Stokes matrices, Poisson Lie groups and Frobenius manifolds

By P. Boalch. 2000

100 citations recorded by the source corpus.

We point out, and draw some consequences of, the fact that the Poisson Lie group G* dual to G=GL_n(C) (with its standard complex Poisson structure) may be identified with a certain moduli space of meromorphic connections on the unit disc having an irregular singularity at the origin. The Riemann-Hilbert map for such connections, taking the Stokes data, induces a holomorphic map from the dual of the Lie algebra of G to the Poisson Lie group G*. The main result is that this map is Poisson. First this leads to new, more direct, proofs of theorems of Duistermaat and Ginzburg-Weinstein (enabling one to reduce Kostant's non-linear convexity theorem, involving the Iwasawa projection, to the linear convexity theorem, involving the `diagonal part'). Secondly we obtain a new approach to the braid group invariant Poisson structure on Dubrovin's local moduli space of semisimple Frobenius manifolds: it is induced from the standard Poisson structure on G*.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/095f58d386879fdf4c639a003a6ba615df955b17

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Stability conditions and Stokes factors (2008, background): “Unlike [4, 5] and earlier treatments however (see, e.”
Twisted wild character varieties (2015, background): “These examples are partly responsible for the recent resurgence of interest in the Stokes phenomenon (cf. e.g. the introduction to [9, 10]).”
Geometry of Multiplicative Preprojective Algebra (2007, background): “Such a map was considered by Hitchin [16] and Hausel [14] (Boalch [3, 4] considered its generalization to the case of irregular singularity).”
Quasi-Hamiltonian geometry of meromorphic connections (2002, unknown): “µ= C−1b−1 − b +C, ω= 1 2 (D,E)+ 1 2 (D,γ)− 1 2 (9) (E,γ) where D= b−C,E= b +C,γ= C∗θand G∗ := {(b −,b +,Λ) ∈ B− ×B + ×t δ(b −)δ(b +) = 1,δ(b +) = exp(πiΛ)} is the Poisson Lie group dual to G(cf. e.g. [7], [9] Appendix B). Considering two poles of order two on P1 leads to the following statement, whi”