Frobenius manifolds : caustic submanifolds and discriminant almost duality - Riley, Andrew (2007)
The concept of a Frobenius manifold was invented by Boris Dubrovin as a geometric interpretation of solutions of the WDVV equations with additional constraints. The theory of Frobenius manifolds contains a rich mathematical structure transcending many disparate fields of study. In this work, consideration will be restricted to so called semisimple Frobenius manifolds and their submanifolds. Chapter 1 introduces the concept of a Frobenius manifold and gives constructions of the closely linked Coxeter group and Hurwitz space based classes. The concept of almost duality is also introduced; this is the notion that from any Frobenius manifold, one may construct a second solution to the WDVV equations adhering to most of the axioms of a Frobenius manifold. Chapter 2 introduces submanifold geometry and natural submanifolds, on which the induced multiplication coincides with that on the ambient manifold. Such submanifolds are classified in terms of caustics and discriminants. Caustic submanifolds of an arbitrary genus zero Hurwitz space are then considered in chapter 3, extending the idea contained within the main example of [25]. Chapter 4 constructs dual WDVV solutions for An Coxeter type and genus zero Hurwitz Frobenius manifolds, including their discriminants. The result of section 4.2 appeared in [21]. It also draws a link, via a twisted Legendre transformation, between certain almost dual solutions. This idea was published in [22]. Finally, chapter 5 deals with the Hurwitz space H,₁,n, which may be thought of in terms of a Jacobi orbit space. In particular, almost dual solutions of the WDVV equations are constructed on the discriminants, giving a generalised version of the result published in [21].
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Keywords: Mathematics