On the geometry of WDVV equations and their Hamiltonian formalism in arbitrary dimension

2025/09/17 by Stanislav Opanasenko, Opanasenko, S., Raffaele Vitolo +1
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2509.13757

openalex publication_date 2025/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is known that in low dimensions WDVV equations can be rewritten as commuting quasilinear bi-Hamiltonian systems. We extend some of these results to arbitrary dimension N and arbitrary scalar product η. In particular, we show that WDVV equations can be interpreted as a set of linear line congruences in suitable Plücker embeddings. This form leads to their representation as Hamiltonian systems of conservation laws. Moreover, we show that in low dimensions and for an arbitrary η WDVV equations can be reduced to passive orthonomic form. This leads to the commutativity of the Hamiltonian systems of conservation laws. We conjecture that such a result holds in all dimensions.

Citations

• Dubrovin duality and mirror symmetry for ADE resolutions
• Manifolds with a commutative and associative product structure that encodes superintegrable Hamiltonian systems
• Remarks on integrability of N=1 supersymmetric Ruijsenaars-Schneider three-body models
• Generalized Frobenius Manifolds with Non-flat Unity and Integrable Hierarchies
• Real Topological Recursions and WDVV Relations
• Semisimple flat F-manifolds in higher genus
• Extended V-systems and almost-duality for extended affine Weyl orbit spaces
• Flat pencils of metrics and Frobenius manifolds
• Equations of Associativity in Two-Dimensional Topological Field Theory as Integrable Hamiltonian Nondiagonalizable Systems of Hydrodynamic Type
• Symplectic and Poisson Geometry on Loop Spaces of Manifolds and Nonlinear Equations
• Geometry of 2d topological field theories

Related

• Line geometry of pairs of second-order Hamiltonian operators and quasilinear systems
• On Symmetries of the WDVV Equations
• Dispersionful Version of WDVV Associativity System
• Multi-Dimensional Conservation Laws and Integrable Systems
• Explicit solutions of the WDVV equation determined by the "flat" hydrodynamic reductions of the Egorov hydrodynamic chains