Preprint 3/2021

Frobenius Statistical manifolds, Geometric invariants & Hidden symmetries

Noémie Combe, Philippe Combe, and Hanna Nencka

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Submission date: 01. Mar. 2021
Pages: 17
MSC-Numbers: 53B99, 62B10, 60D99, 53D45
Keywords and phrases: statistical manifold, Frobenius manifold, Gromov–Witten invariants, Paracomplex geometry
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In this paper, we explicitly prove that statistical manifolds have a Frobenius manifold structure. This latter object, at the interplay of beautiful interactions between topology and quantum field theory, raises natural questions, concerning the existence of Gromov–Witten invariants for statistical manifolds. We prove that an analog of Gromov–Witten invariants for statistical manifolds (GWS) exists, and that it plays an important role in the learning process. These new invariants have a geometric interpretation concerning intersection points of paraholomorphic curves. In addition, we unravel the hidden symmetries of statistical manifolds. It decomposes into a pair of totally geodesic submanifolds, containing a pair of flat connections. We prove that the pair of pseudo-Riemannian submanifolds are symmetric to each other with respect to Pierce mirror.

05.03.2021, 16:22