Stokes structures and moduli spaces related to Markov triples and to Hurwitz spaces
Final Report Abstract
Markov triples are triples of natural numbers which together are zeros of a certain polynomial in three variables. They form an orbit of the braid group Br3 of braids with three strands. The upper triangular 3 × 3 matrices with ones on the diagonal and these triples as entries on the upper right side arise as Stokes matrices of a certain meromorphic connection on C×(a parameter space). The part of the parameter space where a certain Stokes matrix appears is a Stokes region. The meromorphic connection comes from the quantum cohomology of P2. The parameter space encompasses a quotient of the corresponding Frobenius manifold. In the project, this parameter space and a bigger parameter space, which covers this one, were constructed. The Stokes regions which together form the bigger parameter space are in 1:1 correspondence with the distinguished bases modulo signs in a Z-lattice of rank three. Also these distinguished bases modulo signs form a Br3 orbit. The Z-lattice comes equipped with three bilinear forms, a monodromy operator, an even and an odd monodromy group, a set of even vanishing cycles and a set of odd vanishing cycles. All these objects were determined. In this part of the project, the Stokes matrices were the starting point. In the second part of the project, the Frobenius manifold from the quantum cohomology of P2 was the starting point. For this a 1-dimensional Landau-Ginzburg superpotential was constructed. The results for the small quantum cohomology are round. The results for the big quantum cohomology leave some wishes open. In any case this part is a step towards a uniformization of the Frobenius manifold. The part of the project on Hurwitz spaces was not realized.
Publications
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“Unimodular bilinear lattices, automorphism groups, vanishing cycles, monodromy groups, distinguished bases, braid group actions and moduli spaces fro upper triangular matrices”, 396 pages.
Claus Hertling & Khadija Larabi
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1D Landau-Ginzburg Superpotential of Big Quantum Cohomology of CP2. Symmetry, Integrability and Geometry: Methods and Applications.
de Almeida, Guilherme F.
