to that, this note contains mainly two new topics, covering some recent developments in the subject. The details will appear elsewhere. First, in Section 5 of this note we give a description of the construction of period mappings using a sequence of Ss-modules ^? ( F \ keZ instead of ^(0)modules J>^F \ keZ. This construction might give another insight into the relationship between Poincare duality of the Milnor fiber and the period mapping of the family F. This part is based on the lectures by the author at R. T. M. S., Kyoto University in the springs '81 and '82. Secondly, in Section 4 of this note, the existence of primitive forms is reduced to the existence of certain good sections of the "principal symbol module" q*Qp i^° the <f(0)-module ^FThe technique of the proof comes from the solution of Riemann-Hilbert problem on P by B. Malgrange [23] [24] [25] and Birkhoff [4]. The author is grateful to Professor Malgrange for helpful discussions at the Institute Fourier in Grenoble, March '83. Combining this result with a recent result by M. Saito [46], we are now able to construct primitive forms for a large class of singularities. For the moment the period mapping associated to a primitive form has been studied explicitely only for the cases of simple singularities and simple elliptic singularities (cf. [33] [42] [43]) which might give another approach to studying universal family of such singularities by E. Brieskorn [6] [7],

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Period Mapping Associated to a Primitive Form

This paper is devoted to the theory of WDVV equations of associativity. This remarkable system of nonlinear differential equations was discovered by E. Witten [85]and R. Dijkgraaf, E. Verlinde, and H. Verlinde [24]in the beginning of the 1990s. It was first derived as equations for the so-called primary free energy of a family of two-dimensional topological field theories. Later it proved to be an efficient tool in the solution of problems of the theory of Gromov-Witten invariants, reflection groups and singularities, and integrable hierarchies.

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Painlevé Transcendents in Two-Dimensional Topological Field Theory

Main mathematical applications of Frobenius manifolds are in the theory of Gromov Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifolds establishes remarkable relationships between these, sometimes rather distant, mathematical theories. 1991 Mathematics Subject Classification: 32G34, 35Q15, 35Q53, 20F55, 53B50 WDVV equations of associativity is the problem of finding of a quasihomogeneous, up to at most quadratic polynomial, function F (t) of the variables t = (t, . . . , t) and of a constant nondegenerate symmetric matrix (

Geometry and analytic theory of Frobenius manifolds

The objective of the article is to show that the orbit space of a finite reflection group acting on the complexification of the real vector space carries naturally a complex vector space structure Q together with a nondegenerate bilinear form J on it defined over the real number field. For details of the results, one is referred to Theorems I, II and III in the introduction. This structure on the invariants is called the flat structure. Originaly, the flat structure (published later in [7], Oda [24]) is defined on the deformation space of an isolated hypersurface singularity via Gauss-Manin connection and higher residue theory in general. On the other hand, the deformation of a simple singularity is described in terms of a simple Lie algebra. Namely, the deformation space is given by the quotient space of the Cartan algebra by the action of the Weyl group (Brieskorn [2], Slodowy [33]). This suggests that the flat structure is described in terms only of the Weyl group in that case. Actually, such a description is achieved in the present article so that the above mentioned flat structure on the quotient space of a finite reflection group is obtained. The key fact in the construction of the flat metric is the regularity of the eigenvector for exp (2n*f^\/K) of a Coxeter element with the Coxeter % = h (cf. [6], [15], [34]). Since the present article was written (cf. the Footnote *)), there have been several developments in the study of the flat structure and its generalization, as will be summarized below. Nevertheless the present article

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On a Linear Structure of the Quotient Variety by a Finite Reflexion Group

(1980). On a certain generator system of the ring of invariants of a finite reflection group. Communications in Algebra: Vol. 8, No. 4, pp. 373-408.

Kyoji Saito

Papers

On a certain generator system of the ring of invariants of a finite reflection group