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Frobenius manifolds for elliptic root systems. (English) Zbl 1191.53057
The author considers the problem of establishing the structure of a Frobenius manifold on \(M\) such that its intersection form coincides with the tensor \(I^*\) and its Euler field coincides with the vector field \(E\). Here, \((M, I^*, E)\) is a triple consisting of a complex manifold \(M\), a holomorphic symmetric tensor \(I^*\) on the cotangent bundle of \(M\) and a vector field \(E\).
The author obtains the solution for the complex orbit space of the elliptic Weyl groups for the elliptic root systems of codimension 1 with the tensor descended from the standard holomorphic metrics and with the vector field derived from the canonical \(\mathbb C\)-action.

MSC:
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
20F05 Generators, relations, and presentations of groups
32N10 Automorphic forms in several complex variables
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